Testing the Extended SRM against the 238 U( 3 He, 4 He) 237 U ⇤ surrogate probabilities

. Since its introduction in the seventies’, the so-called surrogate-reaction method (SRM) has motivated the development and improvement of theories in connection to direct reactions. A recent work by Bouland and Noguère, Phys. Rev. C 102 , 054608 (2020), has shown that the inclusion of experimental probabilities in the neutron cross section evaluation process can be better achieved using tools resulting from the e ↵ orts made over the two last decades. In particular, the authors have put forward a new prescription, named after the SRM as extended SRM (ESRM), to convert, with reasonable conﬁdence, probabilities measured in direct reactions to pseudo-experimental neutron-induced reaction cross sections. Applied to the 174 Yb( 3 He,p � ) 176 Lu ⇤ transfer reaction, the ESRM has demonstrated much more precision than the early use of the SRM. In addition to the ‘direct’ analysis of reaction probabilities measured in direct reactions using the right modeling as developed in Phys. Rev. C 100 , 064611 (2019), it is worth to try converting the measured probabilities in pseudo experimental neutron-induced cross sections for neutron reactor data applications. In the present paper, ﬁrst results, before conversion, are shown by applying the two ESRM equations to a ﬁssile isotope.


Introduction
The last decades have seen considerable changes in the nature of the uncertainties related to nuclear reactor energy calculations. These uncertainties, mainly driven in the past by numerous approximations embedded in the various equations, have become less and less sensitive to the approximations, while the question of the uncertainties on the model parameters has become an issue. One way to constrain the parameters, and so to reduce nuclear data uncertainties, is to compare the prediction of an 'observable' with its measurement. In this framework, neutron-induced reaction cross sections are a fundamental observable to build a very large experimental database. However since the seventies', an alternative type of observable, the so-called reaction deexcitation probability, has also contributed to enrich the fitting experimental database, especially for the fission channel. Indeed the measured fission probability is well-suited to assign fission barriers of fissile isotopes, since their innerand outer-barrier heights (V A and V B respectively) at the ground state are significantly lower than S n , the neutron emission energy. Figure 1 illustrates this statement.
The present paper focuses on the use of an elaborated technique, recently described and named after the surrogate-reaction method, the ESRM [1] as Extended Surrogate-Reaction Method. It provides a rigorous framework for surrogate-reaction data analysis. Following recent advanced studies, as the ones of Refs. [2][3][4]  of two di↵erent expressions of width fluctuation correction factor for the calculation of cross sections and decay probabilities, respectively, brings more confidence in the simultaneous analyses at low energy. In the Ref. [1], demonstration was made of the capability of the method by application to the 174 Yb( 3 He,p ) 176 Lu ⇤ transfer reaction. Here, first application of the ESRM equations (re-viewed in next Section 2) is made to the decay of a heavy nucleus, the 237 U ⇤ , whose fission reaction opens above S n . Comparisons are made in this work with the data by Marini et al. [5], in which the fission and -emission probabilities for the 238 U( 3 He, 4 He) 237 U ⇤ transfer reaction were measured simultaneously.

Extended Surrogate-Reaction Method (ESRM) equations
The ESRM procedure relies on the equations coupling two distinct observables that share the same excited compound system A ⇤ (CS). An important issue is the use of a unique set of nuclear structure parameters for best reproduction of deexcitation probabilities and cross sections. Reference [1] provides the ESRM equations for a medium-mass nucleus. These equations are still valid for a fissile nucleus carrying a single-humped fission barrier. If not, the equations become more complicated [6]. For easier display, one recalls below the ESRM equations of Ref. [1]. It reads for a deexcitation channel c 0 , where the excitation energy, E x , and the incidentneutron energy, E n , are related as E n = (E x S n )(A + 1)/A with A, being the mass of the target nucleus. A ⇤ n (E n , J ⇡ ) is the neutron-induced partial CS formation cross section, B J ⇡ c 0 is the partial branching ratio (BR) for the deexcitation of the CS into channel c 0 . The BR is defined as the ratio of the transmission coe cient of the exit channel c 0 to the sum of the transmission coe cients of all open channels.
is the fraction of the CS formed by direct reaction in a specific (J, ⇡) state of excitation energy E x and defined as the ratio is the customary in-out-going channel width fluctuation correction factor (WFCF), that contains the usual elastic enhancement for W J ⇡ n,n ground [7]. This elastic channel enhancement is strongly supported by the flux borrowed from the inelastic channels (across W J ⇡ n,n 0 ) and, to a lesser extend, from the radiative channel. The expression for W J ⇡ surr,c 0 (E x ), the surrogate-dedicated WFCF (labelled SWFCF), was recently derived in Ref. [6]. The need for a specific definition of the WFCF comes from the fact that the usual WFCF is suited to the Hauser-Feshbach formalism applied to a compound nucleus A ⇤ . In case of surrogate measurements, A ⇤ is formed in a direct or semi-direct reaction and therefore, at an early stage, is not statistically equilibrated. The nature of the in-going and out-going channel widths and their possible interferences are expected to be di↵erent. For the SWFCF no significant elastic scattering correlation is expected [6], leading to a much different pattern. We have drawn the SWFCF overall pattern (spin-parity integrated) for the 237 U ⇤ in Fig. 2 as a function of the incident-neutron energy and the observed exit channel for an entrance spin-parity distribution identical to the one of the neutron-induced reaction to magnify at best the profile di↵erences with the classic WFCF.   W surr, for the -emission or W surr,n ground for a neutron-emission with the residual nucleus left in its ground state (W surr,n 0 otherwise) . To magnify the comparison, the SWFCF plot uses the entrance spin-parity distribution corresponding to the one applied in WFCF treatment (i.e.; the neutron-incident distribution).
We emphasize that in the ESRM equations, we indeed use a spin-parity distribution relevant to the surrogate reaction treated. The role of the radiative and fission channels that now carry the enhancement pattern and deviates from the customary WFCFs profile [7], is well visible (up to +150%). As expected, the usual high-energy pattern is recovered when the total number of opened deexcitation channels increases; all the SWFCF tending thus to unity. From the features presented in Fig. 2, one understands the importance of correctly treating the width fluctuations in the calculation to reproduce with the minimum of biases the measured -ray emission and fission probabilities.

Solving the ESRM equations for a heavy nucleus
Since the paper [1] has demonstrated the e cacy of the ESRM equations for a medium-mass nucleus, demonstration must be made that it works when the fission channel is in competition with the other deexcitation channels. The ultimate goal, by using the ESRM equations, is to provide the unknown model parameter that is missing for the final agreement between the calculation and the experiment, either in terms of cross sections and deexcitation probabilities. Ideally this approach must be carried out for all the open channels over the widest energy range. Although this is a goal for the future, no neutron-emission probabilities are yet simultaneously measured with the fission and the -emission channels. Therefore, this comparison focuses on those observed channels. An example of poorly-known data involved in Eq. (1) and/or (2) is the spin-parity entrance distribution for the surrogate reaction, F A ⇤ surr (E x , J ⇡ ). The best knowledge of the other quantities must ensure the The choice has been made to study with the ESRM equations the 237 U ⇤ CS, for which the neutron-induced fission reaction threshold lies at about 600 keV neutron energy (Fig. 3). This CS acts as a 'non-fissile' nucleus below 6.0 MeV excitation energy and as a 'fissile' above. Figure 4 displays the neutron-emission, and fission probabilities as a function of the excitation energy of the 237 U ⇤ CS. One observes a slow increase of the fission probability from 5.5 MeV, reaching the fission plateau at 9 MeV. A better knowledge of the n+ 236 U fission cross section is of high interest, since there is still a large spread among the various evaluated curves that do not fully agree with the experimental data points (Fig. 3). The n+ 236 U capture cross section is easier to fit, since the average capture cross section is well reproduced by an adequate choice of the average total capture width using the well-established and simple prescription of Gilbert-Cameron [9] for the level densities. For nuclear energy industry, there is a renewed interest in the n+ 236 U reaction since there is some will to increase the amount of 236 U material in fresh UO 2 fuel. Beyond the economical argument, recent neutron-induced cross section and probability measurements make the 237 U ⇤ CS a good candidate for testing the ESRM equations. Exhaustive description of present calculations is beyond the aim of this proceedings, but we present below some preliminary results.
On the ground of the rigorous approach supplied by the ESRM equations and two di↵erent spin-parity population distributions relevant for the neutron-incident reaction or the surrogate reaction, neutron-induced average cross sections and surrogate-reaction probability calculations have been performed, simultaneously, for the 237 U ⇤ CS. Preliminary results are shown in Figs. 3 and 5 for the cross sections and Fig. 6 for the deexcitation probabilities. The latter results do not carry the biases brought by the early use of the surrogate-reaction method (as shown in Figure 5 of Ref. [10]), that was solely based on the Weisskopf-Ewing hypothesis of the independence of the decay-channel probability on the CS spin-parity distribution [11]. Indeed the agreement between the calculations and the experimental data for the channel is now reached although the exact pattern of the entrance direct-reaction J ⇡ distribution remains an open question. Beyond the use of the most appropriate direct-reaction J ⇡ distribution, the capture right above S n will be deeply impacted by the absence of SWFCF treatment (Fig. 2) in the energy range of the resonance fluctuations. In addition, we expect that a possible application of the WFCF formulation in Eq. (2) instead of the newly-defined SWFCF [6] will bias the fitted model parameters, included the parameters of the poorly known direct-reaction CS J ⇡ distribution.
Assuming that the various terms of the ESRM equations (Eqs. (1) and (2)) are all calculated (or predicted) for the 237 U ⇤ with a reasonable precision (either set from reference nuclear data or fine-tuned in the work) except for the spin-parity entrance distribution of the surrogate reaction, F A ⇤ surr (E x , J ⇡ ), we can evaluate the influence of the distribution used as input to Eq. (2). In the present case, the experimental database refers to the measurement by Marini et al. [5] of the 238 U( 3 He, 4 He) reaction. Figure 6 shows two calculations, one using a J ⇡ distribution from the literature [13] and the other resulting from a pick-up calculation using the in-house QPVR module of the AVXSF-LNG code [12]. The two distributions di↵er in terms of mean value and standard deviation. From Figure (6), it is clear that the input of the J ⇡ distribution from the literature [13], although not perfect [14], shows a better agreement with the experimental data than the calculation based on the QPVR prediction.

Conclusions
By using the ESRM equations (Eqs. (1) and (2)), our actual capability to infer, with reasonable accuracy, any unknown quantity among n,c 0 , P A ⇤ surr,c 0 , B J ⇡ c 0 and F A ⇤ surr , with the other variables, A ⇤ n , W J ⇡ n,c 0 and W J ⇡ surr,c 0 , assessed by the formalism, is on the right path. Demonstration has been made above that the knowledge of all quantities except one, which in present case is the spin-parity distribution corresponding to the 238 U( 3 He, 4 He) reaction, can provide valuable feedback on the missing information.
Our new capability of simultaneously and coherently analyse deexcitation probabilities and neutron-induced partial cross sections is a major asset for the di cult task of evaluating the neutron-induced reaction cross sections within the nowadays nuclear data target accuracy. In the past that kind of comparative analysis was generally made using various model approximations, as the ones involved in the early use of the surrogate-reaction method [15]. Using the ESRM equations ensures our capability to extract the best non-biased information from surrogate-reaction data. Reference [6] also gives the ESRM transformation to convert surrogate-reaction data into pseudo-reaction neutron-induced cross section data as an easy complement to the existing experimental cross section database needed for an evaluation based solely on cross section data. We expect to be able to make this transformation in a close future.