Generation of particle self-shielded neutron
cross-sections for the Monte-Carlo code TRIPOLI-4®

Gilles Noguere, Pierre Leconte, David Bernard, Mathilde Pottier, Carlos

Paradela, Stefan Kopecky, Peter Schillebeeckx

► To cite this version:

Gilles Noguere, Pierre Leconte, David Bernard, Mathilde Pottier, Carlos Paradela, et al. Generation of particle self-shielded neutron cross-sections for the Monte-Carlo code TRIPOLI-4®. EPJ N - Nuclear Sciences & Technologies, 2024, 10, pp.18. 10.1051/epjn/2024019. hal-04810744

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# Generation of particle self-shielded neutron cross-sections for the Monte-Carlo code TRIPOLI-4®

Gilles Noguere<fnref n="1" />,*, Pierre Leconte<fnref n="1" />, David Bernard<fnref n="1" />, Mathilde Pottier<fnref n="1" />, Carlos Paradela<fnref n="2" />, Stefan Kopecky<fnref n="2" /> and Peter Schillebeeckx<fnref n="2" />

Received: 9 June 2024 / Received in final form: 8 September 2024 / Accepted: 16 October 2024

**Abstract.** Self-shielding effects of micrometric particles randomly distributed in nuclear materials were the subject of extensive studies. These effects are known as a double-heterogeneity problem due to “microscopic” heterogeneities, involving the decrease of the neutron flux inside the particles, and “macroscopic” heterogeneities, affecting the neutron flux distribution over the entire volume containing those particles. The present study aims to take advantage of the capabilities of the GELINA facility (JRC-Geel, Belgium) in terms of non-destructive analysis of materials to validate experimentally any models developed to solve the double heterogeneity problem. In order to complement past experiments carried out at the GELINA facility, new transmission experiments were carried out on long cylindrical samples containing microspheres of Gd₂O₃ with diameters of 195 and 380 µm dispersed in UO₂ pellets. The analysis of the experimental transmission spectra with the resonance shape analysis code REFIT and Monte-Carlo neutron transport code TRIPOLI-4® demonstrates that the particle self-shielding model proposed by Doub can reproduce the huge attenuation of the neutron absorption in the Gd resonances by reducing significantly the computational cost of Monte-Carlo simulations.

## 1 Introduction

The double-heterogeneity problem due to the random distribution of micrometric particles in nuclear fuel arose in the early days of the nuclear industry. Models must handle “microscopic” heterogeneities, involving the decrease of the neutron flux inside the particles, and “macroscopic” heterogeneities, affecting the neutron flux distribution over the entire nuclear fuel pellet containing those particles. Several studies propose analytic and stochastic solutions for deterministic [1–6] and Monte Carlo [7–9] codes. They are of particular interest for plate fuels with dispersed particles [10] or TRISO fuel particle concepts loaded in high-temperature reactors [11].

The present work focuses on the particle self-shielding effects associated to gadolinium, which is used as burnable poison. A wide experimental campaign was carried out in the zero-power reactor MINERVE of CEA Cadarache from 1979 to 1983 to measure by the oscillation technique the reactivity worth of various samples containing gadolinium oxide Gd₂O₃ [12]. A series of oscillations was designed to measure enriched UO₂ pellets containing 3% of Gd₂O₃ dispersed in the form of microspheres with diameters of 80, 120, 195, and

380 µm. Experimental results reported in Figure 1 illustrate the sizeable decrease of the reactivity worth as a function of the diameter of the Gd₂O₃ grains. For a diameter of 380 µm, the measured reactivity worth represents 65% of the nominal value. In practice, calculating these experimental values with a Monte Carlo neutron transport code by randomly generating Gd₂O₃ microspheres is time consuming or even impossible in the case of diameters of 80 µm. The generation of particle self-shielded neutron cross-sections can be seen as a suitable solution for reducing the computational cost of Monte Carlo simulations.

Our work aims to complement past studies focused on the generation of particle self-shielded neutron cross-sections for Monte Carlo codes [14,15]. We followed the strategy reported by Becker et al. [16]. The authors used the Markovian Levermore-Pomraning model [17] to analyze the neutron transmission of samples made by mixing tungsten and sulphur powder with nominal grain sizes ranging from 50 to 250 µm. The experiments were carried out at the GELINA facility (JRC-Geel, Belgique). Here, we decided to explore the performances of another analytic model which was proposed by Doub [18]. This model is well adapted to the samples used in the MINERVE reactor. Indeed, it relies on the assumption of spherical particles having all the same radius. In this model, the volume

<footnote n="1">¹ CEA, DES, IRESNE, DER, Cadarache, F-13108 Saint Paul Les Durance, France</footnote>

<footnote n="2">² EC-JRC-Geel, B-2440 Geel, Belgium</footnote>

<footnote>* e-mail: gilles.noguere@cea.fr</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Experimental reactivity worth of UO₂ pellets containing 3% of Gd₂O₃ dispersed in the form of microspheres with diameters of 80, 120, 190 and 380 µm [13]. Measurements were carried out at the zero-power reactor MINERVE of CEA Cadarache by the oscillation technique [12]. The experimental values Δρᵢ are normalized to the nominal value Δρ₀ measured in the case of UO₂ pellets containing 3% of homogeneously mixed Gd₂O₃ powder.</figcaption></figure>

fraction occupied by the microspheres depends explicitly of their number and volume. By contrast, the Levermore-Pomraning is a more generic model in which the line segments in a particular component along a trajectory of a particle have an exponential decaying length probability distribution. Consequently, the volume fraction is expressed in terms of mean chord length. The transmission experiments performed in the framework of this study were also carried out at GELINA using the enriched UO₂ samples containing microspheres of Gd₂O₃ which were designed for the MINERVE experiments carried out in 1982. The experimental transmission was analyzed with the resonance shape analysis code REFIT [19] and the Monte Carlo code TRIPOLI-4® [20]. For that purpose, The particle self-shielded neutron cross-sections for the gadolinium isotopes were generated with the processing tool CADTUI [21] and converted in pointwise evaluated nuclear data file (PENDF).

Governing equations involved in the particle self-shielding formalism proposed by Doub are presented in Section 2. The sample characteristics and the neutron transmission experiments are described in Sections 3 and 4. Results and discussions related to the REFIT and TRIPOLI-4® analysis are reported in Sections 5 and 6, respectively.

## 2 Particle self-shielding formalism of Doub

The particle self-shielded neutron cross-section $\bar{\sigma}_{x,i}(E)$ for a reaction $x$ and isotope $i$ is defined as the product of the particle self-shielding factor $f(E)$ times the Doppler

broadened cross-section $\sigma_{x,i}(E)$:

$$ \bar{\sigma}_{x,i}(E) = f(E)\sigma_{x,i}(E). \quad (1) $$

The factor $f(E)$ can take different forms depending on the assumptions used to derive suitable analytic formulation of the double-heterogeneity problem. One of the first model was proposed by Doub [18] and validated against transmission measurements performed on a sample containing boron-carbide and aluminum microspheres, with diameters ranging from 53 to 87 µm.

The self-shielding factor $f(E)$ obtained by Doub is expressed as follows:

$$ f(E) = \frac{1}{\frac{2}{3}y(E)\left(\frac{V}{g}\right)} \ln\left(\frac{1}{1-\left(\frac{V}{g}\right)\left(1-\bar{t}(E)\right)}\right), \quad (2) $$

in which $V$ is a volume fraction occupied by the microspheres weighted by a packing factor $g$. The collision probability $\bar{t}(E)$ is derived from expressions given by Case et al. [22]:

$$ \bar{t}(E) = \frac{2}{y(E)^2} \left(1 - (1 + y(E))e^{-y(E)}\right), \quad (3) $$

in which the intermediate parameter $y(E)$ depends on the volume density $\rho_i$ and radius $r_k$ of the microspheres $k$ containing isotope $i$:

$$ y(E) = 2r_k\rho_i\sigma_{x,i}(E). \quad (4) $$

In this work, the volume fraction $V$ is defined as follows:

$$ V = \frac{N_k V_k}{V_p}, \quad (5) $$

in which $N_k$ stands for the number of microspheres of volume $V_k$ in a cylindrical sintered UO₂ pellet of volume $V_p$. The volume density $\rho_i$ is then given by:

$$ \rho_i = \frac{Nm_i}{M_iV_p}, \quad (6) $$

with $N$ the Avogadro number, $m_i$ the total mass of isotope $i$ in the pellet, and $M_i$ its atomic weight. This expression means that the total mass of isotope $i$ is diluted over the full volume of the pellet.

In equation (2), the packing factor $g$ is defined by Doub as the “fraction of the geometrical sample volume which is occupied in a sample containing all microspheres”. Doub recommends using the theoretical $g$ value obtained for perfectly packed microspheres which is equal to $g = 0.74$. This recommendation can be a subject of debate, especially in the case of the MINERVE samples used in this work which contain microspheres of Gd₂O₃ dispersed in a UO₂ matrix. However, according to the work reported in reference [23], we can assume that the Gd₂O₃ spheres and UO₂ grains of a few micrometers form a compact arrangement as illustrated in Figure 2. This assumption will be verified in Section 5 by comparing the experimental transmissions measured at the GELINA facility with their theoretical analogs.
<figure><img src="image_3.png" /><figcaption>Fig. 2. Schematic representation of the interface between a Gd₂O₃ microsphere of 300 µm diameter and a UO₂ matrix [23]. The first phase, probably UGd₆O₁₁, exhibits gain size of about 4 µm. The second phase, consists of a solid solution of (U₁-x,Gdₓ)O₂-y, corresponds to a diffusion zone of thickness close to 20 µm.</figcaption></figure>

## 3 Sample characteristics

For the experimental validation of the model proposed by Doub, we used existing samples that were measured in the MINERVE reactor of CEA Cadarache [12]. A wide experimental program was designed to measure the reactivity worth of enriched UO₂ pellets containing different amount of gadolinium oxide (Gd₂O₃). Natural gadolinium consists of the isotopes 152, 154, 155, 156, 157, 158, and 160, with abundances of 0.2%, 2.18%, 14.8%, 20.47%, 15.65%, 24.84% and 21.86%. The two most important isotopes for nuclear reactor applications are ¹⁵⁵Gd and ¹⁵⁷Gd [24]. In the JEFF-3.3 library, their thermal capture cross-sections reach 60732 barns and 253251 barns, respectively [25].

Our study only focuses on a series of experiments carried out in 1982 with samples containing microspheres of Gd₂O₃ of various diameters, ranging from 80 to 380 µm. These samples were composed of a stack of four UO₂ pellets sealed in a zircaloy tube. In this work, we report results obtained for two samples containing microspheres of diameter equal to 195(10) and 380(19) µm. These samples are labeled A4 and A5 throughout the text. The Micrographs shown in Figure 3 illustrates the random dispersion of the microspheres in each UO₂ pellet.

A third UO₂ sample containing homogenously mixed Gd₂O₃ powder with grain size of 15(10) µm was used as reference. This statement is motivated by the results reported in Section 5 which demonstrate the weak particle self-shielding effect for this sample, allowing to neglect the Gd₂O₃ grain size in the calculations. This sample is labeled A1.

The characteristics of the samples are listed in Table 1. The pellet density and diameter reported for A1, A4, and A5 are mean values averaged from the analysis of 80, 5,

and 50 pellets of each type, respectively. The pellets are “diabolo shaped”, therefore the pellet height represents an average value deduced from the pellet volume. The ²³⁵U enrichment is well documented for samples A1 and A5. For sample A4, we used the specified enrichment of the raw UO₂ powder close to 5.1%. For the Gd₂O₃ content, a measured value of 3.08% is reported for sample A1 without uncertainty. For sample A4, we used the specified Gd₂O₃ content of 3%. For sample A5, the Gd₂O₃ content was slightly adjusted in order to calculate 814 Gd₂O₃ microspheres in each pellet, as specified by the manufacturer, using the following expression:

$$N_k = \frac{\rho_{Gd_2O_3} V_p}{\rho_{th} V_k}, \quad (7)$$

in which $V_p$ is the volume of the pellet, $V_k$ is the volume of the microsphere, $\rho_{Gd_2O_3}$ represents the density of Gd₂O₃ in the pellet and $\rho_{th}$ stands for the apparent density of the microspheres. For this calculation, the apparent density of the microspheres is taken equal to the Gd₂O₃ theoretical density which is close to $\rho_{th} = 7.41 \text{ g/cm}^3$, assuming negligible the porosity of the microspheres after calcination and sintering processes. The number of microspheres for samples A1 and A4 was calculated using the same theoretical density $\rho_{th}$.

## 4 Neutron transmission experiments

The neutron transmission of the MINERVE samples listed in Table 1 was measured at the GELINA facility. A schematic representation of the transmission setup is shown in Figure 4. Neutrons are detected by a lithium glass detector which is located at a distance $L = 10.861(2) \text{ m}$ from the neutron source. Figure 5 shows how the MINERVE sample is placed in the neutron beam. Due to the sample geometry and beam collimation, part of the incident neutrons reaches the detector without crossing the sample. This fraction of the incident neutron beam is denoted void fraction $\alpha$.

A formalism was developed to account for the geometry of such a MINERVE sample. The experimental transmission $T_{exp}$ as a function of time $t$ can be deduced from the sample in and sample out measurements as follows [26]:

$$T_{\text{exp}}(t) = \frac{[C_{\text{in}}(t) - B_{\text{in}}(t)] - \alpha[C_{\text{out}}(t) - B_{\text{out}}(t)]}{(1 - \alpha)[C_{\text{out}}(t) - B_{\text{out}}(t)]}, \quad (8)$$

in which $C$ represents the count rate normalized to the beam intensity and $B$ is the background contribution. This equation was used in the data reduction procedure through the AGS code [27]. The AGS code was designed to handle all steps needed to convert raw data in transmission $T_{exp}$, such as dead time correction, background subtraction, and normalization. These steps are well documented in the literature [28]. For the background contribution, parameters of exponential functions were determined via the black resonance technique. These background parameters were optimized by fitting the bottom
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Micrographs of two UO₂ pellets containing Gd₂O₃ microspheres of diameter equal to 195 µm (sample A4) and 380 µm (sample A5).</figcaption></figure>

<table><caption>**Table 1.** Characteristics of the MINERVE samples A1, A4, and A5. Each sample is composed of a stack of four UO₂ pellets sealed in a zircaloy tube containing a nearly similar amount of Gd₂O₃, close to 3%.</caption><thead><tr><th>Sample characteristics</th><th>A1</th><th>A4</th><th>A5</th></tr></thead><tbody><tr><td>Matrix</td><td>UO<sub>2</sub></td><td>UO<sub>2</sub></td><td>UO<sub>2</sub></td></tr><tr><td>Number of pellets</td><td>4</td><td>4</td><td>4</td></tr><tr><td>Pellet density<br>(g/cm<sup>3</sup>)</td><td>10.30(3)</td><td>10.27(1)</td><td>10.32(2)</td></tr><tr><td>Pellet diameter<br>(mm)</td><td>8.193(4)</td><td>8.193(1)</td><td>8.191(1)</td></tr><tr><td>Average pellet height<br>(mm)</td><td>10.94(3)</td><td>10.55(5)</td><td>10.57(15)</td></tr><tr><td><sup>235</sup>U content<br>(%)</td><td>5.06(1)</td><td>5.1</td><td>4.95(1)</td></tr><tr><td>Gd<sub>2</sub>O<sub>3</sub> content<br>(%)</td><td>3.08</td><td>3.0</td><td>3.015</td></tr><tr><td>Gd<sub>2</sub>O<sub>3</sub> density<br>(g/cm<sup>3</sup>)</td><td>0.317</td><td>0.308</td><td>0.311</td></tr><tr><td>Microsphere diameter<br>(µm)</td><td>15(10)</td><td>195(10)</td><td>380(19)</td></tr><tr><td>Number of microsphere in each pellet</td><td>13962454</td><td>5955</td><td>814</td></tr><tr><td>Number of microsphere in the sample</td><td>55849816</td><td>23820</td><td>3256</td></tr></tbody></table>

of cobalt and sodium black resonances located at 132 eV and 2.8 keV, respectively. As explained in reference [26], the void fraction α was deduced from the bottom of the <sup>238</sup>U black resonances at 6.7, 20.9, and 36.7 eV. We obtained consistent void fractions of 0.050(4), 0.048(4), and 0.054(4) for samples A1, A4, and A5, respectively. The dispersion of the α values are mainly connected to the uncertainty associated to the background contribution. For example, in the case of sample A5, the void fraction α is close to 0.052, 0.054, and 0.057 at the bottom of the first, second and third <sup>238</sup>U black resonances. Such an

energy dependence of α indicates bias in the extrapolation of the background shapes down to low neutron energies.

Transmission T<sub>exp</sub> of the MINERVE samples A1, A4, and A5 are reported in Figure 6b up to 200 eV. Data are dominated by the s-wave neutron resonances of <sup>238</sup>U. In order to better identify Gd resonances, the UO₂ contribution was removed in Figure 6c as follows:

$$T_{\text{exp}}^{\text{Gd}}(t) = \frac{T_{\text{exp}}(t)}{T_{\text{th}}^{\text{O}}(t)T_{\text{th}}^{\text{U}}(t)}. \qquad (9)$$
<figure><img src="image_5.png" /><figcaption>Fig. 4. Schematic representation of the transmission set-up at flight path 13 of the GELINA facility [26]. The lithium glass detector is located at L = 10.861(2) m from the neutron source.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5. The left-hand plot (a) shows the MINERVE sample placed vertically in the neutron beam. The middle plot (b) is a picture of the MINERVE sample in the neutron beam. The right-hand plot (c) indicates the limit of the pellet, zircaloy tube, and neutron beam. The collimation was defined to shape a neutron beam diameter of 10 mm.</figcaption></figure>

The theoretical transmission $T_{th}^O$ and $T_{th}^U$ of oxygen and uranium were calculated using the total cross-sections retrieved from the evaluated nuclear data library JEFF-3.3 [25]. The comparison of Figures 6a and 6c indicates that the contributions of $^{152}$Gd, $^{154}$Gd, and $^{160}$Gd are negligible. The impact of the particle self-shielding effect is well observed on four resonances of $^{155}$Gd, $^{157}$Gd, $^{158}$Gd, and $^{156}$Gd located at 2.57, 16.79, 22.31, and 33.15 eV. Figure 7 confirms that increase the diameter of the Gd$_2$O$_3$ microspheres up to 380 µm contributes to decrease the absorption. This is fully consistent with the decrease in the reactivity worth observed in Figure 1. The analysis of these data with the neutron resonance analysis code REFIT and neutron transport Monte Carlo code TRIPOLI-4® are reported in Sections 5 and 6.

## 5 Neutron resonance shape analysis

The resonance shape analysis of the transmission spectra measured at the GELINA facility was performed with the REFIT code [19], in which we introduced the particle self-shielding model proposed by Doub. The REFIT

code is well adapted for interpreting non-destructive analysis of materials using the R-Matrix formalism [29,30]. In this work, we have used the neutron resonance parameters coming from the JEFF-3.3 library [25].

Due to the atypical geometry of the MINERVE samples, the theoretical transmission $T_{th}$ as a function of the incident neutron energy $E$ is given by [26]:

$$T_{th}(E) = \frac{\int_{-r}^{r} \sqrt{1 - \left(\frac{x}{R}\right)^2} \exp \left( -2r \sqrt{1 - \left(\frac{x}{R}\right)^2} \sum_i \rho_i \bar{\sigma}_{t,i}(E) \right) dx}{\int_{-r}^{r} \sqrt{1 - \left(\frac{x}{R}\right)^2} dx}, \quad (10)$$

in which $r$ is the sample radius, $R$ is the beam radius, $\rho_i$ defines the volume density and $\bar{\sigma}_{t,i}$ stands for the particle self-shielded neutron total cross-section of isotope $i$. Equation (10) is a generic expression in which the particle self-shielding correction was set to unity for uranium nuclei and calculated with equation (2) for the gadolinium nuclei. Table 2 reports the volume densities calculated from the sample characteristics listed in Table 1.

The beam radius is an important quantity whose value cannot be measured accurately due to the umbra and penumbra regions produced by the collimation of the
<figure><img src="image_7.png" /><figcaption>Fig. 6. Comparison of the Gd neutron cross-sections with the experimental transmission spectra measured at the GELINA facility. The top plot (a) represents the total cross-sections retrieved from the evaluated neutron library JEFF-3.3 [25] times the natural abundance given in parenthesis. The middle plot (b) shows the transmission $T_{\text{exp}}$ of the MINERVE samples (Eq. (8)). The bottom plot (c) shows the transmission $T_{\text{exp}}^{\text{Gd}}$ without the contribution of the $\text{UO}_2$ matrix (Eq. (9)).</figcaption></figure>

incident neutron beam (Fig. 5). In this work, an effective beam radius was deduced from the void fraction measured thanks to the bottom of the $^{238}$U black resonances. The geometrical relationship between the void fraction $\alpha$, the sample radius $r$ and the effective beam radius $R$ is given by [26]:

$$ \alpha = 1 - \frac{2R^2 \arcsin(r/R) + 2r\sqrt{R^2 - r^2}}{\pi R^2}. \quad (11) $$

The void fraction $\alpha$ is an input parameter that allows REFIT to calculate an effective beam diameter according to the sample diameter. Due to the dispersion of the $\alpha$ values (Sect. 4), we also obtained a dispersion of the effective beam diameter that ranges from 9.33(7), 9.29(7), and 9.40(7) mm for samples A1, A4 and A5, respectively. These values are consistent with the neutron beam collimation of 10 mm.

Figure 8 shows REFIT results obtained for sample A1 which is composed of an homogeneous mixture of $\text{UO}_2$ and $\text{Gd}_2\text{O}_3$ powders. Grains of $\text{Gd}_2\text{O}_3$ with 15 µm diameter (see Tab. 1) were introduced in the Doub's model. The nearly flat residual indicates that the volume densities reported in Table 2 provide a pretty good description of the uranium and gadolinium resonances. Figure 9 shows the local discrepancies observed in the vicinity of the broad $^{238}$U resonances mainly due to the spatial distribution of the neutron beam which is not a uniform parallel neutron beam. Such an issue is discussed in Section 6.

Figure 10 highlights the weak particle self-shielding effect in sample A1 for the four Gd resonances located at 2.57, 16.79, 22.31 and 33.15 eV. The differences between the REFIT calculations performed with (green line) and without (red line) $\text{Gd}_2\text{O}_3$ grains are quite difficult to distinguish, because the red and green lines are nearly on top of each other. Even if some slight differences can be observed in the dips located at the Gd resonance energies, these results confirm that the grain size is sufficiently small compared to the neutron mean free path in the powder grains in order to assume that sample A1 is homogeneous. Therefore, sample A1 can be used as a reference by neglecting the $\text{Gd}_2\text{O}_3$ grain size in the calculations. In this case, the particle self-shielding correction is set to unity for sample A1.

Comparisons of the experimental and theoretical transmission spectra for samples A4 and A5 are shown in Figure 11 for the four Gd resonances located at 2.57, 16.79, 22.31, and 33.15 eV. In REFIT, the particle self-shielding corrections were calculated by introducing the number of microspheres contained in the sample volume intercepted by the neutron beam. Given the beam diameter established from the void fraction $\alpha$, we used 5242 microspheres with a diameter of 195 µm for samples A4 and 724 microspheres with a diameter of 380 µm for sample A5. The agreement with the data demonstrates the good performance of the model proposed by Doub. This analytical model is undeniably suitable for such types of inhomogeneous pellets containing $\text{Gd}_2\text{O}_3$ microspheres.

## 6 Monte Carlo analysis

Results presented in Section 5 confirm that the analytical expression (2) of the particle self-shielding correction can reproduce the neutron absorption in micrometric particles. Connecting this model to neutron transport codes, such as the Monte Carlo code TRIPOLI-4® [20], could be useful for handling complex geometry of large volume and reducing the computational cost of Monte Carlo simulations.
<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Experimental transmission $T_{\text{exp}}^{\text{Gd}}$ of the MINERVE samples A1, A4 and A5 around Gd resonances located at 2.57, 16.79, 22.31, and 33.15 eV. The sample characteristics are reported in Table 1. Sample A1 is composed of a homogenous mixture of $\text{UO}_2$ and $\text{Gd}_2\text{O}_3$ powders. Samples A4 and A5 contain $\text{Gd}_2\text{O}_3$ microspheres of diameter equal to 195 and 380 µm, respectively.</figcaption></figure>

<table><caption>**Table 2.** Volume density $\rho_i$ (in at.10$^{24}$/cm$^3$) for the main isotopes $i$, which were used in the REFIT code.</caption><thead><tr><th>Isotope name</th><th>Sample A1</th><th>Sample A4</th><th>Sample A5</th></tr></thead><tbody><tr><td>U234</td><td>9.0456260E-06</td><td>9.0240865E-06</td><td>9.0692683E-06</td></tr><tr><td>U235</td><td>1.1408150E-03</td><td>1.1470952E-03</td><td>1.1178012E-03</td></tr><tr><td>U236</td><td>6.0794813E-05</td><td>6.0650051E-05</td><td>6.0953709E-05</td></tr><tr><td>U238</td><td>2.1065390E-02</td><td>2.1006346E-02</td><td>2.1146115E-02</td></tr><tr><td>O16</td><td>4.6119075E-02</td><td>4.5967016E-02</td><td>4.6205115E-02</td></tr><tr><td>GD152</td><td>2.1820460E-06</td><td>2.1185599E-06</td><td>2.1401440E-06</td></tr><tr><td>GD154</td><td>2.3475088E-05</td><td>2.2792086E-05</td><td>2.3024291E-05</td></tr><tr><td>GD155</td><td>1.5834163E-04</td><td>1.5373473E-04</td><td>1.5530096E-04</td></tr><tr><td>GD156</td><td>2.1759972E-04</td><td>2.1126871E-04</td><td>2.1342111E-04</td></tr><tr><td>GD157</td><td>1.6530018E-04</td><td>1.6049079E-04</td><td>1.6212587E-04</td></tr><tr><td>GD158</td><td>2.6070623E-04</td><td>2.5312105E-04</td><td>2.5569982E-04</td></tr><tr><td>GD160</td><td>2.2655650E-04</td><td>2.1996490E-04</td><td>2.2220590E-04</td></tr></tbody></table>
<figure><img src="image_9.png" /><figcaption>Fig. 8. Comparison of the experimental and theoretical transmission for sample A1 which is composed of a homogenous mixture of UO₂ and Gd₂O₃ powders, with Gd₂O₃ grain diameter of 15 µm. The residuals are the differences between the experimental and theoretical values weighted by the experimental uncertainties.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. Comparison of the experimental and theoretical transmission for sample A1 around the first and third s-wave $^{238}$U resonances, as shown in Figure 8.</figcaption></figure>

The TRIPOLI-4® code was used to simulate the transmission experiments carried out at the GELINA facility. For that purpose, we have introduced in the calculation the response function of the GELINA facility in order to account for the time distribution of the incident neutrons. These time distributions were calculated with REFIT. In addition, a Gaussian beam shape was used to mimic the non-uniform parallel beam passing through the sample. The validity of the TRIPOLI-4® results was tested against the experimental transmission of sample A1, for which no self-shielding correction is needed. As shown in Figure 12, an excellent agreement is achieved between the experimental and theoretical transmission spectra in the low energy range of interest for this work.

The TRIPOLI-4® model was used to simulate the transmission of sample A5 assuming that gadolinium oxide is homogeneously mixed in the UO₂ matrix. Results reported in the left-hand plots of Figure 13 demonstrate the sizeable impact of the micrometric particles on the neutron absorption process.

To solving the observed discrepancies, we have introduced in the TRIPOLI-4® model 814 Gd₂O₃ microspheres randomly distributed in each pellet of sample A5. Therefore, a total of 3256 microspheres were generated in the sample, which is composed of a stack of four pellets (Tab. 1). The volume densities for the gadolinium and oxygen nuclei in each microsphere are given in Table 3.
<figure><img src="image_11.png" /><figcaption>Fig. 10. Comparison of the experimental transmission for sample A1, as given in Figure 7, with theoretical curves calculated by REFIT with and without $Gd_2O_3$ grains.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 11. Comparison of the experimental transmissions for samples A4 and A5, as given in Figure 7, with theoretical curves calculated by REFIT using the particle self-shielding model proposed by Doub.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 12.** Same plot as Figure 8 for $E < 75$ eV, in which the experimental transmission for sample A1 is compared to TRIPOLI-4® simulations. The experimental response functions introduced in the simulations correspond to the same time distributions as used by REFIT. A Gaussian neutron beam shape was used to mimic the non-uniform parallel beam passing through the sample.</figcaption></figure>

The TRIPOLI-4® results are reported in the middle plots of Figure 13. The pretty good agreement with the experimental transmission confirms the sample characteristics listed in Table 1 and also the number of microspheres specified by the manufacturer. The introduction of thousands of volumes into the TRIPOLI-4® model remains a robust computational solution for sample A5. By contrast, for sample A4, the number of volumes that have to be generated in the whole sample reaches 23820, which will result in an excessive simulation time of up to 40 times that for sample A5.

For using the correction presented in equation (2), the particle self-shielded neutron cross-sections for the gadolinium isotopes were generated with the processing tool CADTUI [21] and converted in pointwise evaluated nuclear data file (PENDF) usable by TRIPOLI-4®. Figure 14 illustrates the huge impact of the model in the thermal and resonance energy ranges of the ¹⁵⁵Gd neutron capture cross-section, in the case of sample A5. Results simulated with TRIPOLI-4® by using self-shielded neutron cross-sections are reported in the right-hand plot of Figure 13. They are almost equivalent to those obtained with the generation of random microspheres.

Figure 15 summarizes the agreement achieved between the experimental transmission spectra and the TRIPOLI-4® simulations using Doub's model. The resid-

uals are nearly equivalent to those obtained with REFIT. These results confirm the suitability of Doub's model for micrometric spherical particles of Gd₂O₃ with diameters up to 380 µm. Consequently, such a particle self-shielding correction could be conveniently introduced in the processing of the gadolinium evaluations to avoid excessive time calculations due to the individual description of the microspheres in TRIPOLI-4®.

## 7 Conclusions

Oscillation experiments carried out in the early 1980s in the MINERVE reactor of CEA Cadarache provided a wide range of integral results on gadolinium oxide Gd₂O₃. Thanks to this program, unique well characterised UO₂ fuel pellets containing micrometric particles are now available for further experimental studies. The diameter of the microspheres (195 and 380 µm) and the Gd₂O₃ content (close to 3%) allow to use non-destructive methods as the neutron transmission technique available at the GELINA facility. The obtained results confirm that the particle self-shielding model proposed by Doub is suitable to generate particle self-shielded neutron cross-sections for the gadolinium isotopes usable by the Monte Carlo code TRIPOLI-4®. Considering particle effects to the nuclear data processing level helps reduce Monte Carlo time sim-
<figure><img src="image_14.png" /><figcaption>**Fig. 13.** Experimental transmission of sample A5 compared to TRIPOLI-4® results. The left-hand plots show TRIPOLI-4® simulations performed on a homogenous mixture of UO₂ and Gd₂O₃ powders without microspheres. The middle plots show TRIPOLI-4® simulations obtained with 814 microspheres of Gd₂O₃ (380 µm) randomly distributed within each pellet. Therefore, 3256 microspheres were generated in the sample, which is composed of a stack of four pellets (Tab. 1). The right-hand plot shows TRIPOLI-4® simulations performed with particle self-shielded neutron cross-sections.</figcaption></figure>

<table><caption>**Table 3.** Volume density (in at.10²⁴/cm³) of the gadolinium and oxygen nuclei on each Gd₂O₃ microsphere for sample A5.</caption><thead><tr><td>Isotope name</td><td>Volume density</td></tr></thead><tbody><tr><td>GD152</td><td>5.0934916E-05</td></tr><tr><td>GD154</td><td>5.4797262E-04</td></tr><tr><td>GD155</td><td>3.6961257E-03</td></tr><tr><td>GD156</td><td>5.0793714E-03</td></tr><tr><td>GD157</td><td>3.8585570E-03</td></tr><tr><td>GD158</td><td>6.0855947E-03</td></tr><tr><td>GD160</td><td>5.2884473E-03</td></tr><tr><td>O16</td><td>3.6913592E-02</td></tr></tbody></table>

Future works will consist of combining the microscopic experiments, performed at GELINA with the time-of-flight technique, and the integral experiments, carried out at the MINERVE reactor with the oscillation technique, to infer more precise information on the quality of the gadolinium evaluations of the JEFF library.

<figure><img src="image_15.png" /><figcaption>**Fig. 14.** Particle self-shielded neutron capture cross-section of ¹⁵⁵Gd in the case of sample A5.</figcaption></figure>

The experimental campaign carried out at GELINA provided a set of accurate transmission spectra that can be delivered at the nuclear data community via the experimental database EXFOR. These data sets could
<figure><img src="image_16.png" /><figcaption>**Fig. 15.** Same plot as Figure 11, in which the experimental transmission spectra are compared to TRIPOLI-4® simulations using the particle self-shielding model proposed by Doub. For samples A4 and A5, the number of microspheres used in the TRIPOLI-4® simulations are the same as in REFIT (see Tab. 1). The residuals obtained with TRIPOLI-4® and REFIT are given for comparison.</figcaption></figure>
be used for the experimental validation of any particle self-shielding model implemented in Monte Carlo tool. In addition, this work confirms that the GELINA facility can offer optimal non-destructive capabilities for exploring more complex nuclear materials, such as TRISO particles.

### Acknowledgments

The authors wish to express their appreciation for the experimental work performed at EC-JRC-Geel in the framework of the EUFRAT project. Special thanks go to Francois-Xavier Hugot and Cedric Jouanne of CEA Saclay for our support and help in using TRIPOLI-4®. TRIPOLI-4® is a registered trademark of CEA. The authors would like to thank Electricite De France (EDF) for partial financial support.

### Funding

This work is carried out in the framework of the SINET and EUFRAT projects funded by the CEA and EC-JRC-Geel, respectively.

### Conflicts of interest

The authors declare that they have no competing interests to report.

### Data availability statement

The data that support the findings of this study are available on request from the corresponding author.

### Author contribution statement

G. Noguere: methodology, software, writing, review, editing. M. Pottier, P. Leconte, D. Bernard: data analysis, review, editing. C. Paradela, S. Kopecky, P. Schillebeeckx: experimental support, raw data analysis, review, editing.

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Cite this article as: Gilles Noguere, Pierre Leconte, David Bernard, Mathilde Pottier, Carlos Paradela, Stefan Kopecky, Peter Schillebeeckx. Generation of particle self-shielded neutron cross-sections for the Monte Carlo code TRIPOLI-4®, EPJ Nuclear Sci. Technol. **10**, 18 (2024)