Synthesis of superheavy elements: Uncertainty analysis to improve the predictive power of reaction models

Background: Synthesis of superheavy elements is performed by heavy-ion fusion-evaporation reactions. However, fusion is known to be hindered with respect to what can be observed with lighter ions. Thus some delicate ambiguities remain on the fusion mechanism that eventually lead to severe discrepancies in the calculated formation probabilities coming from different fusion models.

Purpose: In the present work, we propose a general framework based upon uncertainty analysis in the hope of constraining fusion models.

Method: To quantify uncertainty associated with the formation probability, we propose to propagate uncertainties in data and parameters using the Monte Carlo method in combination with a cascade code called kewpie2, with the aim of determining the associated uncertainty, namely the $95\%$ confidence interval. We also investigate the impact of different models or options, which cannot be modeled by continuous probability distributions, on the final results. An illustrative example is presented in detail and then a systematic study is carried out for a selected set of cold-fusion reactions.

Results: It is rigorously shown that, at the $95\%$ confidence level, the total uncertainty of the empirical formation probability appears comparable to the discrepancy between calculated values.

Conclusions: The results obtained from the present study provide direct evidence for predictive limitations of the existing fusion-evaporation models. It is thus necessary to find other ways to assess such models for the purpose of establishing a more reliable reaction theory, which is expected to guide future experiments on the production of superheavy elements.

Figure 1

Theoretical calculations of ER cross sections for the one-neutron evaporation channel of cold-fusion reactions. The calculated results can be found in Refs. [10, 15, 16, 17, 18]. The figure is taken from Ref. [14].

Figure 2

Calculated formation probabilities for the selected set of cold-fusion reactions. The theoretical results can be found in Refs. [10, 15, 16, 17, 18, 20]. The figure is adapted from Ref. [14].

Figure 3

Average of the survival probability for the $1n$ channel over different energy intervals to take into account the loss of beam energy in the target. The solid red curve indicates the calculated survival probability without taking an average.

Figure 4

Typical values of the reduced friction parameter that can be found in the literature. The figure is adapted from Ref. [41].

Figure 5

Estimated distribution of the empirical formation probability for the reaction ${}^{208}\mathrm{Pb}\left({}^{58}\mathrm{Fe},1n\right){}^{265}\mathrm{Hs}$. The solid line represents the mean value and the dotted lines refer to the lower and upper bounds of the $95\%$ confidence interval, respectively. The cumulative distribution function (CDF) of the empirical formation probability used to define the confidence interval is indicated.

Figure 6

Impact of input distributions on uncertainty associated with the empirical formation probability. The pessimistic case is considered in the upper panel and the optimistic input distributions for $\beta$ and $E_d$ in the lower one. See text for details.

Figure 7

Impact of correction factors on the empirical formation probability. Note that the uncertainty interval is due to all parameters under the pessimistic hypothesis.

Figure 8

Impact of level-density parameters on the empirical formation probability. Uncertainty interval is due to all parameters under the pessimistic hypothesis.

Figure 9

Impact of capture models on the empirical formation probability. Uncertainty interval is due to all parameters under the pessimistic hypothesis.

Figure 10

Impact of fission barriers on the empirical formation probability. Uncertainty interval is due to all parameters under the pessimistic hypothesis.

Figure 11

Systematic comparison of the calculated mean values of the empirical formation probability for the cold-fusion reactions leading to the synthesis of the elements from $Z=104$ to $Z=108$ (cf. Table 1). The deduced formation probabilities are based upon two simple capture models (see text). Lines correspond to theoretical predictions from Refs. [10, 15, 15, 16, 17, 18, 20, 62] as in Fig. 2.

Figure 12

Same as Fig. 11 but the deduced formation probabilities are based upon two different fission-barrier models (see text).

Figure 13

Same as Fig. 11 but the extreme case is presented for both capture and fission-barrier models so as to have the largest discrepancy between the empirical formation probabilities.