5 Initial conditions
Nuclear fission can proceed from a variety of initial states; see section 2.1. In addition to spontaneous fission, the process can be induced by a variety of nuclear reactions that lead to the initial state of the fissioning (mother) nucleus. Figure 3 displays some characteristic time intervals for the preparation of the initial state. The reaction types are roughly ordered by the amount of energy they may bring into the compound system, with the most energetic reactions (“Fast probes”) on the left. On the right, labeled “CN fission”, are the more gentle probes such as neutron capture that proceed through the compound nucleus.
5.1 Neutron-induced fission
Due to its importance in applications, low-energy neutron-induced fission is probably the best experimentally studied and phenomenologically parameterised fission process. Proceeding through narrow neutron resonances, it creates the nucleus in a long-lived excited state that has enough time to thermalise the absorbed energy, thus forming a compound nucleus.
The incident neutron creates a compound nucleus at an excitation energy that can range from somewhat below the barrier to energies above the barrier. As the neutron energy is raised, the resulting excitation may make it possible for the nucleus to evaporate one or more neutrons before fissioning (referred to as a multi-chance fission). At still higher energies, non-equilibrium emission grows important and a thermalised compound nucleus is established only after the loss of one (or possibly more) nucleons.
The representation of the initial compound state in terms of elementary excitations is impractical because the level density is prohibitively high. Thus a direct description in terms of QRPA modes, GCM excited states, etc., would not be feasible and it may be preferable to adopt an approach based on statistical quantum mechanics. However, for the neutron-induced fission close to the neutron drip line, where level densities at the neutron separation energies are small, the process could still be dominated by the direct component rather than the compound one.
Besides the energy, all the quantum numbers of the formed compound nucleus may affect the subsequent fission process, especially when the energy is close to the barrier height. This level of detail is retained in the -matrix theory, discussed in section 3.9.
5.2 Fission induced by fast probes
Fission can be induced by fast probes such as photons (photofission), charged particles, and high-energy neutrons. Surrogate reactions, such as fission following multi-nucleon transfer reactions, are also included here. In these processes, the nucleus is created in an excited state above or below the fission barrier. This state may exhibit specific well-defined structures, such as the giant dipole resonance, which have substantial widths.
Present theoretical descriptions of fission induced by fast probes most often assume the creation of a compound nucleus at a given thermal excitation energy (cf the process of neutron-induced fission discussed in section 5.1). However, for fast probes such an assumption might be ill-founded because the nuclear system may not have sufficient time to thermalise before undergoing fission. Therefore, a non-thermal description of fission at high excitation energies is very much desired Dobaczewski (2019).
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, multi-chance fission is likely to happen.
5.3 Formation by electromagnetic and weak-interaction fields
5.3.1 Photofission
In photofission, a nucleus decays through the fission channel after absorbing a high-energy photon – a -ray. The characteristics of the excited state resulting from photo-absorption – the initial state for the fission process – determines the evolution of the system, for instance, by determining whether enough excitation energy is available to surmount the fission barrier. Thus, the knowledge of excited states above both the ground state (for fissile nuclei) and shape isomers, as well as multipole transition probabilities between these states, is in principle needed to model photo-absorption as a function of the photon energy. If the photon is absorbed through the dipole operator on an even-even nucleus, the angular distribution of the fission fragments gives information about the mixing of the quantum number in the fission process. In general and outside of the giant dipole absorption peak, theories such as the QRPA are needed to sort out the multipoles.
5.3.2 Coulomb excitation
Another electromagnetic excitation method to study fission of heavy nuclei in a relativistic accelerator beam is Coulomb excitation. Here, the process can be treated as excitation by virtual E1 photons, so the considerations in the previous paragraph apply. While the energy transferred is not precisely known, the theory for its distribution is well established.
5.3.3 decay and electron capture
Fission of nuclei far from stability can sometimes be studied when the nuclide is formed by decay of a progenitor nuclide. In terms of the excitation energy, -delayed fission is intermediate between SF and Coulomb-excitation induced fission. Importantly, this process makes it possible to study low-energy fission in proton-rich heavy nuclei that are not accessible by other techniques Andreyev et al. (2013). As in Coulomb excitation, the excitation energy given to the nucleus is not known precisely. Thus the theory of -decay strength function is required to model the whole process. In this regard, the QRPA (in its charge-exchange formulation) is very valuable.
The process of -delayed fission also plays an important role in nucleosynthesis, because it helps to terminate the rapid-neutron-capture process. Fission may occur from the compound nuclei created by neutron capture or from the -decay daughters of those nuclei Mumpower et al. (2018). The latter can happen whenever the decay populates a daughter state with an excitation energy above (or near) the height of the fission barrier. Since it is important to know the spin and the parity of the initial state before fission, the description of -delayed fission requires a microscopic model of the charge-exchange process to provide -strength distributions; for the recent QRPA work see Mustonen et al. (2014); Mustonen and Engel (2016); Shafer et al. (2016). The QRPA applications used to describe -decay are often limited to allowed transitions. Thus it would be necessary to extend many current QRPA codes to enable computation of all possible final states in daughter nuclei.
Because -delayed fission often involves odd-odd nuclei, one should employ a formalism that can be extended to such systems without introducing any additional approximation. Therefore, both the underlying HFB solver as well as the QRPA implementation should break time-reversal symmetry, that is, extend beyond the equal filling approximation. This last point is essential to differentiate between low- and high-spin states in odd-odd nuclei, and thus distinguish between decays from potential isomeric states and the ground state. Once the fissioning daughter state has been determined, one should be able to calculate the corresponding potential energy surface for the particular energy, spin, and parity.
5.4 Heavy-ion reactions
In the search for superheavy nuclei, the experiment uses a heavy-ion reaction to fuse together two large nuclei, hoping that the combined system equilibrates and then decays as a compound nucleus. Cross sections can be estimated for this reaction mechanism, but a crucial ingredient is the probability to form a compound nucleus. The reaction is called fusion-fission in that case; if there is no equilibration it is called quasifission. The understanding of this distinction requires a combination of statistical and truly dynamical approaches which are not necessarily confined to a collective subspace. Quasifission leads to the formation of products that may have similar properties to fission products, but are produced without the formation of compound nucleus. Fusion-fission occurs after the formation of a composite system which fissions due to its excitation, resulting in a fragment distribution that is peaked at equal mass breakup of the composite system. This difference in fragment distributions indicates that quasifission is the faster process and corresponds to a system that is not yet fully equilibrated. As a dynamical process, quasifission is amenable to a description using the TDHF approach Simenel and Umar (2018). A number of TDHF studies of heavy-ion reactions have been reported in recent years Wakhle et al. (2014); Oberacker et al. (2014); Umar et al. (2016); Godbey et al. (2019); Sekizawa (2019); Godbey and Umar (2020). In general, the TDHF results agree well with the experimental quasifission yields, and shed light on some of the underlying reaction dynamics in relation to target/projectile combinations.
Quasi-fission and fusion-fission could be used to map out the non-adiabatic collective landscape between the fusion entrance channel and the fission exit channel. The calculated time scales indicate that while fast quasifission events dominate, much slower events resulting in a fracture with equal mass fragments have also been observed.
One of the open experimental questions is how to distinguish quasifission from fusion-fission. This is important for calculation of the evaporation residue formation probability in superheavy element searches. A collaborative effort between theory and experiment is needed to find ways to address these issues. One may try to “calibrate” the experimental quasifission yields with the help of theoretical simulations thus allowing the extraction of the fusion-fission yield. Study of angular distributions (now routinely measured with large angular acceptance detectors Banerjee et al. (2019)) may be one of the ways to approach this task.
Theoretical studies of quasifission have taught us that the dynamics may be dominated by shell effects Simenel and Umar (2018); Sekizawa (2019). Despite the apparent strong differences between fission and quasifission, it is interesting to note that similar shell effects are found in both phenomena Scamps and Simenel (2018, 2019); Godbey et al. (2019). Quasifission can then potentially be used as an alternative mechanisms to probe fission mode properties. For instance, this could provide a much cheaper way than fusion-fission to test the influence of the Pb shell effects in super-asymmetric SHE fission. Note that this approach would only provide information on the properties of fission modes (mass asymmetry, TKE, excitation energy), but not directly on their competition. Indeed, the latter is likely to be determined near the saddle point, a region of the PES which is not necessarily explored by the quasifission paths.