New 23Na evaluation in the resolved resonance range taking into account both differential and double differential experiments

In 2012 CEA produced a entire new evaluation of sodium nuclear data for the release of the JEFF3.2 evaluated nuclear data library. During the evaluation process performed with the CONRAD code, several differential measurements (total and discrete inelastic cross-sections) have been used. However double differential data (elastic angular distribution) that were yet available in the EXFOR database were not incorporated in the analysis at that time. The experimental elastic angular distribution were discarded because of it was impossible to obtain a good agreement for both angle-integrated cross-sections and double differential ones. The underlying cause of this disagreement is expected to be due to the attribution of quantum numbers to resonance and related channel amplitudes. Indeed these numbers are imposed during the analysis but impact differently angular distributions and angle-integrated cross-sections. An automated search for an accurate set of quantum numbers has been implemented in order to produce a reliable quantum numbers set. In this paper we present a new evaluation of Na-23 taking into account both differential and double differential measurements. The analysis performed with the CONRAD code reached the level of agreement with experimental data for the total and inelastic cross-sections but this time with a significant improvement for the elastic angular distributions. This new evaluation produced in the ENDF-6 format has then been tested and validated on critical facilities calculation (MASURCA and ZPPR) in different configurations (nominal and voided) in order to assess its performances.


Introduction
In 2012, the nuclear data team at CEA-Cadarache produced an 23 Na evaluation for the release of the JEFF-3.2 library [1]. This new evaluation was based on several angle-integrated differential (energy-dependent) experiments (for the total and discrete inelastic channels), however double differential (energy-and angle-dependent) experiments such as Kinneys angular elastic cross-section measurements [2] were not taken into account at that time. The main difficulty was to achieve a good agreement on both angle-integrated cross-sections and double differential ones. We present here investigation work on the 23 Na evaluated data. This paper focuses on the resonance energy range, where the impact of the resonance quantum numbers on the angular differential data has been studied. A new set of resonance parameters have been produced and tested on a set of sodium void benchmarks that are particularly sensitive to 23 Na nuclear data. We also shows that all experimental conditions must be included in the analysis, namely the Doppler-broadening and the resolution function. Additionally we attempted to change the usual boundary conditions convention from B c = S c to B c = − as recommended in Ref. [3], S c being the usual energy-dependent shift-factor of the R-matrix theory [4]. * e-mail: pierre.tamagno@cea.fr

Status of the JEFF-3.23 Na evaluation
The latest evaluation of 23 Na relied on a set of experimental data, for the resonance resonance range the following data were considered 1. Larson 1976 total cross section measurement [5] which We note that other good-resolution data are also available, namely the angular elastic cross section [8] from Kopecky (1997) but that are believed to require a special angular-dependent multiple scattering correction, which is currently unavailable in the CONRAD code [9] used for the present work. The major identified defect of the JEFF-3.2 23 Na evaluation lays in elastic scattering angular distribution. Indeed although some experimental data were available in 2012, elastic scattering angular cross section measured by Kinney [2] in 1976 were not incorporated in the analysis. This lack of experimental constraints could lead to a misassignment of the resonance quantum numbers J π , , s. Therefore even though angle-integrated cross section were in a reasonable agreement with experimental data, the angular-differential cross sections were incorrect, especially in the backward direction. In the present work we have included the Kinney data and made an automatic search for the resonance quantum numbers. This procedure improved strongly the agreement with angular distributions (see Fig. 1) while keeping a good agreement with integrated data. We can see in Fig. 1 that the agreement is still not perfect meaning that either a deeper investigation should be carried out (for instance allowing several neutron channels for each resonance), or that the model may be improved as explained in Sec 4.   Now that we expect a better agreement between experimental data and angular cross section obtained for the new set of resonance parameters, we produced and processed a new ENDF-6 file mostly based on JEFF-3.2 but in which the resonance parameters have been replaced as well as the angular distribution. The former angular distribution present in JEFF-3.2 were produced using the TALYS reaction code [10] that uses statistical models. Therefore the resonant structure observed in the resolved resonance range in the angular distribution were not present in the JEFF-3.2 evaluation, as shown in Fig. 2. With our new file the angular distribution exhibit resonant structures consistent with the model used for the modeling of angleintegrated cross sections. This consistency was already present in the ENDF/B-VII 23 Na evaluation but was one of the defect of the JEFF-3.2 one.

Benchmarking the evaluation
Having obtained a new set of resonance parameters, the new evaluated data have been tested on a set of integral benchmarks known to be highly sensitive to 23 Na data. These are the ZPPR-10A sodium void effect described in the IRPhEP database [11] (see Fig. 4) and the Racine and Pre-Racine (see Fig. 3) sodium void effect measurements performed at the MASURCA facility at CEA/Cadarache [12,13].
The C/E results for these benchmarks are shown in Tab. 1, it can be observed that the new evaluation of the sodium resonances improve significantly the global trend of JEFF-3.2 that overestimates the void effect.

Remaining open issues
While working on the Larson data we realized that some small resonances are impacted by the energy resolution and Doppler-broadening. These experimental conditions were not considered in the previous work that led to JEFF-3.2. Indeed for light isotopes with large resonant structure one expect that the Doppler-broadening has a limited impact. Additionally, when working with high-resolution experimental data, typically obtained with a long flight-path one also expect that their is no need to use a resolution  10 11 12 13 14 15 16 17 18 18 17 16 15 14 13 [12,13] taken from Ref. [14].

A.2. ZPPR-10A 199
Pour la géométrie RZ, on pourra noter que le coeur interne est composé (par   . XY view of the ZPPR-10A core [11] taken from Ref. [14]. function to analyze the data. We tested these hypotheses on the Larson data with a 300 K Doppler broadening and for which the reported [5] energy resolution is It was found that these experimental conditions impact the narrow resonances as illustrated in Fig. 5 where the Doppler broadening (Dop.) the the energy resolution function (RF) are successively added. In the JEFF-3.2, the ENDF convention for the choice of boundary condition parameter [4] B c = S c (E) was applied. However this choice is justified mainly for neutroninduced reactions on actinides for which = 0 implies  B c = 0. Indeed in a rigorous implementation of the Rmatrix theory, the boundary condition B c must be energyindependent, ensuring that the many-body nucleus wave function can be properly expanded on eigen functions. In the case of 23 Na, many resonance are attributed quantum numbers with > 0, therefore the usual ENDF conven-tion is not justified. To investigate the possible impact of this B c = S c (E) we restarted "from scratch" a new resonance range analysis starting from 0 eV to 700 keV with the boundary condition choice B c = − . With this new boundary condition choice, the eigen energies E λ do not correspond anymore to the observed resonance energies. Finding good initial guess of the resonance parameters becomes very tedious therefore we limit the analysis below 700 keV. The first results of this attempt are shown in Tab. 2 where the χ 2 for all listed experiments are given. The results obtained with the new boundary choice B c = − seem better on average than those obtained with either the JEFF-3.2 or the new resonance parameters set presented here. In particular it seems to improve even further the angular distribution at forward angles. It must be kept in mind that these results are not directly comparable as for obtaining the resonance parameters for JEFF-3.2 or for the new evaluation, the inelastic cross section measurement of Rouki [7] was also included whereas it was not for the set with B c = − . In future we plan to extend this work by using the alternative Brune parameterization [15] which allows to work back with eigen energy parameters that matches the resonance energies but with a consistent boundary condition choice. This parameterization would allow to extend our analysis range and confirm whether improvements are truly expected for angular distributions.

Conclusion
We showed here that a new evaluation of the 23 Na resonance parameter considering differential and angularintegrated experimental cross sections can lead to a significant improvement of the C/E agreement in integral void effect benchmarks. However in our analysis we saw that some remaining issues still need to be addressed for instance the use of a more physical boundary condition B c S c (E) or the inclusion of the experimental conditions such as the Doppler-broadening and the consideration of the energy-resolution of the experimental data. More comprehensive analysis work is planned in near future.