6 Forces for fission dynamics

In a microscopic approach to fission, the effective inter-nucleon interaction or energy density functional is the only ingredient of the theory that includes adjustable parameters. Therefore, the choice of a functional ultimately determines the quality of the microscopic description of phenomena related to fission, and the level of quantitative agreement with data.

Several types of EDF have been proposed over the years, many of which have also been applied to fission. These functionals can be non-relativistic or relativistic (or covariant), and this choice leads to different equations of motion for nucleons; they can be functionals of local or non-local densities; they can be strictly defined as the expectation value of a corresponding generating many-body operator or not; and finally the couplings (parameters) can be constants or include a medium (density) dependence.

The two most widely used non-relativistic EDFs Bender et al. (2003); Schunck (2019) are the finite-range Gogny EDF, which is constructed including the HFB expectation value of a density-dependent interaction, and the Skyrme EDF, which includes momentum- and density-dependent zero-range terms in the interaction. Other types of local non-relativistic EDFs that were recently developed and applied to detailed studies of fission processes are the Barcelona-Catania-Paris-Madrid Baldo et al. (2013), SEI Behera et al. (2016), and SeaLL Bulgac et al. (2018) EDFs. Similarly, there are several varieties of relativistic EDFs in use Schunck (2019); Agbemava et al. (2017, 2019), either with finite-range (meson-exchange) or contact interaction potentials, with non-linearities in the meson and/or nucleon fields, or including density-dependent couplings. Two relativistic point-coupling (contact) functionals, in particular, have successfully been applied to studies of fission dynamics: PC-PK1 Zhao et al. (2010) and DD-PC1 Nikšić et al. (2008).

One important challenge is to increase the predictive power of novel nuclear EDFs compared to traditional functionals such as Gogny or Skyrme, which apparently cannot be improved further Kortelainen et al. (2014). For instance, the density-matrix expansion Carlsson and Dobaczewski (2010); Gebremariam et al. (2010, 2011); Stoitsov et al. (2010) can be used to construct nuclear EDFs that are guided by first principles Dyhdalo et al. (2017); Navarro Pérez et al. (2018); Zhang et al. (2018). Extensions of time-tested EDFs have been subject to recent studies. For example, higher-order gradient terms have been added to the Skyrme EDF Carlsson et al. (2008); Davesne et al. (2013); Becker et al. (2017). The Gogny family of functionals have been extended to include additional density-dependence and tensor interactions Chappert et al. (2015); Pillet and Hilaire (2017); Bernard et al. (2020). One hopes that such extensions would augment the parameter space to be optimised for a better description of fission properties.

Over the past decade it has been realised that exact projection techniques (see section 3.4.2) and exact GCM are ill-defined for EDFs that are not strictly constructed from an effective Fock-space Hamiltonian Anguiano et al. (2001); Dobaczewski et al. (2007); Lacroix et al. (2009); Bender et al. (2009); Duguet et al. (2009); Robledo (2010b). The basic dilemma that one faces in this context is that a suitable form of an effective Hamiltonian that reaches the descriptive power of conventional EDFs has not yet been identified. As a first step in this exploration, a scheme for a systematic construction of flexible two-body interactions by combining finite-range Gaussians and gradients, has been proposed Dobaczewski et al. (2012); Bennaceur et al. (2017). Limiting oneself to a two-body interaction, however, will inevitably lead to an unrealistically small effective mass Davesne et al. (2018), such that one always has to add three-body, and perhaps even higher, interactions. The computationally simplest form of such terms is provided by contact three-body forces with gradients Sadoudi et al. (2013). It turns out, however, that when added to two-body interactions of various forms, they do not offer sufficient flexibility, which makes this quest even more challenging.

Many advanced methods for beyond-DFT modelling of fission dynamics often include explicit correlation energies that were implicit in the effective interaction obtained from the EDF optimisation. This inconsistency may degrade the descriptive and predictive power of the model and should be avoided. A better strategy would be to optimise the EDF parameters using data sensitive to large deformations. This issue is most obvious in the case of corrections for quantal zero-point motion related to symmetry breaking and shape fluctuations, such as those for the centre-of-mass, rotational, and shape-vibrational motion. For instance, the inertia that determines the former is the mass number $A$, which becomes ambiguous whenever one considers the separation of a single nucleus into fragments Goeke et al. (1983); Skalski (2006). The rotational correction increases with deformation and therefore lowers fission barriers, etc. To further complicate matters, one form of quantal correction is transformed into other forms when changing deformation Goeke et al. (1983); Skalski (2006), such that from this point of view many quantal corrections have to be treated simultaneously. The same considerations also apply to exact projections and full GCM.

Static and dynamic pairing correlations play a crucial role for the calculation of deformation energy surfaces, the dynamic fission path, and collective inertia. This means that the pairing part of the effective interaction or EDF might have to be tailored in such a way to reproduce both ground-state properties and selected features that determine fission data, see section 6.2.

Once the form and the framework for which the parameters of an EDF are to be adjusted are decided, the next question concerns the selection of fit observables. Most of the fission observables (lifetimes, fission fragment distributions, …) are computationally expensive and cannot be systematically considered during the optimisation. Therefore, one has to identify properties that encapsulate the essence of the relevant physics probed by fission and, at the same time, can be computed in a reasonable time.

First of all, the EDF has to be capable of describing the structure of the initial state of the fissioning nucleus and the final state of the fragments. At low excitation energy, the requirements for this are the same as for standard nuclear structure applications. One of the most important constraints on the EDF specifically relevant for fission studies is its ability to describe states at very large deformation. Two different types of properties control the general features of fission dynamics: on the one hand the surface and surface-symmetry energy coefficients that determine the average resistance of the nucleus against deformation Nikolov et al. (2011); Jodon et al. (2016), and on the other hand the evolution of shell structure that generates the minima and maxima associated with the multi-humped structure of the deformation energy landscape Brack et al. (1972).

There is some direct information about the excitation energy of highly-deformed states that is available and that can be used to inform the parameter fit. On the one hand, there are barrier heights data Capote et al. (2009); Smirenkin (1993), which have to be interpreted with some caution as in one way or the other the available values were obtained via intermediate models Capote et al. (2009). On the other hand, there are also measured excitation energies of some fission isomers Singh et al. (2002). For a very limited number of fission isomers there is also information about their quadrupole deformation from $E2$ transition moments Metag et al. (1980); Thirolf and Habs (2002), and some information about their shell structure can be obtained from the quantum numbers of bandheads. Additional data on such states would clearly be of great help for fine-tuning the nuclear EDF.

To date, the EDFs most commonly used in fission studies have been adjusted to fission isomer excitation energies Kortelainen et al. (2012) or fission barriers Bartel et al. (1982); Berger et al. (1991); Goriely et al. (2007), with the exception of the relativistic functionals PC-PK1 Zhao et al. (2010) and DD-PC1 Nikšić et al. (2008) that combine information on deformed heavy nuclei and the nuclear matter equation of state. Some authors suggest paying more attention to the nucleus-nucleus interaction between pre-fragments near scission Adamian et al. (2016).

A technical issue that needs to be addressed is that many parameterisations of the nuclear EDF exhibit so-called finite-size instabilities, meaning that homogeneous infinite nuclear matter is unstable against a transition to an inhomogeneous phase that is either polarised in spin or isospin or both, see Pastore et al. (2015) for a review. Finite-size instabilities can be triggered by gradient terms in the EDF, but also by finite-range terms in non-local EDFs Martini et al. (2019); Gonzalez-Boquera et al. (2020). Many parameterisations of the Skyrme EDF exhibit such instability in one or the other spin channel, which becomes an issue when working with time-reversal breaking configurations or when calculating certain RPA modes. Such instabilities are also sometimes found in isovector channels of some Skyrme and Gogny parameterisations. All of these instabilities can be efficiently and unambiguously detected with linear-response calculations of infinite nuclear matter. Such test can be easily incorporated into fit protocols, as already done for the UNEDF2 Kortelainen et al. (2014) and SLy5sX Jodon et al. (2016) Skyrme parameterisations.

Irrespective of the choices that will be ultimately made for the form of the EDF and the protocol for the adjustment of parameters, it is clearly desirable to have just one or a few standard EDFs for fission studies that are used by as many groups as possible in order to eliminate possible dependencies upon the parameterisation when comparing results obtained with different approaches to treat the many-body problem. For TDDFT treatments, it is also important that EDFs are fitted without the centre-of-mass corrections Goeke et al. (1983); Skalski (2006); Kim et al. (1997); Simenel (2012); Kortelainen et al. (2012).

As discussed in the previous section, nuclear density functionals have to be calibrated to experimental data. This empirical wisdom is built into the quality measure ${\chi }^2(𝒑)$ which is a scalar function of the $N_p$ model parameters $𝒑$. The common use of ${\chi }^2$ is to deduce the optimal parameterisation ${𝒑}_0$ by minimising ${\chi }^2$.

Systematic uncertainties can be revealed by comparing predictions of different models; for fission applications, see Kortelainen et al. (2012); Agbemava et al. (2017). In the context of statistical uncertainties related to model parameters, much information can be unraveled by employing ${\chi }^2$ in connection with the tools of statistical analysis Dobaczewski et al. (2014); McDonnell et al. (2015); Schunck, McDonnell, Sarich, Wild and Higdon (2015); Schunck, McDonnell, Higdon, Sarich and Wild (2015); Nikšić et al. (2015); Reinhard (2018).

Computing the probability distribution of the parameters $𝒑$ rather than a single point gives immediately access to two important new pieces of information, the uncertainty of a predicted observable, and the correlation between two observables. Uncertainties are important to control the quality of a prediction. This is mandatory when using the results in further calculations as done, e.g., in nuclear astrophysics, and it is an extremely useful indicator for model development because it reveals deficiencies of parameterisations. Correlations add another world of information. They allow a sensitivity analysis to check the impact of a certain model parameter on an observable Kortelainen et al. (2010, 2012) and they indicate the information content of a new observable as compared to previous ones Reinhard and Nazarewicz (2010); McDonnell et al. (2015). In the context of the present report, it is particularly interesting to apply correlation analysis for the very different observables discussed here, e.g., relating fusion cross sections and fission properties.

There is still more potential in statistical analysis of DFT. So far, the evaluation of uncertainties and correlations has mostly been based on a Taylor expansion of the ${\chi }^2$ and of observables around the optimal parameterisation ${𝒑}_0$. This runs easily beyond validity, particularly for fission properties Higdon et al. (2015). Thus one needs to evaluate the integrals $\int d^{N_p}p\mathrm{\dots }$ in detail which grows quickly infeasible. Here one can take advantage of modern techniques of supervised learning. Employing the posterior probability distribution computed with emulators, one can propagate theoretical statistical uncertainties in predictions of various computed quantities, including binding energies and PESs McDonnell et al. (2015); Neufcourt et al. (2018, 2020); Lasseri et al. (2020). One can teach the emulators to improve the predictions of selected observables in a given region of the nuclear chart by one order of magnitude at practically no extra cost. This is particularly desirable if the output of nuclear DFT calculations is used as input in other chains of calculations as, e.g., in nuclear astrophysics simulations.

To estimate uncertainties, both systematic and statistical, uniform model mixing Erler, Birge, Kortelainen, Nazarewicz, Olsen, Perhac and Stoitsov (2012); Agbemava et al. (2017) can already provide a very valuable information. More advanced techniques involve Bayesian model averaging Neufcourt et al. (2019, 2020), which allows to maximise the “collective wisdom" of relevant models by providing the best prediction rooted in the most current experimental information. This will be an important part of future collaborative projects in fission theory.