Future of Nuclear Fission Theory 1 Introduction 3 Basic concepts of fission theory

2 Main Features of Fission

To set the stage for the subsequent specialised considerations, we begin with a brief presentation of the main features of the nuclear fission phenomenon. Figures 1 and 2 present schematic illustrations of the evolution leading from a single nucleus to two pre-fragments, nascent fragments, primary fragments, which subsequently appear in detectors as fission fragments, see caption of figure 2.

Figure 2: Schematic representation of the different stages of a fission process, starting from the initial nucleus (on the left), approaching the scission point as pre-fragments, dividing into two excited nascent fragments, which after getting fully Coulomb accelerated become primary fragments, then promptly emit neutrons and photons and undergo $\beta$ decays, and finally become fission fragments in the exit channel. The associated time scales are indicated on the axis underneath.

Fission is a time-dependent transformation which can be conveniently separated into distinct stages, each characterised by its own time scale, as shown in figure 2. The process proceeds from some initial state through a complicated collective evolution ending with the emergence of two excited nascent fragments. They in turn undergo a sequence of prompt and/or delayed de-excitations decays ending with two product nuclei in their ground or isomeric excited states¹¹ 1 In this paper, we are not concerned with the extremely rare phenomena of ternary and quaternary fission..

The most obvious physical attribute during the evolution of the fissioning nucleus is its overall elongation, correlated with the different stages as shown in figure 1. Initially the elongation is that of the equilibrium shape of the mother nucleus. From this, the collective evolution proceeds through a sequence of shapes whose time-dependent elongations exhibit a diffusive behaviour. Eventually the system finds itself beyond the outer saddle point and then evolves toward scission, as its shape takes on a binary form and the elongation grows ever larger. At scission the system divides into nascent fragments which are then accelerated apart.

2.1 Spontaneous and induced fission

It is useful to distinguish spontaneous fission (SF) which occurs in nuclei in their ground states from induced fission brought about by a reaction or decay process bringing in energy from the outside. SF is one of the main decay modes of superheavy nuclei and is therefore of great interest in the experimental search for them. While SF primarily occurs from the nuclear ground state, it has also been observed from isomeric states.

On the theory side, the relatively long lifetimes are due to the existence of a potential barrier that must be penetrated. Consequently SF is an inherently quantal process; see section 3.6. An interesting aspect of SF is its dependence on the number parity of the nucleus: in odd- $A$ nuclei it is typically hindered by $\sim 3-5$ orders of magnitude relative to their even-even neighbours. Fission of odd-odd nuclei is believed to be even more hindered, but credible data are scarce.

In addition to a SF, fission can be induced by a variety of nuclear reactions. The fission-induced processes include: neutron capture (responsible for energy production in fission reactors), electron capture and beta decay, photofission, and reactions involving charged particles and heavy ions. In all these processes, the fissioning nucleus is created in an excited state, which may lie above or below the fission barrier.

Theoretical descriptions of fission induced by fast probes often assume the creation of a compound nucleus at a given thermal excitation energy. However, as discussed later, that assumption might be ill-founded for fast probes because the nuclear system may not have sufficient time to thermalise before undergoing fission. This becomes increasingly important at higher energies where pre-equilibrium processes play an increasingly significant role and may lead to the emission of one or more nucleons before equilibrium is reached. Moreover, as the excitation energy of the compound nucleus is increased, neutron evaporation competes ever more favourably with fission and as a result, one or more neutrons may be evaporated before fission occurs (multi-chance fission). In addition, for non-thermalised systems one should develop approaches using fixed energy rather than fixed temperature.

2.2 Important observables

When talking about fission observables, it is important to remember that what is often considered “experimental” is often the result of an indirect process, in which a quantity of interest is extracted from measurements with the help of some model or model-dependent assumptions.

Nuclear fission is a very complex transformation and there are many quantities of interest that are directly measurable and subject to theoretical modelling. [A set of key fission observables suitable for validation of theoretical models was proposed in Bertsch et al. (2015).] We list here some of the most important ones, with their common designations:

Spontaneous-fission half-lives ( $T_{\rm SF}$ ).: Measured SF lifetimes (or half-lives) span a range from microseconds or smaller to billions of years. To describe such a range is a significant challenge to theory.
Total and differential fission cross sections.: For instance, the neutron induced fission cross section $\sigma$ (n,f) and its energy and angular dependence or the threshold energy for fission observed in a photo-fission cross section that is closely related to the height of a fission barrier.
Yields ( $Y(A)$ , $Y(Z)$ , $Y(Z,A)$ ).: They describe probabilities for producing fission fragments of given mass and/or charge. Such data are particularly important in nuclear astrophysics. Yields refer to primary, independent or cumulative distributions (see figure 2).
Fission spectrum.: This includes the average number of neutrons per fragment, their energies, the average number of photons per fragment and their energies, multiplicity distributions, angular correlations, etc.
Total kinetic energy (TKE).: The post-acceleration kinetic energy of the fission fragments, its distribution, and its dependence on fragment mass.
Beta-decay spectrum of fission products.: This is particularly important for the fundamental theory of beta decay and includes the neutrino spectrum.

Correlations between the above quantities (e.g., between fragment mass and TKE), as well as with other quantities (e.g., with the spin of the fissioning nucleus) are also very important. We wish to emphasise that the fission observables should be accompanied by uncertainties. This is crucial in the context of nuclear data evaluation and applications in general.

In this context, it is useful to mention some important unobservables (physical concepts that cannot be observed directly). Arguably, the most celebrated quantity that belongs to this group is the fission barrier. Fission barrier height can be defined theoretically as the energy difference between the ground state and the highest saddle point in a computed potential energy surface (PES) that has the lowest energy for all possible paths leading to fission from the ground state. Fission barriers inferred from measured cross sections are plagued with ambiguities because the extraction procedure is often based on a simplistic picture of a fission pathway. Another unobservable concept is that of a compound nucleus; it is based on a model that assumes the full thermalisation of the system and ignores pre-equilibrium processes. Other useful yet unobservable quantities include: scission point at which the nucleus breaks into nascent fragments, shell energy on the path to fission, pairing energy at the barrier, and pre-fragments that are formed in the pre-scission region.