To lay the groundwork for discussing the promising ideas for future development, we recall here some of the basic theoretical tools at our disposal. In time-dependent formalisms, basic distinctions can be made between dynamics based on inertial motion, dynamics based on diffusion motion in a statistical framework, and dynamics that combine inertial and diffusive motion. The relevant computational methodologies are often referred to by their acronyms; the ones used here are listed in Table 1.
| Acronym | Meaning |
|---|---|
| ATDDFT | Adiabatic TDDFT |
| ATDHFB | Adiabatic TDHFB |
| BCS | Bardeen-Cooper-Schrieffer |
| CHF | Constrained HF |
| CHFB | Constrained HFB |
| CSE | Collective Schrödinger equation |
| DDD | Dissipative Diabatic Dynamics |
| DFT | Density Functional Theory |
| EDF | Energy Density Functional |
| GCM | Generator Coordinate Method |
| GOA | Gaussian overlap approximation |
| HF | Hartree-Fock |
| HFB | Hartree-Fock-Bogoliubov |
| HO | Harmonic Oscillator |
| MM | Microscopic Macroscopic |
| PES | Potential Energy Surface |
| QRPA | Quasiparticle RPA |
| RPA | Random Phase Approximation |
| SF | Spontaneous Fission |
| TDDFT | Time-Dependent DFT |
| TDGCM | Time-Dependent GCM |
| TDHF | Time-Dependent HF |
| TDHFB | Time-Dependent HFB |
| TDRPA | Time-Dependent RPA |
| TKE | Total Kinetic Energy |
Before going into details of different concepts discussed below, we want to touch upon one specific term that is abundantly used in the theory of nuclear fission, namely, the concept of adiabaticity. First, a disclaimer is in order, because in the rigorous (electronic) time-dependent density functional theory (TDDFT), see, e.g. Burke et al. (2005), the term "adiabatic" has a different meaning than here. Indeed, there it denotes an approximation of the time-dependent functional that is local in time and thus disregards memory effects. In this sense, all time-dependent approaches to fission, which we discuss below, are adiabatic, and releasing this constraint in nuclear physics probably belongs to the future not covered by the present report at all.
In nuclear physics, the term "adiabatic" has several interwoven, although not fully identical facets. First, it may mean that the collective motion proceeds through a sequence of local ground states, each corresponding to the system being constrained to a given set of collective coordinates and intrinsic quantum numbers. Adiabatic motion then means that a time-dependent wave function acquires collective kinetic energy through infinitesimal admixtures of local excited states, whereas non-adiabatic corrections correspond to significant admixtures of those. A dissipative motion (section 3.5) means a constant irreversible flow of energy away from the local ground state.
Provided the local ground states are well defined and do not cross with excited states, this constitutes a coherent physical picture. However, this picture breaks down in situations where several local ground states (characterized by different intrinsic quantum numbers) coexist and compete energetically (e.g., different one-quasiparticle states in odd- nuclei). Then, the system may proceed diabatically, along a fixed configuration, or adiabatically, by changing the configuration, depending on the Landau-Zener probability of a diabatic transition (section 3.7).
In the context of the time-dependent Hartree-Fock (TDHF) or TDDFT, adiabaticity denotes a very specific approximation of the time-dependent one-body density matrix, which is assumed to have the time-odd part much smaller than its time-even part Baranger and Vénéroni (1978). In essence, this approximation holds only when the motion is appropriately slow. Another commonly used definition of adiabaticity involves a separation of variables into slow and fast coordinates Tully (2012). Many concepts of fission theory, such as the Collective Schrödinger Equation (CSE), are based on the division of degrees of freedom into “collective” and "non-collective”.
All those definitions are connected by the fact that, in practice, the local ground states can only be considered within the mean-field picture, which means the TDDFT interpretation of the one-body evolution. A weak mixing with low-lying excited states is then equivalent to the requirement of the slow motion. In the following, the notions of adiabaticity and dissipation are discussed in many places, as undoubtedly they constitute pivotal points of the theoretical description of fission.
It is important to understand the various time scales associated with the different stages of fission in order to anticipate the kind of dynamics that would be needed in the theory. One of the most intriguing questions about fission dynamics is the time it takes for fission to occur.
There are in fact several time scales that affect the duration of the fission process. Fission that goes through the compound nucleus is delayed by the compound nucleus lifetime, which is much longer than the dynamics time scales. At excitation energies below the fission barrier, the fission lifetime is largely dominated by the tunnelling probability and can vary by many orders of magnitude. The next scale is that of the collective motion from the outer turning point to scission, see figure 1. The slower this is, the more valid will be diffusive and statistical modelling of the dynamics. Finally, the time it takes to scission plays a special role affecting particularly the TKE and excitation energies of the fragments. At higher energies, the distinctions between the different stages are less clear, but the basic dynamics taking the system from a highly excited compound nucleus to a scission configuration is governed by a similar time scale.
One of the most difficult questions to investigate experimentally is fission time scales since they involve the early stages of fission dynamics. They are not generally accessible directly but must be inferred from the analysis of products at later stages of fission. Experiments attempting to measure fission times Hinde (1993); Jacquet and Morjean (2009); Frégeau et al. (2012); Sikdar et al. (2018) often need to be complemented by a model description of, e.g., the emitted neutrons and their dependency on angular momentum or excitation energy. See section 7.7 for a discussion of this topic. As a result, it is likely that different experimental methods probe different characteristics of the fission time distribution. Theoretically, in addition to dynamics, statistical processes such as particle emission and thermal fluctuations may be important. In general, one needs theoretical approaches accounting for fluctuations in order to predict the entire fission time distribution instead of the average or most likely time.
The mean-field approximation provides the backbone of microscopic nuclear theory for all but the lightest nuclei. In the context of nuclear fission, the great advantage of the mean-field theory is that it is directly formulated in the intrinsic, body-fixed reference frame of the nucleus, in which the concept of deformed nuclear shape and its dynamical evolution is naturally present.
Briefly, the self-consistent many-body wave functions are directly or indirectly composed of Slater determinants of orbitals, with the orbitals computed as eigenstates of one-body mean-field potential. If the mean-field potential is determined by the expectation value of a Hamiltonian in the Slater determinant, we arrive at Hartree-Fock (HF) approximation. If a pairing field is included, we arrive at the Hartree-Fock-Bogoliubov (HFB) approximation. As in electron density functional theory (DFT) of condensed matter and atomic physics, the Fock-space Hamiltonian is often replaced by an energy density functional (EDF) defined through one-body densities or density matrices. As is common practice in the nuclear physics literature, we will use these notions interchangeably, where HFB and HF are used to distinguish between nuclear DFT with (HFB) and without (HF) treatment of pairing correlations. The use of an EDF instead of a Hamilton operator sometimes necessitates to take different intermediate steps in formal derivations, but leads to self-consistent equations that for all practical purposes coincide with those of HF (or HFB if pairing is present).
Another approach in common use, the macroscopic-microscopic (MM) method, avoids the delicate issues of constructing an EDF that reproduces the systematic properties of heavy nuclei. Here the basic properties of the nucleus are derived from its size and shape, expressed in some parameterisation of the surface. The orbitals are constructed with a potential derived from the shape of the nuclear surface, and its energy is computed using the liquid drop model together with shell corrections determined by the orbital energies. The first quantitative theoretical understanding of fission came from this approach Brack et al. (1972); Bjørnholm and Lynn (1980), see also its review in Krappe and Pomorski (2012), and it has been successfully applied to calculate mass and charge yields.
In HF and HFB, wave functions representing different nuclear shapes are constructed by constraining the single-particle density matrix in some way. This is often implemented by adding fields with Lagrange multipliers, but it can also be done more directly; see section 8.1.2. Typically, in nuclear DFT the nuclear shape is defined by several parameters that are taken as collective variables.
The potential energy surface (PES) represents the lowest possible energy of the evolving system consistent with the specified values of the collective variables. As mentioned above, the PES is generally multi-dimensional. Although the PES alone does not suffice for predicting the dynamical evolution, it is nevertheless very useful because its topography makes it possible to understand and anticipate the main features of the dynamics. The local minima, saddle points, and the scission surface are key features that often make it possible to predict isomeric properties, threshold energies, and fission fragment yields.
For a given point in the collective space, the potential energy of the corresponding nuclear configuration and its internal structure can be obtained either by minimising the total energy in the CHF (Constrained HF) or CHFB (Constrained HFB) framework or by calculating the MM energy for the specified shape. The first method results in an optimised shape within the given constraints while the second method can miss aspects of the shape beyond the defined shape parameterisation. There are important consequences in both methods for defining the collective space variables and for the continuity of the resulting surface Möller et al. (2001); Dubray and Regnier (2012); Schunck et al. (2014).
While the standard PES describes the configuration having no excited orbitals or quasi-particle excitations, some approaches need energy in the presence of internal excitations. In the MM method it requires the calculation of shell and pairing corrections at finite excitation Ignatyuk et al. (1980), while the self-consistent method may employ a temperature-dependent DFT formalism Egido et al. (2000); Pei et al. (2009); Sheikh et al. (2009); Schunck, Duke and Carr (2015); Zhu and Pei (2016).
The PES is usually presented as a function of a few multipole moments in the CHF and CHFB framework, but multipole moments control the shape only loosely and do not provide sufficient discrimination between intrinsic configurations at large elongations. When needed, other types of constraints can provide additional discrimination power. For example one can define a neck-size parameter to be added to the multipole moments Warda et al. (2002). More drastically, the entire density distribution can be constrained. Such a density-constrained method Cusson et al. (1985); Umar et al. (1985) has been used successfully within TDHF (Time-dependent HF) approach to calculate heavy-ion interaction potentials Umar and Oberacker (2006); Simenel and Umar (2018). For a sequence of shapes in the collision, the instantaneous density of the evolving system obtained in TDHF is used as a constraint for a static HF calculation, yielding the lowest-energy configuration compatible with the constraint. This eliminates both the collective kinetic energy and the internal excitation and may therefore be interpreted as the potential energy. While this information is important for going beyond TDHF and TDHFB (Time-dependent HFB), there is no simplification in the dynamics when taking the -dependent density as a collective variable. See Sect. 4.1 for additional discussion of collective variables.
It is also possible to introduce constraints that depend more on the wave function than on the shape. In particular, one can get a high discriminatory power in the space of axially symmetric configurations by requiring a certain filling of the orbitals with respect to their axial symmetry Bertsch et al. (2018). See also constraints pertaining to the strengths of pairing correlations, discussed in section 4.1.
The time-dependent version of HF is an established approach to nuclear dynamics and has been extensively used to model heavy ion collisions Simenel (2012); Simenel and Umar (2018); Sekizawa (2019). In principle it can be easily generalised to the HFB approximation, but one is only now reaching the computational power to carry out calculations without introducing artificial constraints and approximations Bulgac et al. (2016); Hashimoto and Scamps (2016); Scamps and Hashimoto (2017); Magierski et al. (2017); Bulgac, Jin, Roche, Schunck and Stetcu (2019). These approaches have an important property that they respect energy conservation and the expectation values of conserved one-body observables such as particle number. Their strong point is that they usually give a good description of the average behaviour of the system under study. Their weak point is that, since TDHF equations emerge as a classical field theory for interacting single-particle fields Kerman and Koonin (1976), the TDDFT approach can neither describe the motion of the system in classically-forbidden part of the collective space nor quantum fluctuations. As a consequence, the real-time TD approach cannot be applied to SF theory. Moreover, the fluctuations in the final state observables, some being due to non-Newtonian trajectories Aritomo et al. (2014); Sadhukhan et al. (2017), are often greatly underestimated in time-dependent approaches.
While the symmetry-broken product wave function of HFB already provides a very good description for many properties, it is deficient if a self-consistent mean-field symmetry is weakly broken. In such cases, it is advisable to extend the method beyond a single-reference DFT. One way of doing this is to use the small amplitude approximation to the TDHFB, i.e., the quasiparticle random phase approximation (QRPA). The QRPA is a vertical expansion that accounts for selected correlations coming from excited states of the system. Another way of enriching the DFT product state is through a multi-reference DFT Bender et al. (2019). This represents a horizontal expansion Dönau et al. (1989). Two commonly used beyond-DFT methods belong to this category. One is the generator coordinate method (GCM). The GCM wave function is a superposition of single-reference DFT states computed along a collective coordinate (or coordinates). The second group contains various projection techniques, in which the projection operation is applied to an HFB state in order to restore internally-broken symmetries. The most advanced multi-reference DFT approaches combine the virtues of the vertical and horizontal expansion by employing the GCM based on the projected HFB states, which often contain contributions from multi-quasiparticle excitations.
A microscopic Hamiltonian treated in the CHF or CHFB approximations can be mapped onto a collective Schrödinger equation (CSE) in the coordinates defined by constraints. This mapping is the essence of the GCM. Typically the mapping is carried out using the Gaussian overlap approximation (GOA) to determine the kinetic energy operator. Examples of such calculations for low-energy fission can be found in Goutte et al. (2004, 2005); Erler, Langanke, Loens, Martínez-Pinedo and Reinhard (2012); Regnier et al. (2016); Zdeb et al. (2017); Tao et al. (2017); Regnier et al. (2019); Zhao et al. (2019). With several coordinates, the GCM produces much wider distribution in the mass yields than can be realised in the evolution in time of a single CHF or CHFB configuration. On the other hand, the underlying wave function is composed of zero-quasiparticle configurations and so underestimates the non-collective internal energy.
To take into account non-adiabatic effects during the fission process, the inclusion of excitations built on the zero-quasiparticle vacuum becomes essential. Several experimental observables attest to the importance of two-quasiparticle (2-qp) excitations, which include the pair-breaking mechanism and the coupling of pairs to the collective degrees of freedom. From this point of view, the inclusion of explicit 2-qp components into the GCM wave function is of interest Bernard et al. (2011). One of the major advantages of the model is the nonlocal nature of the couplings between collective modes and intrinsic excitations. The development of this approach, however, poses several problems related to the truncation of the 2-qp space; keeping track of excitations along the collective path; and evaluation of overlap kernels. So far, the model presented in Bernard et al. (2011) has not yet been be applied to fission problems.
The nuclear Hamiltonian commutes with particle number, angular momentum, and parity symmetry operations. The density functional of nuclear DFT is usually symmetry-covariant Carlsson et al. (2008); Rohoziński et al. (2010). Still, due to the spontaneous breaking of intrinsic symmetries in mean-field theory, several symmetries are usually broken in a nuclear DFT-modeling of fission. There are well-established projection methods to restore broken symmetries based on the generalised Wick’s theorem Mang (1975); Stoitsov et al. (2007); Bender et al. (2019); Sheikh et al. (2019) that have been applied to calculations of the fission barrier of Pu, either combining parity and particle-number projection Samyn et al. (2005), or combining angular-momentum and particle-number projection with shape mixing Bender et al. (2004). The methods are straightforward in principle for models based on a Fock-space Hamiltonian. Difficulties can arise in EDF realisations of nuclear DFT, as discussed in Anguiano et al. (2001); Dobaczewski et al. (2007); Bender et al. (2009); Duguet et al. (2009); Sheikh et al. (2019)). However, these problems do not concern the calculation of one-body observables such as the average particle number in the fission fragments Regnier and Lacroix (2019); Bulgac (2019).
While the self-consistent DFT dynamics is very powerful, it largely ignores the internal degrees of freedom that can bring large fluctuations of observables and dissipate energy Kubo (1966); Yamada and Ikeda (2012). There are several ways that the additional degrees of freedom can be taken into account in the equation of motion.
A simple diffusion master equation assumes the presence of first-order time derivatives. This approach has been remarkably successful in describing mass and charge yields Randrup and Möller (2011). While the utility of this ansatz has received some support from recent microscopic calculations Bulgac, Jin, Roche, Schunck and Stetcu (2019), its quantitative validity still needs to be derived.
More generally, one can consider time-dependent models that combine time-even inertial dynamics with time-odd dissipative dynamics. A common classical formulation is with a multidimensional Langevin equation Sierk (2017); Usang et al. (2019). In this approach, the dissipated energy goes into a heat reservoir characterised by a temperature. Recently, a hybrid Langevin-DFT approach has been applied to explain SF yields Sadhukhan et al. (2016). While this is reasonable in a phenomenological theory, there is so far no microscopic justification of this approach. It is to be noted, however, that the predicted fission yield distributions are found insensitive to large variations of dissipation tensor Randrup et al. (2011); Sadhukhan et al. (2016); Sierk (2017); Matheson et al. (2019). The corresponding quantum dynamics requires an equation of motion for the density matrix of the system. One formulation is with the Lindblad equation; see also Bulgac, Jin and Stetcu (2019).
Tunnelling motion in SF is usually treated via a quasiclassical, one-dimensional formula for the action integral which is based on two main quantities that can be obtained in nuclear DFT: the PES and the collective inertia (or mass) tensor. The fission path is computed in a reduced multidimensional space, using between two and five collective coordinates describing the nuclear shape and pairing; see section 4.1. The mass tensor requires the assumption of a slow, near-adiabatic motion; see section 4.2. The pairing gap makes this assumption most credible for even-even nuclei, but even in such systems one can expect non-adiabatic effects due to level crossings Schütte and Wilets (1975a, b); Strutinsky (1977); Nazarewicz (1993). The following questions are relevant for making progress in SF studies.
Usually, SF trajectories in the collective space are determined by considering several shape-constraining coordinates. It is better to assume that the collective motion happens in a large space parameterised by the Thouless matrix characterising a HFB state. One approach to determine the collective path in that way has been proposed in Marumori et al. (1980) and Matsuo et al. (2000). There the equations of motion have a canonical form (involving both coordinates and momenta), and constraining operators are dynamically determined.
The current barrier-penetration methodology is based on a minimisation of the collective action along one-dimensional paths, although our experience with above-barrier fission evolution suggest that the use of several degrees of freedom is important. It may be possible to generalise the one-dimensional quasiclassical WKB-like formula by a more general solution to a few-dimensional tunnelling problem Scamps and Hagino (2015).
The admixtures of non-adiabatic states may be crucial to understand fission hindrance in odd nuclei. The excitations to higher configurations can be induced by crossings of single-particle levels and by the Coriolis coupling; see section 3.7.
An alternative approach is provided by the formalism of imaginary-time TDHFB Reinhardt (1979); Levit et al. (1980); Puddu and Negele (1987); Negele (1989); Skalski (2008). Configuration mixing can be performed according to well defined equations, and spontaneous fission lifetimes could be determined without having to define collective inertia. Non-self-consistent solutions using a phenomenological Woods-Saxon potential and omitting pairing have already been obtained Brodziński et al. (2018). If the simplified approach with pairing gives the proper order of magnitude for the fission hindrance and its weak dependence on particle numbers, the next step would be to incorporate the requirement of self-consistency.
In the original framework for a microscopic theory of fission above the fission barrier, Hill and Wheeler Hill and Wheeler (1953) proposed a model based on time-dependent diabatic evolution of mean-field configurations interrupted by possible jumps to other configurations at the points of level crossings. At those intersections the probability to switch orbitals would be computed by the Landau-Zener formula Wittig (2005). This viewpoint has been pursued further in the later literature, especially in the context of MM models Schütte and Wilets (1978); Nörenberg (1983); Matev and Slavov (1991) but the challenges of implementing a microscopic theory has prevented the actual calculation of macroscopic parameters such as friction coefficients.
In the present era, computational resources are available to carry out this program using DFT and effective interactions to compute the interaction matrix elements at level crossings. Thus, we may now make theoretical predictions of the balance between inertial and dissipative dynamics that can be used as inputs to more macroscopic models such as the ones solved with the Langevin equation. The steps to carry out this program could follow the strategy of the dissipative diabatic dynamics (DDD) approach Nörenberg (1983, 1984); Berdichevsky et al. (1989); Matev and Slavov (1991); Mirea (2014, 2016). This would involve the construction of diabatic PES, computing the interaction matrix elements between the configurations that cross each other, and obtaining information about the time-dependence of the motion along the path. This can be achieved by adding constraints on the velocity fields in the time-dependent evolution of the configurations so that energy is conserved, see section 8.1.2. With these additional tools one can explore the probability that there will be some excitation of the nucleus along the fission path. Namely, the probability of exciting the system from the adiabatic path to a 4-qp excited state can be computed using the Landau-Zener formula. To get an actual dissipation rate, one would need to track a large number of level crossing along the diabatic path. There are many issues that need to be studied carefully at this point such as (i) non-orthogonality of the configuration basis; (ii) validation of the level density against compound nucleus level density in the first well; (iii) breaking down of the assumptions inherent in the Landau-Zener formula at low velocities; and (iv) development of reliable statistical approximations to deal with the large number of level crossings.
The nuclear shape evolution generally rearranges the nucleons and it is important to understand the associated collective kinetic energy. Beyond the outer turning point, while the electrostatic repulsion tends to accelerate toward the scission point, dissipative couplings damp the motion. To connect with experiment, collective kinetic energy beyond the saddle point is particularly important, because any relative motion at scission adds to the fragment kinetic energy generated by the Coulomb repulsion following scission. While in an adiabatic description all the energy difference between the saddle point (or the outer turning point for the low-energy fission) and the scission point is converted into collective kinetic energy, for strongly non-adiabatic motion, the system will irreversibly convert most of that energy into intrinsic excitations, endowing the nascent fragments with little collective motion.
For the low-energy fission, where the motion is fairly adiabatic and the dynamics of the system is governed by a CSE, the corresponding kinetic energy can be calculated on the basis of the associated inertia tensor. For a quantitative description of the collective kinetic energy it is therefore essential to understand: the relevant collective coordinates (see section 4.1); the inertia tensor (see section 4.2); the role of non-adiabatic effects in general and during the descent to scission in particular (see section 3.12); and the role of dissipation, especially near scission. In the time-dependent approaches, the kinetic energy can be obtained by computing the collective current as the local collective kinetic energy density , where is the current density.
While most models agree that the pre-scission kinetic energy forms only a small part of the final fragment kinetic energy, there is no general consensus about its quantitative magnitude Bonneau et al. (2007); Borunov et al. (2008); Simenel and Umar (2014); Bulgac, Jin and Stetcu (2019). In general the TDDFT calculations suggest that the evolution beyond the fission barrier is strongly dissipative, and this impacts the predicted kinetic energy Bulgac, Jin and Stetcu (2019).
It should be noted that TDHF models for high-energy fission are too diabatic, as the absence of pairing leads to artificial fission hindrance Goddard et al. (2015). The inclusion of pairing by allowing occupation number evolution solves this hindrance problem Matev and Slavov (1991); Tanimura et al. (2015); Scamps et al. (2015); see also section 3.10.
The calculation of collective kinetic energy and inertia for nuclei with an odd number of protons and/or neutrons sometimes leads to diverging quantities. While a solution to this problem is still missing, a natural strategy would be to relax the adiabatic approximation. Note that this is also mandatory when the two nascent fragments start accelerating close to the scission point. In this context, the TDDFT is arguably the most suited method, since it naturally allows the investigation of non-adiabatic effects in macroscopic transport coefficients Tanimura et al. (2015). The most important challenge for the TDDFT method is the inclusion of dissipation along the fission path, together with consistent fluctuations in such a way that the fluctuation-dissipation theorem is satisfied. The real challenge for this microscopic approach will be to properly describe the energy exchange between collective and intrinsic degrees of freedom (see section 3.12 for more discussion).
Nuclear fission can be naturally formulated in the language of reaction theory. Indeed, the SF process can be viewed as a decay of a Gamow resonance, while the induced fission can be expressed as a coupled-channel problem. The description of fission cross sections in induced fission, for instance, is clearly in the domain of reaction theory.
There are two general frameworks for the reaction theory of many-particle systems, namely -matrix theory and -matrix theory. The -matrix framework has been extensively used in the past to construct phenomenological treatments of induced fission Bjørnholm and Lynn (1980). But this approach is not well adapted to microscopic calculations and has never been applied at a microscopic level. In contrast, the -matrix theory is closely allied with the configuration-interaction Hamiltonian approach that has been very successful in nuclear structure theory. The -matrix theory has been applied to a broad range of physics subfields, but in nuclear physics only as a framework for statistical reaction phenomenology Kawano et al. (2015). There are severe challenges to implementing the theory microscopically. Some of these challenges are similar to those discussed in section 3.7 in the context of microscopic DDD implementations.
First, one needs to construct a basis of non-orthogonal CHFB configurations that effectively span the important intermediate states in the fission dynamics. This may be contrasted with present approaches that rely heavily on an adiabatic approximation or TDDFT implementations. Another challenge is the need for microscopic calculation of the decay width of internal configurations to continuum final states of the daughter nuclei. Tools based on the GCM should be powerful enough to estimate the needed widths Bertsch and Younes (2019); Bertsch and Robledo (2019). It would take a large computational effort, and to date no implementations of the GCM have be validated. However, there is some experience for nuclear decays releasing an alpha particle IdBetan and Nazarewicz (2012) as well as simple reactions involving light composite particles Wen and Nakatsukasa (2017).
The -matrix reaction theory might be applied as a schematic model for testing the approximations made in other approaches Bertsch (2020). In particular, the importance of pairing in induced fission is not well understood. As mentioned in the next subsection, fission does not occur on a reasonable time scale in pure TDHF at low energies; adding pairing via TDHFB lubricates the dynamics.
It is often said that pairing acts as a lubricant for fission. What is meant by this assertion is that if pairing is removed from the treatment, then the evolution from the ground state to scission takes place through diabatic configurations which are often disconnected. As a consequence, mean-field time evolution is sometimes unable to find the path to scission Goddard et al. (2015). As realised early Moretto and Babinet (1974); Negele (1989); Nazarewicz (1993), the pairing interaction mixes those configurations and enables smooth transitions between them Nakatsukasa and Walet (1998). The stronger the pairing is, the easier these transitions are, and the faster fission occurs.
Pairing also plays an important role in the traditional WKB treatment of SF. The half life is proportional to the exponential of the action, which in turn is proportional to the square root of the effective collective inertia. The latter is proportional to the inverse of the square of the pairing gap, so the stronger pairing correlations the smaller action and shorter half lives. Indeed, numerous MM studies Urin and Zaretsky (1966); Łojewski and Staszczak (1999); Staszczak et al. (1989) demonstrated that pairing can significantly reduce the collective action; hence, affect predicted spontaneous fission lifetimes. Implications of the pairing strength being a collective degree of freedom for fission are very significant, especially for the SF half-lives Staszczak et al. (1989); Giuliani et al. (2014); Sadhukhan et al. (2014); Zhao et al. (2016); Bernard et al. (2019).
Apart from possible tunnelling, the fission path traverses the PES at finite intrinsic excitation energy.22 2 For clarity, the excitation energy is the difference between the total energy and the PES energy computed in CHFB constrained to the same shape parameters. The collective kinetic energy is subtracted out to obtain the intrinsic part. It can also be thought of as the energy of the quasiparticle excitations in the fissioning nucleus. Because the intrinsic energy is fairly high, and the collective evolution is fairly slow, the system has the character of a compound nucleus. Therefore the intrinsic energy is often referred to as the statistical energy and characterised by a local temperature. Any dynamical model of fission must therefore take into account statistical excitation energy parameterised by a local temperature. Furthermore, it is of interest to study how the fission process develops as a function of total energy, as is conveniently done in experiments inducing fission by projectiles at variable energies. However, in the microscopic frameworks, the concept of the finite temperature is plagued by a number of conceptual and technical difficulties:
In the context of the MM approaches, an effective, deformation-dependent temperature can easily be defined following the recipes given in Ignatyuk et al. (1980) and Diebel et al. (1981). Given the local temperature at each point of the collective space, one can construct an auxiliary potential energy surface by damping the shell correction accordingly Randrup and Möller (2013). This maintains a micro-canonical description of the process where the total energy is constant, yet an effective PES exists and can be used for dynamics. Such an approach is more difficult in the DFT framework. First of all, many EDFs have an effective nucleon mass well below unity, adversely affecting the relationship between excitation energy and derived temperature. Secondly, the connection between the experimental excitation energy and the finite-temperature PES has not been clearly defined and, in principle, calculations of dynamics should be carried out without its help Pei et al. (2009); Sheikh et al. (2009); Schunck, Duke and Carr (2015); Zhu and Pei (2016). In any case, it is important to have a good definition of temperature to describe the disappearance of fission barriers, the increase in fluctuations, and the damping of pairing and shell effects.
At finite excitation energy the fissioning system displays statistical fluctuations in addition to its inherent quantum fluctuations. Therefore a large variety of outcomes is possible and, consequently, fluctuations of observables are significant. As is obvious from the wide spreads in mass and charge yields, it is essential that the theoretical framework allows the development of large fluctuations in the final outcome. Moreover, because the possible final outcomes exhibit a very large diversity, it is not feasible to express them as fluctuations around an average. Rather, the only practical approach would provide an ensemble of outcomes whose further fate (the primary fragment de-excitations process) can then be followed individually and specific observables can be extracted much as an ideal experiment would be analyzed. This can be achieved in probabilistic treatments using Monte-Carlo simulations.
The adiabatic approximation has often been employed to describe spontaneous fission and low-energy induced fission. In these formulations, the coupling of the adiabatic collective states to the other internal degrees of freedom is a continuing challenge. Nevertheless, it is important to assess how such couplings affect decay rates and branching ratios of the fission channel to other channels. There are models available for the coupling, e.g. Brink et al. (1983); Caldeira and Leggett (1983), but they have never been validated in a microscopic reaction-theory setting. It is worth noticing, however, that the classical Langevin equation can be derived using the model by Caldeira and Leggett Abe et al. (1996). This fact might be utilised to extend the Langevin approach to the quantal (tunnelling) regime. That would be an important step for the theory of low-energy nuclear dynamics.
Another problem is that the adiabatic approximation breaks down at level crossings. In that situation, a possible approach to treat dissipation is with the DDD approach (see section 3.7).
A challenge for microscopic theory is to include adiabatic dynamics together with couplings to internal degrees of freedom. Such a method should include a consistent treatment not only for intermediate states but also for the collective inertia. Current methods to compute inertial-mass tensor rely on the adiabatic approximation Giannoni and Quentin (1980); Matsuo et al. (2000); Hinohara et al. (2007, 2008); Wen and Nakatsukasa (2020). A challenging problem is to develop a microscopic theory for the large-amplitude collective motion that takes into account non-adiabatic transitions.
Ideally, the dynamic equations would provide a time-dependent statistical density matrix rather than the time-dependent wave function produced by TDHF, TDHFB, etc. An ambitious framework for such a theory has been proposed in Dietrich et al. (2010). It would require major additions to the present coding algorithms as well as availability of high-performance computing resource to implement.
In the theory of heavy ion reactions, it has been long recognised that there are two distinct mechanisms that arise in a semi-classical approach to dissipation Sierk and Nix (1980); Randrup and Swiatecki (1984). The one-body dissipation operates at the level of TDDFT. It is fast when it is present because the relevant time scale is the time it takes a nucleon to transverse the nucleus. The two-body dissipation is associated with nucleon-nucleon collisions which are largely blocked at the Fermi surface; its time scale is much longer. Quantum mechanically, it requires theoretical frameworks beyond mean-field theory, for example, the inclusion of quasiparticle excitations in the time-dependent wave functions.
In the semi-classical theory, the one-body dissipation can be encapsulated in two formulas, the wall formula for the internal dissipation in a large nucleus, and the window formula for heavy ion reactions. Both have been used very successfully for many years. However, the assumptions required for the validity of the wall formula may become questionable for low-energy fission dynamics: time scales are long and shape changes are highly correlated into low multipoles.
With the improvements in the computational capabilities for carrying out TDDFT, it should be possible to map out the region of validity of the semi-classical reductions much better. We now have credible evidence that the one-body dissipation in a quantum framework is adequate to dissipate the collective kinetic energy Wada et al. (1993), but still not capable to produce a statistical equilibrium.
The presence of dissipation has two distinct but fundamentally related effects on the evolution of the collective variables. One is the average effect of the dissipative coupling which acts as a friction force resisting the evolution; this part is well described by the DFT. The other arises from the remainder of the dissipative effect which appears as a random force on the collective variables. These two forces are related by the fluctuation-dissipation theorem Kubo (1966), often referred to as the Einstein relation.
As a consequence of the fluctuating force, the system is continually faced with a multitude of trajectory branchings, a situation that is very hard to encompass within the usual microscopic frameworks. That mean-field approaches, such as TDHF, are not suitable for describing collective fluctuations has become especially apparent after the advent of the variational approach by Balian and Vénéroni Balian and Vénéroni (1981) who also proposed an alternative treatment of one-body fluctuations equivalent to time-dependent random phase approximation (TDRPA) Balian and Vénéroni (1984). The practical applications of this method to fission are still limited Scamps et al. (2015); Williams et al. (2018) and further developments of the formalism are required. A more radical approach would be to develop treatments that automatically endow the collective variables with fluctuations by making their evolution explicitly stochastic, as discussed in section 3.12.2.
A quasiparticle HFB vacuum is not expected to be a good approximation for long-time evolutions. A simple estimate leads to the conclusion that the lifetime of such state is of the order of 100-200 fm, whereas the time it takes from the saddle to scission might exceed several thousand of fm.
For the long time evolution, the mean-field state is expected to couple to the surrounding many-body states leading both to the breakdown of the mean-field picture, and to a dispersion beyond mean field in the collective space Goeke and Reinhard (1980); Goeke et al. (1981). This dispersion is usually described by the TDGCM (Time-dependent GCM).
However, there are a number of limitations in current applications of the TDGCM to fission that all employ the GOA. For the moment, most implementations assume that the collective motion stays in the adiabatic PES. With this assumption, the manifestation of non-adiabaticity, and henceforth a proper description of the transfer of energy from collective motion to internal excitation, cannot be achieved. Extending the TDGCM approach beyond the adiabatic limit Bernard et al. (2011); Regnier and Lacroix (2019) to incorporate dissipation and internal excitation, will require broadening the CSE picture for the collective degrees of freedom; see, for instance, Dietrich et al. (2010).
Once the energy is transferred from the collective to the internal degrees of freedom, it should be understood how the energy is subsequently being redistributed so that internal statistical equilibrium is approached.
The onset of equilibration in interacting many-body systems is a long-standing problem and several theories have been proposed to treat this process Abe et al. (1996); Lacroix et al. (2004); Simenel (2010). In most treatments, it is assumed that repeated in-medium Pauli-suppressed two-body collisions lead the internal degrees of freedom towards statistical equilibrium on a time scale that is relatively short compared with that of the macroscopic evolution.
One example is the extended TDHF approach Wong and Tang (1978, 1979); Lacroix et al. (1999) or its extensions based on the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, generically called time-dependent density matrix Cassing and Mosel (1990); Peter et al. (1994). These approaches have rarely been used in nuclear reactions Tohyama and Umar (2002); Assié and Lacroix (2009) and specific technical problems seem to strongly jeopardise the obtained results Wen et al. (2018). However, a promising step forward has been achieved with the purification technique, opening new perspectives Lackner et al. (2015, 2017).
Apart from these technical issues, this approach has the advantage that it leads naturally to the Boltzmann-like description. However, during fission, especially as the system passes through the barrier region, the excitation energy is sufficiently low to cause quantal and thermal fluctuations to coexist. This may lead to non-Markovian effects in the macroscopic evolution, which obviously would complicate the treatment. An extension of the TDGCM approach has been proposed for including thermal fluctuations Dietrich et al. (2010), while quantum approaches treating the thermalisation process have proven to be rather complicated. To circumvent them, approximate treatments have been suggested, including the relaxation-time approximation; see Reinhard and Suraud (2015) for recent developments.
Microscopic treatments of dissipation are discussed in section 3.7 and the previous section. Such a level of detail can be avoided by a macroscopic transport approach, treating its parameters phenomenologically. The equations here describe the evolution of just a few collective properties, typically the shape of the fissioning system, as the initial compound nucleus evolves into two separate fragments. Because the retained collective degrees of freedom are coupled dissipatively to the internal system, the macroscopic evolution has a stochastic character and the natural formal framework is the Langevin transport equation. This treatment has been very successful Sierk (2017); Usang et al. (2019) in calculating a variety of fission observables. A particular advantage of the Langevin dynamics is that it automatically allows the collective trajectory to undergo dynamical branchings, thereby making it possible for the system to evolve from a single shape to a large variety of final configurations.
Once the collective degrees of freedom have been identified, the Langevin equation requires three ingredients: the (multi-dimensional) PES, the associated inertia tensor, and the dissipation tensor describing the coupling to the internal system and giving rise to both the collective friction force and the diffusive behaviour of the collective evolution. It is straightforward to apply microscopic theory to determine the first two of these key quantities. For example, recent calculations of spontaneous-fission mass and charge yields Sadhukhan et al. (2016, 2017); Matheson et al. (2019), employed DFT to obtain the PES and the inertia tensor as a function of several collective coordinates, then performed a WKB action minimisation for the tunnelling, and a subsequent Langevin propagation until scission using a schematic dissipation tensor and random force. Such a hybrid approach can be extended to the calculation of other fission observables, such as the shapes and kinetic energies of the fragments.
The microscopic justification for the parameterised dissipation tensor remains a problem. As we have seen previously, TDDFT includes one-body dissipation mechanisms. However, dissipation cannot take place without fluctuations but it is not clear how to include the fluctuations in the microscopic treatments. Fluctuations inherent in individual configurations of HF or HFB can be addressed by the Stochastic Mean-Field approach, which makes a statistical assumption on the origin of fluctuations, see Ayik (2008); Lacroix and Ayik (2014); Tanimura et al. (2017) and references therein. In this approach, the noise only stems from the initial conditions. However, as it is well known in open quantum systems theory, complex initial fluctuations can lead to a stochastic dynamics with Markovian and non-Markovian noise continuously added in time during the evolution. Understanding the connection between initial fluctuation in collective space with the microscopic Langevin approach on one side, and the link with current phenomenological Langevin approaches on the other, should be addressed in the near future.
In parallel, attempts have been made to reformulate quantum theories leading to thermalisation as a stochastic process between quasiparticle states Reinhard and Suraud (1992); Lacroix (2006) and important efforts are being made nowadays in condensed matter physics to apply these methods Slama et al. (2015); Lacombe et al. (2016). For the moment, such reformulation have been essentially made assuming jumps between Slater determinants, and equivalent formulations including superfluidity is desirable for fission. A specific problem is again that at very low excitation energy, stochastic approaches might face the difficulty of exploring rare processes.
As is clear from the discussion in previous sections, dissipation plays a key role in fission dynamics. Langevin transport treatments of the collective evolution Karpov et al. (2001); Sierk (2017) employ the simple wall and window formulas (see section 3.12.1) in various variants, for example the chaos-weighted wall friction Pal and Mukhopadhyay (1998). In many calculations, the dissipation tensor was phenomenologically adjusted to reproduce experimental results. In recent transport studies Usang et al. (2016, 2017), both the dissipation tensor and the inertial-mass tensor were derived microscopically within the locally harmonic linear response approach as outlined in Ivanyuk and Hofmann (1999), but a validation of this method still remains to be carried out.
In general, one-body dissipation is rather insensitive to the local nuclear temperature (whereas two-body dissipation is strongly energy dependent, especially at low energy where the Pauli blocking is effective). Recent studies Sadhukhan et al. (2016) have shown the importance of dissipation in fission, even at energies relevant to spontaneous fission Dagdeviren and Weidenmüller (1987). As a consequence, the shape evolution acquires the character of Brownian motion and many resulting observables, most notably the fragment mass distribution, are rather independent of the specific dissipation strength employed Randrup et al. (2011); Sierk (2017); Sadhukhan et al. (2016).
One observable that is somewhat sensitive to the dissipation strength is the final fragment kinetic energy, a quantity that has proven to be difficult to treat reliably in models. By contrast, the time elapsed from the crossing of the fission barrier until scission is quite sensitive to the dissipation, being roughly inversely proportional to its strength. However, this quantity is difficult to measure directly, though somewhat equivalent experimental information can be obtained from quasi-fission processes Williams et al. (2018); Banerjee et al. (2019).
It is an important challenge to derive the dissipation tensor from microscopic models. For this, the TDDFT method (including pairing) might be a suitable tool. In its basic form, by energy conservation and by the knowledge of the kinetic energy and excitation of the nascent fragments after scission, one can determine the total energy dissipated from the initial condition. (The excitation of the nascent fragments is initially partly given in the form of distortion energy which will gradually be converted to additional internal excitation as the fragment shapes relax to their equilibrium forms.) Two existing approaches might be useful for obtaining information on dissipation in TDDFT. The first is the density-constrained TDHF method of section 3.2.2. More systematic application of this approach to disentangle the collective energy from the excitation energy without imposing the adiabatic approximation is desirable. An alternative approach, called Dissipative-Dynamics TDHF, consists in making a macroscopic mapping of the collective evolution Washiyama and Lacroix (2008); Washiyama et al. (2009) which, however, requires a somewhat ambiguous choice of the relevant collective coordinates. This approach has not yet been applied to the fission problem, although a first step in this direction has been made Tanimura et al. (2015).