© G. Noguere and S. Xu, Published by EDP Sciences, 2022

1. Introduction

The propagation of the thermal scattering law uncertainties to neutron cross sections and neutronic parameters is addressed in a few works reported in the literature. Methodologies rely on Monte-Carlo or direct perturbations of the model parameters involved in the calculation of the thermal scattering laws. Owing to its importance for criticality and reactor applications, they were mainly applied to H₂O [1–7]. The present work aims to complement the reported results by including the effects of the thermal scattering laws of UO₂.

Part of the model parameters used in the description of the elastic scattering cross section at low neutron energies comes from abinitio or molecular dynamic simulations. Therefore, the relationship between condensed matter and neutronic physics is not straightforward. Several calculation steps handled by dedicated processing tools are needed to link the model parameters and the neutronic parameters. In the case of UOX integral benchmarks with UO₂ fuel moderated by water, this relationship can be simplified by assuming some approximations. The originality of this work is to use the effective temperature as an uncertainty indicator for quantifying the impact of the thermal scattering laws on the neutron multiplication factor. The effective temperature has the advantage of taking into account the vibration dynamics or atom binding effects of materials [8–11], but neglects some complex molecular behaviour, such as in the case of the water molecule. Fortunately, this specificity of the water molecule plays a minor role in the uncertainty quantification process associated to UOX benchmarks [4].

In this study, the relevance of using the effective temperature as an uncertainty indicator is illustrated with the MISTRAL-1 benchmarks [12] carried out at the EOLE facility (CEA Cadarache, France). The neutron multiplication factor was measured as a function of temperature up to 354 K (80°C). Such a unique set of integral results is suitable for testing the relationship between the effective temperature and the thermal scattering laws of H₂O and UO₂. For this purpose, the nominal calculations were performed with the Monte-Carlo code TRIPOLI-4^® [13] by using the evaluated nuclear data library JEFF-3.1.1. The perturbed calculations also rely on JEFF-3.1.1 in which the thermal scattering laws of H₂O and UO₂ were replaced by candidate files for the JEFF-4 library, which are available in the test library JEFF-4T1 [14].

The definition of the effective temperature is given in Section 2. Section 3 presents the methodology used to propagate the model parameter uncertainties of the thermal scattering laws to the neutron multiplication factor in the case of the MISTRAL-1 core. Uncertainties obtained on the neutron cross sections of H₂O and UO₂, as well as their contributions on the k_eff uncertainty are discussed in Section 4.

2. Effective temperature

The evaluated nuclear data files used in this work come from the working library JEFF-4T1, which contains new files for H₂O and UO₂. The evaluation of ¹H in H₂O is a slightly modified version of the file available in the US library (ENDF/B.VIII.0) [15]. The evaluation was modified to include a refined temperature grid, including the freezing point (273.15 K), critical point (674.1 K), and a grid with 5 K interval between 285 K and 650 K. Extrapolated temperature points between 650 K and 1000 K (every 50 K) were also added. The model relies on a Gaussian expansion, following the work of Maul et al. [4]. The evaluations for ²³⁸U and ¹⁶O in UO₂ are based on experimental results measured at the Institute Laue-Langevin of Grenoble (France). Detailed explanations are given in references [16, 17]. The evaluated nuclear data files for UO₂ contain thermal scattering laws for 31 temperatures ranging from 284 K to 1700 K.

Fig. 1. Elastic scattering cross section of 1H in H2O as a function of the neutron incident energy. The top plot (a) shows the neutron cross-section over the full energy range. The middle plot (b) highlights the discontinuity at the thermal cut-off energy (4.95 eV) between the thermal scattering law and the Free Gas Model when the thermodynamic temperature T is introduced in the Free Gas Model. The bottom plot (c) shows the continuous behaviour of the cross-section when T = Teff.

Fig. 2. Neutron elastic scattering cross sections of 238U and 16O in UO2 reconstructed with the evaluated nuclear data files available in the test library JEFF-4T1.

The CINEL code [18] was used to generate the thermal scattering laws for H₂O and UO₂ by starting from model parameters describing the dynamics and structure of the materials. The key parameter is the phonon density of states ρ(ω), from which the effective temperature is calculated as follows [9, 19]:

(1)

where T is the temperature of the material under consideration, k_B represents the Boltzmann constant and ℏω stands for the neutron energy transfer. The effective temperatures at T = 294 K (20°C) and T = 354 K (80°C) are listed in Table 1.

For ¹H in H₂O, the importance of the effective temperature can be illustrated by the behaviour of the elastic scattering cross section around the thermal cut-off energy of 4.95 eV (Fig. 1). Above this energy, nuclear data processing codes use a Free Gas Model to reconstruct the neutron cross sections at a given temperature T. A continuous behaviour between the thermal scattering law and the Free Gas Model treatments can only be achieved by introducing T = T_eff in the Free Gas Model.

For UO₂, the effective temperature is related to the Doppler broadening of the ²³⁸U resonances. As shown in Figure 2, the first resonance of ²³⁸U is very close to the thermal cut-off energy of 4 eV, below which dominate the different components of the elastic scattering of ²³⁸U and ¹⁶O in UO₂. For reactor applications, it is useful to link the effective and Debye temperatures θ_D by introducing a Debye phonon spectrum in equation (1). The first-order approximation leads to:

(2)

Butland [20] derived a similar expression by introducing a phonon density of states for ²³⁸U in UO₂ calculated by Thorson and Jarvis:

(3)

The combination of equations (2) and (3) gives a Debye temperature of θ_D = 249 K, which is equal to the Debye temperature reported in reference [21] derived from specific heat measurements at low temperature.

Starting from the work of Butland, Meister [22] approximated the phonon density of states of ²³⁸U in UO₂ by an Einstein model with acoustic and optical vibration modes, whose characteristics were deduced from transmission experiments performed at the GELINA facility [23]. Equivalences between effective resonance integrals calculated with Free Gas and Crystal Lattice Model provided the following parameterization:

(4)

According to this expression, Butland’s effective temperature has to be increased by 8.6 K. Figure 3 compares the effective temperatures calculated with equations (3) and (4). Results obtained with the phonon density of states of the JEFF-4T1 library lie between these two trends.

Fig. 3. Ratio Teff/T for 238U in UO2 calculated with equations (3), (4) and (1) as a function of T.

3. Perturbation scheme

The integral benchmarks selected in this study are part of the MISTRAL program carried out in the EOLE reactor of CEA Cadarache [12]. A detailed description of the experiments can be found in reference [24]. This program was the subject of the work reported in reference [25], in which the impact of the thermal scattering law of H₂O on the isothermal reactivity coefficients α_iso is studied. The present work focuses on the integral results measured with the MISTRAL-1 core, which was a homogenous UO₂ configuration that serves as a reference for the whole MISTRAL program. The cylindrical core (Fig. 4) consists of a regular lattice using 750 standard PWR fuel pins (3.7% enriched in ²³⁵U) in a square pitch of 1.32 cm with 16 guide tubes dedicated to safety rods. The moderation ratio is 1.7 (representative of light water reactor moderation). The reactivity excess was measured as a function of the temperature up to 354 K with a fine temperature step of 5 K. In the MISTRAL-1 configurations, the concentration of the soluble boron was adjusted in the moderator in order to compensate for the reactivity loss due to the temperature increase. The experimental uncertainties of about ±25 pcm mainly come from the kinetic parameters, the measurements of the doubling time and of the boron concentration. In comparison, the technological uncertainties are close ±250 pcm. Results calculated with the Monte-Carlo code TRIPOLI-4^® are shown in Figure 5 as a function of T. For each configuration the statistical uncertainty due to the Monte-Carlo calculations is lower than ±5 pcm. The slope of the solid line represents the calculation bias on the reactivity temperature coefficient α_iso. For JEFF-3.1.1, its value is close to −0.36 pcm/K, which remains in the lower limit of the experimental uncertainties (±0.3 pcm/K).

Fig. 4. Radial cross section of the MISTRAL-1 core composed of 750 UOX fuel pins in light water.

Fig. 5. Differences in reactivity for the MISTRAL-1 core expressed in terms of C-E obtained with the Monte-Carlo code TRIPOLI-4® and the JEFF-3.1.1 library up to 354 K (80°C). The solid line represents the best-fit curve reported in reference [25]. The present calculations account for the thermal expansion of the materials, while in the past study [25], the thermal expansion was taken into account as a correction factor of the Monte-Carlo results. The reported uncertainties only take into account the uncertainties of the kinetic parameters, measurements of the doubling time and boron concentration.

As indicated by Figure 5, a small variation of the C-E values is expected between the room temperature and 354 K. Such a small variation in conjunction with a demanding computational time makes an uncertainty analysis via repetitive Monte-Carlo perturbations difficult. Therefore, the uncertainty analysis will be performed by direct perturbations of the parameters involved in the thermal scattering law models. In addition, we will limit ourselves to two extreme temperatures at 294 K and 354 K.

Two types of model parameters can be distinguished. Those that are linked to the neutron-nucleus interaction and those that depend explicitly on the studied material. The neutron scattering lengths or equivalently the neutron elastic scattering cross sections at zero Kelvin are well documented in the literature. Their direct perturbation does not cause any problems. By contrast, the range of variation of the parameters describing the temperature-dependent dynamic structure factors for H₂O and UO₂ is poorly known. A solution to overcome this lack of information has been investigated in reference [2]. The reported results indicate that the sources of uncertainties impacting the calculated k_eff can be lumped into a single parameter which is the energy interval δ used to reconstruct the phonon density of state. This approximation was successfully used by Rochman on criticality benchmarks [7]. However, it supposes that the other parameters which drive the intramolecular vibrations or H-bond modes have negligible contributions to the k_eff uncertainty. This assumption was also confirmed by Maul in his work on the OPALE reactor [4], but cannot be generalized, especially for well-thermalized neutron spectrum or cold neutron sources. The aim of our perturbation scheme is to apply a scaling factor Δ to this energy interval δ as follows:

(5)

and to use Δ as an intermediate parameter for propagating T_eff uncertainties to neutron cross sections and k_eff values via equation (1). Final results will be summarized in terms of calculation biases expressed in pcm/K.

Fig. 6. Comparison of the experimental and theoretical total cross section of H2O. For clarity, the EXFOR data [27] and the ISIS data [26] measured at room temperature are shown separately. The grey zones represent the uncertainties.

4. Results and discussions

4.1. Uncertainties on the thermal scattering laws for H₂O

The covariance matrices associated with the thermal scattering laws of H₂O were evaluated in the framework of a few studies [1, 2, 4–6] in the case of the evaluations available in the JEFF-3.1.1 and ENDF/B.VIII.0 libraries. Although different methodologies were used, they all converged to the same conclusions. At room temperature, the relative uncertainty on the total cross-section of H₂O is lower than 0.5% in the eV energy range and remains below 5% around the thermal energy (25.3 meV). The relative uncertainties start to increase well beyond 5% in the cold and ultra-cold neutron energy ranges, where the translational diffusion behaviour of the water molecule plays a major role in the calculation of the dynamic structure factor. The assessment of the uncertainties at elevated temperatures is less documented because of the lack of accurate experimental values. Fortunately, a recent experimental program carried out at the Vesuvio facility of ISIS (UK) provided a unique set of accurate data for water at 283, 293 and 353 K [26].

Figure 6 summarizes the agreement between the total experimental cross-sections of H₂O and the JEFF-4T1 library. At room temperature, many experimental values are available in the EXFOR data base [27]. The theoretical curve starts to deviate from the data below 1 meV. The right-hand plots of Figure 6 highlight the excellent agreement between the theory and the ISIS data in the thermal energy range. This good agreement validates the position, and the shape of the broad libration band observed in the phonon density of states of H₂O calculated by molecular dynamic simulations. The accuracy of the ISIS data also reveals systematic biases in some of the EXFOR data. This trend is confirmed by the two structures observed in Figure 7, which compares the distributions of the EXFOR and ISIS data with respect to JEFF-4T1.

Fig. 7. Distributions of the ratios of the theoretical total cross-section of H2O (C) to the experimental data (E). The theoretical curves were calculated with the JEFF-4T1 evaluation of H2O.

Fig. 8. Correlation matrices of the total cross section of H2O calculated at 294 and 354 K.

As introduced in Section 3, covariance matrices for H₂O at 294 and 354 K can be established by direct perturbations of two model parameters. In the eV energy range, the behaviour of the total cross-section mainly depends on the elastic scattering cross section of ¹H at zero kelvin σ_s . The uncertainty on this parameter, recommended by the Neutron Data Standard of IAEA [28], is close to 0.2%:

We decided to apply a direct perturbation of the same order of magnitude, which is equivalent to a perturbation of 41 mbarns on σ_s . In the thermal energy range, the behaviour of the total cross section is driven by the phonon density of states or equivalently by the effective temperature T_eff (Eq. (1)). In order to reach a statistically significant convergence of the results, a perturbation of 5 K was applied to T_eff. According to our calculation scheme, this perturbation leads to a variation of 3% and 3.7% of the scaling factor Δ (Eq. (5)) at 294 and 354 K, respectively. The perturbation scheme was complemented by an iterative procedure in order to optimize the uncertainty of the effective temperature, given the constraint to obtain an uncertainty on the thermal total cross section σ_th that reproduces the standard deviation of the EXFOR data between 0.02 and 0.06 eV. The following sets of effective temperatures were found:

and

In Figure 6, the grey zones represent the resulting uncertainty bands calculated over the full neutron energy range. Around 5 eV, the relative uncertainty is close to 0.2%. In the thermal energy range, the uncertainty increases up to 3% and reaches 10% below 0.1 meV. At 25.3 meV, we obtained:

and

The correlation matrices associated with the total cross-section of H₂O at 294 and 354 K are shown in Figure 8. Our perturbation scheme provides matrices with a two-block structure which is consistent with previous studies reported in the literature [4].

Fig. 9. Comparison of the experimental and theoretical total cross section of UO2 at room temperature.

4.2. Uncertainties on the thermal scattering laws for UO₂

Evaluation works on UO₂ are scarce in the literature. Despite the latest studies performed in the framework of the US library (ENDF/B.VIII.0) [29], no uncertainty analysis on the thermal scattering laws was undertaken. One reason can be the lack of experimental data covering the low neutron energy range. Only two rather old experimental total cross-sections of UO₂ are available in the EXFOR library. In Figure 9, the comparison with the theoretical curve seems to confirm the poor quality of the existing data around thermal energy, making them unreliable for uncertainty analysis. Consequently, results provided in this work only comes from roughly estimated prior model parameter uncertainties.

The model parameters of interest are still the effective temperature as well as the neutron scattering lengths b_coh associated with the reactions n+¹⁶O and n+²³⁸U. Values for b_coh were taken from the compilation reported in reference [30], and the order of magnitude of the uncertainty was derived from values reported by Mughaghab [31]:

For the effective temperature, the difference between the prescriptions of Butland and Meister (Fig. 3) provides a confidence range of variation of T_eff for ²³⁸U. A conservative estimate of ±10 K was used in this work:

Fig. 10. Elastic scattering cross sections of UO2 calculated at 294 and 354 K with the JEFF-4T1 evaluations. The solid black lines and the grey zones represent the uncertainties.

Fig. 11. Relative uncertainty calculated on the elastic scattering cross sections of UO2. The dotted black line indicates the thermal cut-off energy (4 eV).

Figure 10 shows the uncertainty band calculated for the UO₂ elastic scattering cross-section by applying a perturbation of 0.01 fm and 5 K on b_coh and T_eff, respectively. The perturbation of the effective temperature of ²³⁸U leads to a non-negligible variation of the scaling factor Δ (Eq. (5)) ranging from 10% to 15% at 294 and 354 K, respectively. This perturbation was simultaneously applied to the partial phonon density of states of ²³⁸U and ¹⁶O in UO₂. The relative uncertainty as a function of the incident neutron energy is shown in Figure 11. A reliable value of ±1.9% is obtained at the cut-off energy of 4 eV. The uncertainty at the thermal energy slightly increases from ±3.9% to ±4.8% with the temperature:

and

4.3. Uncertainty on the multiplication factor k_eff

This work aims to illustrate the use of the effective temperature as an uncertainty indicator for quantifying the impact of the thermal scattering laws on the neutron multiplication factor. The nominal calculations are performed with the JEFF-3.1.1 library and compared to calculations in which the H₂O or UO₂ evaluations of the test library JEFF-4T1 are perturbed. The propagation of the model parameter uncertainties to the neutron multiplication factor of the MISTRAL-1 program follows the same methodology as presented in Sections 4.1 and 4.2. The parameters and uncertainties used in this study are summarized in Table 2. We applied perturbations of 5 K to T_eff, 41 mbarns to σ_s and 0.01 fm to b_coh.

The nominal results calculated with TRIPOLI-4^®-pagination

before perturbation are compared in Table 3 with those obtained thanks to the JEFF-4T1 evaluations of H₂O and UO₂. The thermal scattering laws of H₂O and UO₂ coming from JEFF-4T1 have opposite effects on k_eff. The compensation of each contribution is illustrated with the C-E results reported in Figure 12. The calculation bias on the isothermal reactivity coefficients α_iso, given by the slope of the solid line, is improved with the JEFF-4T1 library. It becomes equal to −0.09 pcm/K, compared to −0.36 pcm/K with JEFF-3.1.1. Such an improvement is reflected by the low uncertainty on the k_eff values reported in Table 4. The uncertainties due to the thermal scattering laws of H₂O are close to ±50 pcm, and those due to UO₂ remain lower than 30 pcm. For comparison, Table 5 shows results previously reported in the literature for the MITRAL-1 core. The non-negligible uncertainties associated with the thermal scattering laws of H₂O in JEFF-3.1.1 (> 100 pcm) progressively decrease to the present results. This trend is explained by the intense work performed on the modelization of the dynamical structure factor of H₂O and on the total cross-section measurements at elevated temperatures.

Fig. 12. Differences in reactivity for the MISTRAL-1 core obtained with TRIPOLI-4®. The solid blue line is the best-fit curve obtained with the JEFF-3.1.1 library alone (see Fig. 5). The solid red line is the best-fit curve obtained when the thermal scattering laws of H2O and UO2 are replaced by those of JEFF-4T1. The slope of each line gives the calculation bias on the isothermal reactivity temperature coefficient αiso. For clarity, the uncertainties are shortly discussed in Section 3 and not reported in the plot.

The results of Table 4 can be conveniently translated into pcm/K. This uncertainty indicator represents a calculation bias on k_eff relatively to the effective temperature uncertainty. In our case, it has the advantage of providing rather similar results for H₂O and UO₂. At room temperature, it is close to 2.6 pcm/K on average for both H₂O and UO₂, while it is dominated by H₂O at elevated temperatures. Consequently, an average estimate of about 2.3 pcm/K can be safely used from 294 to 354 K. This estimate provides a k_eff uncertainty of ±51 pcm, resulting from the quadratic sum of each contribution given effective temperature uncertainties of ±20 K and ±10 K for H₂O and UO₂, respectively. The consistency of the obtained results over the studied temperature range can be seen as a step forward to store thermal scattering law uncertainties in evaluated nuclear data files. Providing uncertainties on the effective temperatures instead of covariance matrices associated to temperature-dependent phonon densities of states can be easily handle in a modern nuclear data format [32].

The major conclusion of this study is that in the future evaluated nuclear data libraries, uncertainty budget analysis associated with the low neutron energy scattering process of H₂O and UO₂ will be a marginal contribution compared to the uncertainties on the capture process. Indeed, uncertainties in the UOX reactivity due to the ¹H(n, γ) and ²³⁸U(n, γ) reactions are expected to be as large as ±150 pcm, if the covariance database COMAC [33] is used in the calculations.

5. Conclusions

The propagation of the model parameter uncertainties involved in the description of the thermal scattering law of H₂O to neutronic parameters is the subject of a few works reported in the literature. The present uncertainty analysis provides new results as a function of the temperature for H₂O and complements past uncertainty analysis with UO₂. The obtained results confirm that the accuracy of the latest evaluations of H₂O and UO₂, available in the working library labelled JEFF-4T1, will imply a modest impact of about ±50 pcm on the k_eff uncertainty of UOX benchmarks. In comparison to capture reactions, the low neutron energy scattering process will become a marginal source of uncertainties. This achievement can be explained by the intense works performed on the modelization of the dynamical structure factor of H₂O and UO₂ in association with new measurements performed at elevated temperatures. This conclusion is only valid for UOX benchmarks and cannot be generalized, especially for well-thermalized neutron spectrum or cold neutron sources.

The results reported in this work also probe that the specificity of the neutron spectrum for UOX cores makes it possible to use the effective temperature as an uncertainty indicator to quantify the impact of the thermal scattering laws on the calculated reactivity. The corresponding uncertainty indicator, expressed in pcm/K, represents a calculation bias on k_eff relatively to the effective temperature uncertainty. In the framework of the MISTRAL-1 program, it has the advantage to be close to 2.3 pcm/K on average for both H₂O and UO₂ contributions. This approach simplifies the relationship between the condensed matter and neutronic physics, and reduces the complexity of the uncertainty analysis to a single parameter. Therefore, providing uncertainties about the effective temperatures in the evaluated nuclear data files is a recommendation that could avoid dealing with the covariance matrices associated to the temperature-dependent phonon densities of states.

Acknowledgments

TRIPOLI-4^® is a registered trademark of CEA. The authors would like to thank Electricite De France (EDF) for partial financial support.

Conflict of interests

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

Funding

This work is carried out in the framework of the SINET project funded by the CEA.

Data availability statement

This article has no associated data generated and/or analyzed.

Author contribution statement

G. Noguere: conceptualization, methodology, software, validation, writing, review, editing, visualization. S. Xu: software, review, editing.

References

All Tables

All Figures

Fig. 1. Elastic scattering cross section of 1H in H2O as a function of the neutron incident energy. The top plot (a) shows the neutron cross-section over the full energy range. The middle plot (b) highlights the discontinuity at the thermal cut-off energy (4.95 eV) between the thermal scattering law and the Free Gas Model when the thermodynamic temperature T is introduced in the Free Gas Model. The bottom plot (c) shows the continuous behaviour of the cross-section when T = Teff.

In the text

	Fig. 2. Neutron elastic scattering cross sections of 238U and 16O in UO2 reconstructed with the evaluated nuclear data files available in the test library JEFF-4T1.
In the text

	Fig. 3. Ratio Teff/T for 238U in UO2 calculated with equations (3), (4) and (1) as a function of T.
In the text

	Fig. 4. Radial cross section of the MISTRAL-1 core composed of 750 UOX fuel pins in light water.
In the text

Fig. 5. Differences in reactivity for the MISTRAL-1 core expressed in terms of C-E obtained with the Monte-Carlo code TRIPOLI-4® and the JEFF-3.1.1 library up to 354 K (80°C). The solid line represents the best-fit curve reported in reference [25]. The present calculations account for the thermal expansion of the materials, while in the past study [25], the thermal expansion was taken into account as a correction factor of the Monte-Carlo results. The reported uncertainties only take into account the uncertainties of the kinetic parameters, measurements of the doubling time and boron concentration.

In the text

	Fig. 6. Comparison of the experimental and theoretical total cross section of H2O. For clarity, the EXFOR data [27] and the ISIS data [26] measured at room temperature are shown separately. The grey zones represent the uncertainties.
In the text

	Fig. 7. Distributions of the ratios of the theoretical total cross-section of H2O (C) to the experimental data (E). The theoretical curves were calculated with the JEFF-4T1 evaluation of H2O.
In the text

	Fig. 8. Correlation matrices of the total cross section of H2O calculated at 294 and 354 K.
In the text

	Fig. 9. Comparison of the experimental and theoretical total cross section of UO2 at room temperature.
In the text

	Fig. 10. Elastic scattering cross sections of UO2 calculated at 294 and 354 K with the JEFF-4T1 evaluations. The solid black lines and the grey zones represent the uncertainties.
In the text

	Fig. 11. Relative uncertainty calculated on the elastic scattering cross sections of UO2. The dotted black line indicates the thermal cut-off energy (4 eV).
In the text

Fig. 12. Differences in reactivity for the MISTRAL-1 core obtained with TRIPOLI-4®. The solid blue line is the best-fit curve obtained with the JEFF-3.1.1 library alone (see Fig. 5). The solid red line is the best-fit curve obtained when the thermal scattering laws of H2O and UO2 are replaced by those of JEFF-4T1. The slope of each line gives the calculation bias on the isothermal reactivity temperature coefficient αiso. For clarity, the uncertainties are shortly discussed in Section 3 and not reported in the plot.

In the text