# Testing the extended SRM against the $^{238}$U($^{3}$He, $^{4}$He) $^{237}$U* surrogate probabilities

Olivier Bouland, Paola Marini, Beatriz Jurado

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Olivier Bouland, Paola Marini, Beatriz Jurado. Testing the extended SRM against the $^{238}$U($^{3}$He, $^{4}$He) $^{237}$U* surrogate probabilities. 15th International Conference on Nuclear Data for Science and Technology (ND2022), Jul 2022, Online, United States. pp.03008, 10.1051/epjconf/202328403008 . hal-04299821

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# Testing the Extended SRM against the $^{238}$U($^{3}$He,$^{4}$He)$^{237}$U* surrogate probabilities

Olivier Bouland¹,*, Paola Marini², and Beatriz Jurado²

¹CEA, DES, IRESNE, DER, SPRC, Physics Studies Laboratory, Cadarache, F-13108 Saint-Paul-lez-Durance, France

²Univ. Bordeaux, CNRS, LP2I Bordeaux, UMR 5797, F-33170 Gradignan, France

**Abstract.** 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 still be better achieved using tools resulting from the efforts 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, measured direct-reaction induced probabilities to pseudo experimental neutron-induced cross sections. Applied to the $^{174}$Yb($^{3}$He,$\mathrm{py}$)$^{176}$Lu* transfer reaction, the ESRM has demonstrated much more precision than the early use of the SRM. Beyond 'direct' analysis of direct-reaction induced probabilities 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.

## 1 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 section* measurements are a basic 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 inner and outer barriers heights at ground state ($V_A, V_B$) are significantly lower than $S_n$, the neutron emission energy. Figure 1 illustrates this statement.

The present paper focuses on the use of an elabo- rated technique, recently described and named after the surrogate-reaction method, the ESRM [1] as Extended Surrogate-Reaction Method. It provides a rigorous frame- work for surrogate-reaction data analysis. Beyond recent- advanced studies as the one of Ref. [2], the use of two

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Predicted and measured fission probabilities for the $^{237}$Np($^{3}$He, df)$^{238}$Pu* transfer reaction as a function of the com- pound nucleus excitation energy. The neutron emission threshold ($S_n$), shown as vertical dashed line, provides the neutron-scaled baseline. The vertical long-dashes lines represent the inner- and outer- barrier heights ($V_A, V_B$).</figcaption></figure>

different concepts of width fluctuation correction factor for the calculation of cross sections and decay probabil- ities still bring more confidence in the resulting simulta- neous analyses at low energy. In the Ref. [1], demonstra- tion was made of the capability of the method by appli- cation to the $^{174}$Yb($^{3}$He,$\mathrm{py}$)$^{176}$Lu* transfer reaction. Here, first application of the ESRM equations (reviewed in next

<footnote>*e-mail: olivier.bouland@cea.fr</footnote>
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. [3] in which the fission- and $\gamma$-emission probabilities for the $^{238}$U($^3$He, $^4$He)$^{237}$U* transfer reaction were measured simultaneously.

## 2 Extended Surrogate-Reaction Method (ESRM) equations

The ESRM procedure relies on the equations coupling two distinct observables that share the same $A^*$ excited compound system (CS). An important issue is the use of an 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 [4]. For easier display, one recalls below the ESRM equations of Ref. [1]. It reads for a $c'$ deexcitation channel,

$$ \sigma_{n,c'}(E_n) = \sum_{J^\pi} \sigma_n^{A^*}(E_n, J^\pi) B_{c'}^{J^\pi}(E_n) W_{n,c'}^{J^\pi}(E_n), \quad (1) $$

$$ \mathcal{P}_{\text{surr},c'}^{A^*}(E_x) = \sum_{J^\pi} \mathcal{F}_{\text{surr}}^{A^*}(E_x, J^\pi) B_{c'}^{J^\pi}(E_x) W_{\text{surr},c'}^{J^\pi}(E_x). $$

where the excitation energy, $E_x$, and $E_n$ are related as $E_n = (E_x - S_n)(A + 1)/A$ with A, being the mass of the target nucleus. $\sigma_n^{A^*}(E_n, J^\pi)$ is the neutron-induced partial CS formation cross section, $B_{c'}^{J^\pi}$ is the partial branching ratio (BR) for the deexcitation of the CS into channel $c'$. BR is defined as the ratio of the transmission coefficient of the exit channel $c'$ to the sum of the transmission coefficients of all open channels. $\mathcal{F}_{\text{surr}}^{A^*}(E_x, J^\pi)$ is the fraction of the CS formed by direct reaction in a specific $(J, \pi)$ state of excitation energy $E_x$ and defined as the ratio $[\sigma_{\text{surr}}^{A^*}(E_x, J^\pi)/\sigma_{\text{surr}}^{A^*}(E_x)]$. $W_{n,c'}^{J^\pi}(E_n)$ is the customary in-out-going channel width fluctuation correction factor (WFCF) that contains the usual elastic enhancement for $W_{n,n'\text{ground}}^{J^\pi}$ [5]. This elastic channel enhancement is strongly supported by the flux borrowed from the inelastic channels (across $W_{n,n'}^{J^\pi}$) and, to a lesser extend, from the radiative channel. The expression for $W_{\text{surr},c'}^{J^\pi}(E_x)$, the surrogate-dedicated WFCF (labelled SWFCF) was recently derived in Ref. [4]. 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 is expected to be different and their possible interferences. For the SWFCF no significant elastic scattering correlation is expected [4], 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 neutron-incident energy and the observed exit channel for an entrance spin-parity distribution taken identical to the one of the neutron-incident reaction to magnify at best the profile differences

<figure><img src="image_3.png" /><figcaption>**Figure 2.** (Color online) Comparison of the overall pattern of the surrogate-dedicated WFCF (SWFCF) of the $^{237}$U* as a function of the neutron energy and the exit channel ($W_{\text{surr},f}$ for the fission, $W_{\text{surr},\gamma}$ for the $\gamma$-emission or $W_{\text{surr},n'\text{ground}}$ for a neutron-emission with the residual nucleus left in its ground state ($W_{\text{surr},n'}$ 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).</figcaption></figure>

(2) with the classic WFCF. 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 endorse the enhancement pattern and thus deviating from the customary WFCFs profile [5], is well visible (up to +150%). As expected, the usual high-energy pattern is recovered when the total number of deexcitation channels opened 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 a calculation to reproduce with the minimum of biases the measured $\gamma$-ray emission and fission probabilities.

## 3 Solving the ESRM equations for a heavy nucleus

Since the paper [1] has demonstrated the efficiency of the ESRM equations for a medium-mass nucleus, demonstration must be made that it works when fission is in competition. 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 $\gamma$-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, $\mathcal{F}_{\text{surr}}^{A^*}(E_x, J^\pi)$. The best knowledge of the other quantities must ensure the
<figure><img src="image_4.png" /><figcaption>**Figure 3.** (Color online) Neutron-induced fission cross section of the $^{236}$U as a function of the neutron energy. Present calculation based on the ESRM equations and the AVXSF-LNG code is compared with various evaluated curves (JEFF-3.1, JENDL-4.0 and ENDF/B-VIII.0) and some recent experimental data sets [6].</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 5.** (Color online) Neutron-induced capture cross section of the $^{236}$U as a function of the neutron energy. Present calculation based on the ESRM equations and the AVXSF-LNG code is compared with various evaluated curves (JEFF-3.1, JENDL-4.0 and ENDF/B-VIII.0) and some recent experimental data sets [6].</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 4.** (Color online) Partial deexcitation probabilities calculated as a function of the excitation energy in the $^{237}$U* CS. The vertical dashed lines represent the inner- and outer-barrier heights ($V_A, V_B$) evaluated in this work. Uniform spin-parity distribution for the CS population is assumed in this display.</figcaption></figure>

validity of the solution obtained for the missing ingredient.

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 carries the double property to act as a 'non-fissile' nucleus below 6.0 MeV excitation energy and as a 'fissile' above. Figure 4 displays the neutron-emission-, $\gamma$- and fission overall probabilities as a function of the excitation energy in the $^{237}$U* CS. One observes a slow increase of the fission probability from 5.5 MeV, reaching the fission plateau at 9 MeV.

Better knowledge on the n+$^{236}$U fission cross section is of present interest since there is still a large spreading among the various evaluated curves that do not represent at best the experimental data points (Fig. 3). The n+$^{236}$U capture cross section is easier to fit since the average capture cross section if well reproduced by an adequate choice of the average total capture width using the well-established and simple prescription of Gilbert-Cameron [7] 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 not 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 different spin-parity population distributions relevant either to 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 displayed on 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. [8]) that was solely based on the Weisskopf-Ewing hypothesis of reaction decay probability spin-parity independence [9]. Indeed the agreement between the calculations and the experimental data in terms of $\gamma$-channel is now reached although the exact pattern of the entrance direct-reaction $J^\pi$ distribution remains an open question. Beyond the use of the most appropriate direct-reaction $J^\pi$ distribution, the capture
<figure><img src="image_7.png" /><figcaption>**Figure 6.** (Color online) Fission and γ-emission probabilities as a function of the excitation energy in the $^{237}$U* CS. Present *preliminary* calculations, using the same nuclear database as the calculated cross sections, are compared to the data by Marini *et al.* [3] for which the fission and γ-emission probabilities were measured according to the $^{238}$U($^{3}$He,$^{4}$He) surrogate reaction.</figcaption></figure>

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 [4] will bias the fitted model parameters that can be the parameters of the not so well known direct-reaction $J^\pi$ distribution.

Assuming that the various terms of the ESRM equations (Eqs. (1) and (2)) are 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, $F_{surr}^{A^*}(E_x, J^\pi)$, we can evaluate the influence of the distribution used as input to Eq. (2). In present case, the experimental database refers to the measurement by Marini *et al.* [3] using the $^{238}$U($^{3}$He,$^{4}$He) reaction. Figure 6 shows two calculations, one using a $J^\pi$ distribution from the literature [11] and the other resulting from a pick-up calculation using the in-house QPVR module [10] of the AVXSF-LNG code. The two distributions differ in terms of mean value and standard deviation. From Figure (6), it is clear that the input of the $J^\pi$ distribution from the literature [11], although not perfect [12], shows a better agreement with the experimental data than the calculation based on the QPVR prediction.

# 4 Conclusions

By using the ESRM equations (Eqs. (1) and (2)), our actual capability to infer with reasonable accuracy any

unknown quantity among $\sigma_{n,c'}$, $P_{surr,c'}^A$, $B_{c'}^{J^\pi}$ and $F_{surr}^A$ with the remaining variables $\sigma_n^{A^*}$, $W_{n,c'}^{J^\pi}$ and $W_{surr,c'}^{J^\pi}$ 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 analysing deexcitation probabilities and neutron-induced partial cross sections is a major asset for the difficult task of evaluating the neutron-induced cross sections within 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 [13]. Using the ESRM equations ensures our capability to extract the best non-biased information from surrogate-reaction data. Reference [4] 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.

# References

[1] O. Bouland and G. Noguère, Phys. Rev. C **102**, 054608 (2020).

[2] R. Pérez Sánchez et al., Phys. Rev. Lett. **125**, 122502 (2020).

[3] P. Marini et al., EPJ Web of Conferences **122** CNR*15 12004 (2016) 12004, DOI: 10.1051/epjconf/2016122

[4] O. Bouland, Phys. Rev. C **100**, 064611 (2019).

[5] S. Hilaire, Ch. Lagrange and A. J. Koning, Annals of Physics **306**, 209 (2003).

[6] V. Semkova, N. Otuka, M. Mikhailiiukova, B. Pritychenko and O. Cabellos, EPJ Web of Conferences, 146, 07003 (2017).

[7] A. Gilbert and A. G. W. Cameron, Can. J. of Physics, **43**, 1446 (1965).

[8] G. Boutoux et al., Phys. Lett. B **712** (2012) 319.

[9] V.F. Weisskopf, D.H. Ewing, Phys. Rev. **57** (1940) 472.

[10] O. Bouland et al., Phys. Rev. C **88** (2013) 054612.

[11] I.Thompson, J.E.Escher, UCRL-TR-225984 (2006).

[12] J.E.Escher, Private communication at the ND2016 conference (2016).

[13] J.E. Escher, Rev. Mod. Phys. **84**, No. 1 (2012) 353.