# Highlighting cross section library effects on absorption rates and flux for natural elements in a thermal spectrum using TRIPOLI-5

Kévin Fröhlicher, Benjamin Dechenaux, Davide Mancusi, Raphaelle Ichou, Fausto Malvagi

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Kévin Fröhlicher, Benjamin Dechenaux, Davide Mancusi, Raphaelle Ichou, Fausto Malvagi. Highlighting cross section library effects on absorption rates and flux for natural elements in a thermal spectrum using TRIPOLI-5. SNA + MC 2024 - International Conference on Supercomputing in Nuclear Applications + Monte Carlo, Sfen, Oct 2024, Paris, France. pp.07007, 10.1051/epjconf/202430207007. irsn-04890356

HAL Id: irsn-04890356

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Submitted on 16 Jan 2025

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# Highlighting cross section library effects on absorption rates and flux for natural elements in a thermal spectrum using TRIPOLI-5

Kévin Fröhlicher<fnref n="1" />,*, Benjamin Dechenaux<fnref n="1" />, Davide Mancusi<fnref n="2" />, Raphaëlle Ichou<fnref n="1" />, and Fausto Malvagi<fnref n="1" />

**Abstract.** This work presents an investigation of cross section library effects with the new TRIPOLI-5 Monte Carlo code, a joint project between CEA, IRSN and EDF in France. Cross section library effects on the absorption rate and the flux spectrum were highlighted for natural elements subjected to a thermal neutron flux. The numerical benchmark that is presented here is a sphere with a radius of 30 cm filled with $10^{-2}$ atoms·(barn·cm)$^{-1}$ of a single element according to its natural isotopic abundances. For the thermal flux case, a smaller sphere with a radius of 1 cm is placed at the center of the main sphere to obtain a thermal-like flux from a pointwise Maxwellian source using a mixture of hydrogen and boron-10. A total of 12 cross sections libraries have been compared.

## 1 Introduction

In neutron transport simulations, two key ingredients affect the accuracy and thus the predictive power of the results: the method used to solve the transport equation and the nuclear data (especially cross sections), which are part of the equation coefficients, that are used to characterize the medium.

Many so called deterministic solvers are used in the majority of cases. However, the Monte Carlo method has a reputation of giving a "reference" solution, in the sense that no approximations are made when treating the cross sections energy dependence. On the contrary, deterministic methods rely on the multi-group approximation which introduces a bias that is impossible to quantify *a priori*.

Several nuclear data *evaluations* of the cross sections, often called *libraries*, may be used to characterize neutron-matter interaction probabilities for a given problem. These cross sections physically depend on the target nuclide (or isotope) and the incident neutron energy. Today's main libraries include ENDF/B (USA), JEFF (Europe), JENDL (Japan) and TENDL (Switzerland). Each library is updated every few years, which shows that our knowledge of nuclear data is still uncertain. The latest released versions of above mentioned libraries are ENDF/B-VIII.0 [1], JEFF-3.3 [2], JENDL-5.0 [3] and TENDL-2023 [4]. Industrial actors are

<footnote n="1">¹Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PSN-RES/SNC/LN, F-92260, Fontenay-aux-Roses, France</footnote>

<footnote n="2">²Université Paris-Saclay, CEA, Service d'Etudes des Réacteurs et de Mathématiques Appliquées, 91191, Gif-sur-Yvette, France</footnote>

<footnote>*e-mail: kevin.frohlicher@irsn.fr</footnote>
usually reluctant to change for the latest releases, given the cost of validating their simulations over their whole set of applications, and generally use older but "proven" versions of these libraries.

In this work, the focus is on the dispersion of results due to different choices of nuclear data libraries amongst the most recent ones. The same Monte Carlo code and simulation parameters were used for all calculations presented below to be free of any model discrepancies. Thus, the resulting dispersion is expected to be an *indicator* of the uncertainty associated to nuclear data, which must be taken into account for safety considerations.

The benchmark modeled in this work is a sphere composed of a single natural element (i.e., according to natural isotopic abundances), exposed to a flux representative of a thermal reactor. The TRIPOLI-5 [5] Monte-Carlo code was used for the comparisons. Despite being in the early development stages, its results have been shown to be completely consistent with OpenMC [6].

## 2 Benchmark description

The benchmark is a sphere with a radius of 30 cm, filled with a test material made of a single element with a concentration of $10^{22}$ atoms·cm$^{-3}$. This concentration is kept constant for all elements, so as to compare the absorption (or poisoning) power for different elements. For most elements this corresponds to about one collision per history in the test region.

To expose the test material to a PWR-like thermal flux, a smaller driver sphere with a radius of 1 cm is placed at the center of the test sphere. It is filled with a $^1$H and $^{10}$B mixture with a concentration of 0.8 atoms·cm$^{-3}$. The atomic fraction of $^{10}$B is 0.22%. This setup allows to go from a 1 MeV pointwise Maxwellian distribution, which is defined at the center of the geometry, to a flux spectrum similar to that of a standard PWR pincell before entering the test material. Thus, the source flux shape only differs from the spectrum of a real PWR in the absence of resonance dips due to uranium and oxygen. This is shown in figure 1.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Flux generated in the driver sphere compared to the spectrum in a typical pincell of a standard Pressurized Water Reactor (PWR).</figcaption></figure>

Absorption rates integrated over the whole test region and flux spectra in the test region are presented below for 12 nuclear data libraries that have been processed using the first version of the GAIA code [7] developed at IRSN:
* JEF-2.2 (j22 in figures),

* JEFF-3.1.1 (j311 in figures),

* JEFF-3.2 (j32 in figures),

* JEFF-3.3 (j33 in figures),

* ENDF/B-VI.8 (b68 in figures),

* ENDF/B-VII.1 (b71 in figures),

* ENDF/B-VIII.0 (b80 in figures),

* JENDL-4.0 (jdl4 in figures),

* JENDL-5.0 (jdl5 in figures),

* TENDL-19 (tdl19 in figures),

* TENDL-21 (tdl21 in figures),

* TENDL-23 (tdl23 in figures).

## 3 Numerical results

Each calculation is performed as a fixed source calculation using 100 batches of $10^4$ particles (i.e., $10^6$ independent histories). The statistical accuracy is computed as the relative standard error of the mean (SEM) for the integrated absorption rate defined as

$$ \text{relative SEM} = \frac{\sigma}{\sqrt{N}} \frac{1}{\overline{\phi \Sigma_a}} \qquad (1) $$

where $\sigma$ is the observed standard deviation between the N random walks (here 100 independent batches) and $\overline{\phi \Sigma_a}$ is the average absorption rate. The distribution of the relative SEM over all simulations (for all the libraries and elements) is displayed in Figure 2. One can see that all simulation have a relative SEM on the absorption rate below 1.5%.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Distribution of the relative standard error of the mean for the integrated absorption rate in the test volume for all simulations</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3. Absorption rates of elements showing a dispersion (over all libraries) lower than 2%.</figcaption></figure>

## 3.1 Absorption rates

The dispersion of results amongst the different libraries is computed as

$$dispersion = \frac{1}{2} \times \frac{(max - min)}{average} \qquad (2)$$

where *min*, *max* and *average* are the minimum, maximum and average absorption rates observed amongst the libraries.

Figures 3, 4 and 5 present the absorption reaction rates in the test region for each library in the top plot, and the dispersion of those rates (as defined above) in the bottom plot. The dispersion is shown for two sets of nuclear data libraries, one gathering all the above mentioned libraries (named *all libs*) and another one only composed of the most recent libraries, namely, ENDF/B-VIII.0, JEFF-3.3, JENDL-5, TENDL-23 (named *ref libs* in the rest of the article). Only elements for which the absorption rate if above 0.01 absorption per source particle are shown.

One can observe on Figure 3 that there seems to be a consensus regarding the absorption rates of most of the element composing steels that can be found in a reactor: iron (Fe), manganese (Mn), titanium (Ti), cobalt (Co), with less than 1% of dispersion between different nuclear data evaluations. The common neutron absorbers used in reactors also appear in the same figure with relatively low dispersion: boron (B), silver (Ag), indium (In), gadolinium (Gd), hafnium (Hf) with less than 1% dispersion and cadmium (Cd) with about 1.5%. A few elements found in structures are showing higher dispersions (higher than 2 %) even for the most recent evaluations (cf. chromium (Cr), nickel (Ni), copper (Cu) and molybdenum (Mo) in Figure 4).

Figure 5 displays elements for which the dispersion is higher than 5% (even for recent libraries). As one can see, all these elements are not really relevant in the PWR case since their isotopes are not present in natural abundances in nuclear reactors. Other elements such as carbon (C), zirconium (Zr) or nobium (Nb) are not shown since their absorption rate is too low to be relevant here.
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Absorption rates of element showing a dispersion (over all libraries) between 2% and 5%.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Absorption rates of elements showing a dispersion (over all libraries) higher than 5%.</figcaption></figure>

## 3.2 Flux spectrum

The test region flux spectra are also physical quantities of interest. The said flux spectra are given for a few major absorbers used in nuclear reactors, namely silver, indium, cadmium, gadolinium and hafnium in Figures 6, 7, 8, 9 and 10, respectively. The flux spectra obtained for the different libraries under investigation are plotted as plain lines for JEF-2.2, JEFF-3.1.1, JEFF-3.2, ENDF/B-VI.8, ENDF/B-VII.1, JENDL-4.0, TENDL-19, and TENDL-21 on the top plots of the figures. The maximum variation (i.e., $\phi_{min}(E) - \phi_{max}(E)$) envelop amongst the reference libraries (ENDF/B-VIII.0, JEFF-3.3, JENDL-5, TENDL-23) is also displayed, in the form of the gray area in each figure. The bottom plot shows the standard deviation of
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Flux spectrum in the test region for gadolinium. The gray area in the top plot covers the min and max for the libraries chosen as reference. The natural abundances of isotopes in the mixture are also given.</figcaption></figure>

the distribution between libraries for two cases: when considering all 12 libraries (*all libs*) or when considering only the latest libraries (*ref libs*: ENDF/B-VIII.0, JEFF-3.3, JENDL-5, TENDL-23).

These figures show that the standard deviation of the flux between libraries can amount to several tens of percent (e.g., almost 40% for gadolinium on Figure 6, 30% for hafnium on Figure 7, 20% for silver and indium on Figures 8 and 9) in some regions of the spectrum, even for most recent libraries.

For the above mentioned elements, the higher differences can be observed in the beginning of the unresolved resonance range (URR). This is particularly obvious for indium on Figure 9, for which TENDL-19, TENDL-21 and JEFF-2.2 have no URR data for $^{115}$In (amounting for about 95% of the natural indium composition, and whose URR starts around 1.5-1.8 keV for the other libraries).
<figure><img src="image_8.png" /><figcaption>**Figure 7.** Flux spectrum in the test region for hafnium. The gray area in the top plot covers the min and max for the libraries chosen as reference. The natural abundances of isotopes in the mixture are also given.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 8.** Flux spectrum in the test region for silver. The gray area in the top plot covers the min and max for the libraries chosen as reference. The natural abundances of isotopes in the mixture are also given.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Figure 9.** Flux spectrum in the test region for indium. The gray area in the top plot covers the min and max for the libraries chosen as reference. The natural abundances of isotopes in the mixture are also given.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 10.** Flux spectrum in the test region for cadmium. The gray area in the top plot covers the min and max for the libraries chosen as reference. The natural abundances of isotopes in the mixture are also given.</figcaption></figure>
# 4 Conclusions and future work

A study of library dispersion for absorption rates has been carried out for all natural elements submitted to a driving neutron thermal flux very similar to the one in a UOX pincell of a standard PWR for 12 nuclear data libraries. This work is part of the testing of the TRIPOLI-5 code, and aims at roughly quantifying the library dispersion for all the major recent releases of evaluated libraries (ENDF/B-VIII.0, JEFF-3.3, JENDL-5.0, TENDL-23) as well as older versions of those libraries (ENDF/B-VII.1, ENDF/B-VI.8, JEFF-3.2, JEFF-3.1.1, JEF-2.2, TENDL-21, TENDL-19). The above results show an overall satisfying agreement between nuclear data libraries regarding the absorption rate for major elements (with an isotopic composition based on natural abundances) found in nuclear reactors.

Nonetheless, the analysis carried out in this work is not complete, as it does not sufficiently highlight all effects of nuclear data libraries on the behavior of certain important elements used in a reactors. This is either due to the fact that absorption is not the most important reaction when characterizing the behavior of these elements (e.g., one could look at reflective properties for iron), or that the thermal spectrum is not relevant (e.g., the cases of chlorine, fluorine, sodium and lead, which are used in fast reactors). Therefore, this study will be extended to other parameters including different source spectra and different observables (e.g., transmission rates, albedo). In addition, the test materials will be diversified by including isotopic compositions taken from spent fuel burnup calculations (instead of natural abundances) and actinides.

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