2.1 Various predictions

Compound nuclei formed during the collision decay through neutron evaporation or fission. Thus, the 1n survival probability is nothing else than a branching ratio

where Γ_n and Γ_f stand for the neutron evaporation width and the fission width respectively. The latter dominates. They can be calculated by a standard statistical model; see e.g. Ref. [9]. Uncertainty in the survival probability is easily deduced from the uncertainties in the widths, using the usual propagation formula:

Here, we have assumed that Γ_n and Γ_f are independent from each other. When the ratio Γ_f/Γ_n is large, the relative uncertainty in the survival probability reaches its maximum value. This is very intuitive: as the fission decay mode dominates, evaporation events are very rare. Thus, a small change in the fission width will affect much the survival probability.

Uncertainty in the neutron evaporation width and in the fission width contribute equally. However, Γn mainly depends on the neutron separation energy whereas, the fission width mainly depends on the fission barrier, the damping energy of the shell correction energy and the friction coefficient when taken into account. None of these parameters can be directly measured and are not well determined, inducing large uncertainties in the fission width and then, survival probability. As shown in Refs. [5, 6], the fission barrier is the dominating quantity among these three parameters.

Fission barriers can be calculated by various microscopic or macroscopic-microscopic models while the other two parameters are generally adjusted to reproduce experimental data. Comparison between different calculations [7, 8] has shown that predictions in fission barriers, which is the most sensitive parameter, span few MeV. Consequently, calculated survival probabilities can differ by several orders of magnitude.

Even if one restricts the comparison to fission barriers that share the same shell correction energy, the predicted values span 1.5 MeV and can induce 2 orders of magnitude differences in the fusion-evaporation cross-sections [5, 6]. In one case, the fission barriers were estimated by the old method, i.e.

where B_LDM is the fission barrier calculated with a Liquid-Drop-Model and ΔE_shell is the shell correction energy at the ground state. The other case corresponds to fission barriers calculated directly by the same macroscopic-microscopic model.

Although the survival probability is the best understood part of the reaction leading to the synthesis of superheavy elements, the ambiguities in the fission barrier lead to discrepancies that span several orders of magnitude. This is more than for the formation probability that is supposed to be less understood.

Thus, reliable predictions requires accurate fission barrier predictions. How well can we predict such a sensitive parameter?

2.2 Best estimate

Assuming that the model predicting the fission barrier is correct, there are still ambiguities in its parameters that induce uncertainties in the calculated physical quantities. There are few uncertainty analysis of the theoretical evaluation of the fission barrier. In Ref. [10], a very simple micro-macro model was used and lead to an uncertainty in the fission barrier of about 0.5 MeV. Another estimate based on a Bayesian analysis of a DFT model lead to a similar value for the uncertainty in the fission barrier of ²⁴⁰Pu [11].

All these estimates rely on a fitting procedure on nuclear masses and give an uncertainty proportional to the RMS of the fit [12]. Whatever the model, this RMS has remained over few hundreds of keV this last decade. There is little hope to get an order of magnitude better. Thus, the uncertainty in the fission barrier for a given model will also remain at few hundreds of keV.

Constraint from experiments is possible although direct measurement of the fission barrier is not an easy task. The heaviest nucleus which fission barrier has ever been measured is ²⁵⁴No using the gamma multiplicities. Uncertainty at spin 0 is estimated to be about 0.9 MeV [13-15]. This value is extrapolated from measurements at higher spins. The fission barrier can also be extracted by inverting a statistical decay model, namely the Kewpie2 code [9]. The result is model dependent and one has to be careful when using the value in another model. In particular, the extracted fission barrier depends on the value of the other parameters such as the friction coefficient. An analysis based on Bayesian inference was performed to deduce the fission barrier from invented experimental data [16]. The uncertainty in the barrier depends on the experimental uncertainty and the number of data points. Values range from 34 keV for quite accurate data to 0.4 MeV; see Ref. [16] for details.

Consequently, it seems that there is an incompressible value of the uncertainty in the fission barrier that cannot be overcome. Although less important, other parameters also have large uncertainties. The friction coefficient is not better known than 20 years ago. The damping energy of the shell correction energy has not been carefully investigated since the Ignatyuk’s prescription [17] in 1975.

Because of these uncertainties in key parameter together with the amplification effect due to the fact that we want to estimate the cross-section of rare events, it is hardly conceivable to produce predictions with an accuracy lower than about one order of magnitude for cold fusion reactions. It is more for hot fusion reactions as we need fission barriers of several isotopes.

3.1 Common understanding of the phenomenon

Understanding and modelling fusion hindrance is a long standing problem. Back in the 1960’s, the first predictions of fusion-evaporation cross-sections were far too optimistic [18]. Nix and Sierk showed for symmetric reactions that the dynamical trajectory for the fusing system must pass over a conditional saddle point in a multidimensional space in order to form a compound nucleus, in contrast to light systems for which the conditional saddle point lies outside the point of hard contact in heavy-ion reactions [19]. Later, Świątecki [20, 21] first introduced a dynamical model that broke with the idea that the capture process can be understood in terms of the static interaction of two spheres. The model emphasises the role of rapid dynamical deformations away from the configuration of two spheres in contact. It also includes dissipation that leads to the requirement of even higher energies to fuse. Soon, experimental evidences confirmed the model and a lot of efforts were devoted to quantify the reduction of the cross-section due to the fusion hindrance to the synthesis of superheavy elements. For a review, see Ref. [22].

Although dissipation plays a crucial role in understanding the fusion hindrance [23], associated fluctuations were not included so far. The Langevin equation that has been used extensively to study the fluctuation-dissipation dynamics [24] has been adopted to study the dynamical diffusion over the conditional saddle that lies between the contact and the compound configurations [25-29]. This lead to more realistic dependence of the fusion cross section as a function of energy and a better estimate of the extremely low cross-sections.

Very recently, we showed that the initial condition of the formation step slips due to the elimination of the fast variables in a multidimensional description and this affects significantly the formation probability [30].

This point of view is challenged by so-called DNS model that was developed later. It is based on a very different concept, assuming a frozen configuration and exchange of nucleons [31-33]. Thus, it is contradictory with the models mentioned above based on the evolution of collective degrees of freedom.

Although progress have been made in understanding the fusion hindrance, there is still no consensus on the dynamical treatment. Thus, it is almost impossible to perform an uncertainty analysis in such conditions. Moreover, many quantitative ambiguities remain. What is the barrier height of the conditional saddle that has to be overcome? It is clear that a small change in its value will have a great impact on the formation probability. What is the dissipation strength? What is the role of structure effects?

A comparison between models can give some insights on the amplitude of the discrepancies. Figure 1, reproduced from Ref. [3], shows that the formation probability for cold fusion reactions calculated by various models spans 2 or 3 orders of magnitude. Consequently, to improve the predictive power of the models describing the whole reaction leading to the synthesis of superheavy elements, one must constrain the formation probability responsible of the fusion hindrance.

3.2 Strategies to constrain the formation probability

Fusion-evaporation residue cross sections are of no help to assess the formation mechanism because of the survival probability that is also not well constrained as explained in the previous section. To get rid of the decay step, one should focus the step before in the reaction, i.e, the fusion cross-section.

The channel competing with the formation is the so-called quasifission process. It consists in reseparation into two fragments without reaching the compound state. On an experimental point of view, of course, understanding the competition between quasifission and fusion is very important. However, it is very difficult to distinguish the quasifission fragments from the fission ones as they have similar mass distributions. The two processes differ by their time scale and thus, angular distribution [34]. Consequently, one lacks of reliable experimental fusion cross-sections.

One idea is to consider that reaction dynamics models must strive to reproduce experimental data on quasifission. However, constraining models on the dominating channel leads to unavoidable uncertainties. And, as for fission-evaporation competition, these uncertainties in quasifission modelling will have a large impact on the rare fusion events. Thus, reproducing quasifission data will definitively help to constrain models but might not allow to get a better agreement in the formation probability predictions.

Alternatively, hindrance could be constrained by comparing reactions leading to the same compound nucleus, one being hindered and the other not, in order to get rid of the ambiguities in the decay phase of the reaction. Such studies, currently under development, should naturally include uncertainty analysis.