https://doi.org/10.1051/epjn/2016020
Regular Article
Impact of the thermal scattering law of H in H_{2}O on the isothermal temperature reactivity coefficients for UOX and MOX fuel lattices in cold operating conditions
^{1}
CEA, DEN, DER Cadarache, Saint Paul les Durance, France
^{2}
Neutron Physics Department and Instituto Balseiro, Centro Atomico Bariloche, CNEA, Argentina
^{⁎} email: gilles.noguere@cea.fr
Received:
25
November
2015
Received in final form:
24
February
2016
Accepted:
23
March
2016
Published online: 10 June 2016
The contribution of the thermal scattering law of hydrogen in light water to isothermal temperature reactivity coefficients for UOX and MOX lattices was studied in the frame of the MISTRAL critical experiments carried out in the zero power reactor EOLE of CEA Cadarache (France). The interpretation of the core residual reactivity measured between 6 °C to 80 °C (by step of 5 °C) was performed with the MonteCarlo code TRIPOLI4^{®}. The nuclear data from the JEFF3.1.1 library were used in the calculations. Three different thermal scattering laws of hydrogen in light water were tested in order to evaluate their impact on the MISTRAL calculations. The thermal scattering laws of interest were firstly those recommended in JEFF3.1.1 and ENDF/BVII.1 and also that recently produced at the atomic center of Bariloche (CAB, Argentina) with molecular dynamic simulations. The present work indicates that the calculationtoexperimpental bias is −0.4 ± 0.3 pcm/°C in the UOX core and −1.0 ± 0.3 pcm/°C in the MOX cores, when the JEFF3.1.1 library is used. An improvement is observed over the whole temperature range with the CAB model. The calculationtoexperimpental bias vanishes for the UOX core (−0.02 pcm/°C) and becomes close to −0.7 pcm/°C for the MOX cores. The magnitude of these bias have to be connected to the typical value of the temperature reactivity coefficient that ranges from −5 pcm/°C at Begining Of Cycle (BOC) up to −50 pcm/°C at End Of Cycle (EOC), in PWR conditions.
© J.P. Scotta et al., published by EDP Sciences, 2016
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The isothermal temperature reactivity coefficients, or equivalently the reactivity temperature coefficients (RTC), are one of the major reactor safety parameters. They represent the change in reactivity due to a change in temperature [1]. Recent publications deal with RTC for various reactor configurations in “cold conditions” (T < 50 °C) [2–4] up to “hot conditions” (T < 300 °C) [5,6]. The present work focuses on the calculation of RTC for critical assemblies in “cold conditions” for temperatures ranging from 6 °C to 80 °C at atmospheric pressure. The isothermal temperature coefficient α_{iso}(T) is determined from the excess of reactivity ρ(T) measured at given temperatures T. In practice, the experimental results allow estimating Δα_{iso}(T) which represents the calculation error on RTC. The latter is given by the derivative of the difference Δρ(T) between the calculated (C) and measured (E) excess of reactivity with respect to the temperature:(1)with(2)
A series of MISTRAL experiments [7–15] was carried out in the EOLE facility of CEA Cadarache (France) in order to study Δα_{iso} for UOX (MISTRAL1 configuration) and MOX (MISTRAL2 and MISTRAL3 configurations) lattices. Previous interpretations [16,17] were performed with the deterministic code APOLLO2 [18] by using the evaluated nuclear data libraries JEF2.2 and JEFF3.1.1. Results are summarized in Table 1. According to conclusions reported in reference [16], Δα_{iso} is mainly sensitive to the spectral shift of thermal neutrons in the low temperature range (T<40 °C). The contribution of the water density effects becomes sizeable when the temperature increases. In addition, the contribution of the thermal spectrum effects to the calculation errors is strongly dependent on the shape of the ^{235}U and ^{239}Pu neutron crosssections in the thermal region.
The main physical trends observed in the MISTRAL1 experiment between 6°C and 80°C for UOX lattices are confirmed by a sensitivity analysis performed on the critical assembly of the Kyoto University between 27°C and 57°C [19]. However, the reported results mainly emphazise the importance of the thermal scattering crosssection of hydrogen bound to H_{2}O. Such a significant contribution to the calculation errors Δα_{iso} was not reported in the previous interpretations of the MISTRAL programs.
The present work aims at quantifying the impact of the thermal scattering law (TSL) of hydrogen in light water on Δα_{iso}. Reference values were calculated with the MonteCarlo code TRIPOLI4^{®} [20] by using the evaluated nuclear data library JEFF3.1.1 [21]. They are compared to results obtained with JEFF3.1.1 in which the TSL are replaced by those of the US library ENDF/BVII.1 [22] and of the CAB library [23], produced at the atomic center of Bariloche.
2 Thermal scattering law for light water
2.1 Governing equations
In the low energy range (below approximately 5 eV), the neutron scattering in a light water moderator is affected by the intermolecular and intramolecular hydrogen bonds. They modify the energy and angular distributions of secondary neutrons. A description of the model for water is given in references [24,25], and studies that investigate how we can accurately calculate neutrons slowing down in water are reported in reference [26]. The double differential incoherent inelastic scattering crosssection of a single bound atom in molecule (H bound in H_{2}O) can be written as a function of the symmetric scattering law S(α, β):(3)where E and E′ are the incident and secondary neutron energies, Ω defines the scattering angle, σ_{b} represents the characteristic bound crosssection for the material, k is the Boltzmann constant and T is the temperature of the material. The scattering law contains all the dynamic and structural information about the target system. It is a function of the momentum transfer α:(4)and of the energy transfer β:(5)where cos(θ) is the cosine of the scattering angle in the laboratory system and A is the ratio of the mass of the scattering atom to the neutron mass.
Some approximations are customarily used to represent the S(α, β) function over a large dynamical range with simple mathematical expressions. For hydrogenous moderators, like light water, the incoherent neutron scattering dominates the scattering process. This assumption, combined with the Gaussian approximation [27], leads to the following expression for the scattering law:(6)where the function γ(t) is computed as:(7)
The function P(β) is related to the generalized frequency spectrum of the material ρ(β) by:(8)with the condition:(9)
The distribution ρ(β) contains a complete description of the intermolecular and intramolecular vibration modes of the water molecule.
2.2 Frequency spectrum used in the TSL models
The frequency spectrum is a continuous probability density function. For H in light water, ρ(β) can be decomposed into a sum of four components:(10)where ρ_{c}(β) is a continuous distribution that describes the rotational mode of the water molecule, ρ_{t}(β, c) mimics the translational mode that depends on the diffusion constant c and is a sum of two discrete oscillators which define the intramolecular vibrations, namely bending and stretching. The weights satisfy the following condition:(11)
Three different sets of frequency spectra were studied. Two of them stem from the model developed by M. Mattes and J. Keinert [28]. This model will be called IKE model in the text. It was used for establishing the thermal scattering laws available in the JEFF3.1.1 and ENDF/BVII.1 libraries. The third one, called CAB model [23], was developed by J.I. Marquez Damian at the atomic center of Bariloche. Parameters used in each model at 294 K are given in Table 2.
In the JEFF3.1.1 library, the frequency spectra of H in H_{2}O are based on experimental values measured by Haywood and Page at 294 K and 550 K [29]. The symmetric and asymmetric stretching modes are described by a single discrete oscillator at 0.436 eV. For the bending mode, a discrete oscillator at 0.205 eV was used. The IKE model parameters were slightly modified for producing new S(α, β) tables for the ENDF/BVII.1 library. The characteristics of the discrete oscillators remain the same as JEFF3.1.1.
A new approach was used for the CAB model. Molecular dynamic simulations were performed for calculating the temperaturedependent frequency spectra of hydrogen in light water. The characteristics of the discrete oscillator (energies and weights) obtained from the Molecular dynamic simulations and used in the IKE model are nearly similar. In contrast, large differences can be observed between the continuous rotational mode used in each model (Fig. 1). For the translational mode (ω_{t}ρ_{t}), a diffusion model [30] with an effective mass of 116 a.m.u was adopted in the CAB model, while a free gas model with a mass of 52 a.m.u and 46 a.m.u was used in ENDF/BVII.1 and in JEFF3.1.1, respectively. Upon interaction with the incident neutron, a heavier effective mass will reduce the contribution of the translational mode of the water molecule (ω_{t} decreases) and will increase the probability of undergoing a rotation (ω_{c} increases). In the CAB model, special attention has been paid to the description of the translational mode for improving the agreement between the experimental and calculated crosssections in the cold neutron energy range, below the thermal energy of 25.3 meV. The impact of the S(α, β) tables generated with each model was investigated in the frame of the MISTRAL program.
Parameters for the TSL models of H in H_{2}O at 294 K.
Fig. 1 Comparison of the continuous and discrete frequency spectrum for H in H_{2}O at 294 K. 
3 Interpretation of the MISTRAL programs with the MonteCarlo TRIPOLI4^{®}
3.1 Description of the MISTRAL configurations
The MISTRAL experimental programs were designed in the late nineties to evaluate the feasibility of using 100% MOX fuel in light water reactors. The different core configurations were tested in the EOLE reactor of CEA Cadarache (France). Many relevant neutronic parameters were measured during the MISTRAL programs such as critical mass, geometrical buckling, spectral indices, conversion factor, isothermal temperature coefficient, single absorber worth, soluble boron worth and effective delayed neutron fraction.
The present work focuses on the isothermal temperature reactivity coefficient measured in the MISTRAL1, MISTRAL2 and MISTRAL3 configurations (Fig. 2). A detailed description of the experiments can be found in reference [16].
The MISTRAL1 core is a homogenous UO_{2} configuration that serves as reference for the whole MISTRAL programs. The cylindrical core consists of a regular lattice using 750 standard PWR fuel pins (3.7% enriched in ^{235}U) in a square pitch of 1.32 cm with 16 guide tubes dedicated for safety rods. The moderation ratio is 1.7 (representative of LWR moderation).
The MISTRAL2 core is a homogenous 100% MOX configuration with 1572 MOX fuel pins with a fuel enrichment of 7% in AmPuO_{2}. This second configuration is characterized by the same number of guide tubes, pitch and moderation ratio as MISTRAL1.
The MISTRAL3 core is a homogenous 100% MOX configuration with 1388 fuel pins with a fuel enrichment of 7% in AmPuO_{2}. The main differences with respect to MISTRAL2 are the moderation ratio, close to 2.1, and the square pitch which was set to 1.39 cm. The aim of this configuration was to measure the fundamental neutronic parameters in a slightly overmoderated lattice.
The reactivity excess was measured as a function of the temperature from 6 °C to 80 °C with a fine temperature step of 5 °C. In the MISTRAL1 and MISTRAL3 configurations, the concentration of the soluble boron was adjusted in the moderator in order to compensate the reactivity loss due to the temperature increase. In MISTRAL2, the criticality was achieved by adjusting the critical size of the core. MOX pins with enrichment of 8.7% were strategically added at the periphery of the core.
Fig. 2 Radial crosssections of the MISTRAL1 (750 UOX fuel pins), MISTRAL2 (1572 MOX fuel pins) and MISTRAL3 (1388 MOX fuel pins) cores. For MISTRAL2, the given core is the configuration at 20 °C. 
3.2 Processing of the TSL data files for TRIPOLI4^{®}
The MonteCarlo code TRIPOLI4^{®} [20] was used for the interpretation of the MISTRAL experiments. For this purpose, thermal scattering files of H in H_{2}O were generated for each temperature step in a format compatible with the official nuclear data library of TRIPOLI4^{®} based on JEFF3.1.1.
The processing of the TSL data files was performed with the NJOY code [25]. Two modules of NJOY are specifically dedicated to this treatment. The LEAPR module calculates the S(α, β) tables by using the formalism briefly described in Section 2.1. The THERMR module uses the S(α, β) tables for calculating the double differential inelastic crosssections (Eq. (3)). Figure 3 shows the flowchart representing the processing scheme applied to the TSL files of JEFF3.1.1, ENDF/BVII.1 and generated with the CAB model.
Before analyzing the MISTRAL experiments, the processing scheme used in this work to produce thermal scattering laws was tested and validated against the official library of TRIPOLI4^{®}. The differences on the calculated effective multiplication factor (k_{eff}) between the official library and our NJOY treatment were quantified on the MISTRAL1 benchmark at 20 °C. Results are reported in Table 3.
As a first step, we have evaluated the sensitivity of the calculated k_{eff} to the thermal scattering law of hydrogen by considering the hydrogen in water as a free gas. Figure 4 compares the ^{1}H and H in H_{2}0 total crosssections calculated at T = 300 K. The thermal energy cutoff is equal to 4.95 eV. Then, TRIPOLI4^{®} uses the Sampling of the Velocity of the Target nucleus (SVT) up to T_{max} = 400k_{B}T. Beyond this energy, the static Assymptotic Kernel (AK) approximation is applied. The importance of the TSL depends on the size of the neutronic core. A small core yields a high thermal neutron leakage, so a high effect of the thermal neutron models is expected. In our case, the free gas model overestimates the experimental reactivity excess by approximately +800 pcm. Such a large difference confirms the importance of the thermal scattering laws and their processing with the LEAPR and THERMR modules of the NJOY code for a correct interpretation of the MISTRAL experiments.
The two NJOY modules were tested separately. The THERMR module was applied to the S(α, β) tables given with the official TRIPOLI4^{®} library. In order to test the compatibility of the LEAPR calculations, we used the input files for H in H_{2}O reported by Mattes in reference [28]. The input file contains the model parameters listed in Table 2 and the continuous frequency spectra shown in Figure 1. As reported in Table 3, the differences between the k_{eff} values calculated with the TSL files coming from our processing scheme and the official library of TRIPOLI4^{®} are negligible and remain below the statistical uncertainties of ±10 pcm. This good agreement shows that our processing scheme can be safely used for the interpretation of the MISTRAL programs.
Fig. 3 Flowchart of the calculation scheme used to produce S(α, β) tables for the TRIPOLI4^{®} code [20]. The processing of the S(α, β) tables from the three TSL data files of interest for this work is performed with the NJOY code [25]. The crosssections of the JEFF3.1.1 library is used for the neutron transport and only S(α, β) of light water are replaced by taking the needed information from alternatively the JEFF3.1.1, ENDF/BVII.1 and CAB libraries. The CADTOOL package [31] provides an easytouse interface for automated sequential processing schemes. 
Excess of reactivity calculated with the TRIPOLI4^{®} code for the MISTRAL1 configuration at 20 °C. The differences C_{i} − C_{1} are calculated by using the result obtained with the official T4 library as reference.
Fig. 4 Comparison of the ^{1}H and H in H_{2}O total crosssections calculated at T = 300 K. The thermal energy cutoff is equal to 4.95 eV. Then, TRIPOLI4^{®} uses the Sampling of the Velocity of the Target nucleus (SVT) up to T_{max} = 400k_{B}T. Beyond this energy, the static Assymptotic Kernel (AK) approximation is applied. 
3.3 Interpolation of the model parameters
In the JEFF3.1.1 and ENDF/BVII.1 libraries, the thermal scattering laws are tabulated in terms of S(α, β) tables over a broad temperature mesh. Only three (20 °C, 50 °C and 100 °C) and two (20 °C and 77 °C) temperatures, respectively, are reported to map the temperature range of the MISTRAL programs from 6 °C to 80 °C. Such a broad temperature mesh is not adequate for a precise estimation of the isothermal temperature reactivity coefficient ar room temperature.
New S(α, β) tables were generated up to 80 °C with a fine temperature step of 5 °C by interpolating the model parameters and the frequency spectra contained in the LEAPR input files. Results for parameters established by Mattes [28] are shown in Figures 5 and 6. The total crosssections of H in H_{2}O from the JEFF3.1.1 library are compared in Figure 7 to the total crosssections calculated with a fine temperature mesh. High sensitivities to the temperature are observed for cold neutrons (E < 25.3 meV).
Final results were verified by comparing the total crosssection reconstructed from our interpolation procedure to the JEFF3.1.1 total crosssection evaluated at 20 °C and 50 °C. In both cases, the differences remain negligible over the neutron energy range of interest. They reach a maximum of 1.5 barns at 0.01 meV, corresponding to a calculation error of 0.15%.
Fig. 5 Interpolation of the model parameters established by Mattes [28] between 6 °C to 80 °C. The meaning of each parameter is given in Table 2. 
Fig. 6 Continuous frequency spectra for H in H_{2}O for the broad temperature mesh of the JEFF1.1.1 library (top plot) and interpolated over a fine temperature mesh (bottom plot). 
Fig. 7 Total crosssection of H in H_{2}O calculated with the broad temperature mesh of the JEFF1.1.1 library (20 °C, 50 °C and 100 °C) and interpolated over a fine temperature mesh (from 6 °C to 80 °C). 
4 Results and discussions
4.1 Comparison of the TSL data files
The processing scheme used in association with our interpolation procedure allows comparing the thermal scattering laws for each temperature of the MISTRAL programs up to 80 °C. For the sake of clarity, we only report comparison on the total crosssections in Figure 8.
In the cold neutron energy range, large discrepancies are observed between the TSL of JEFF3.1.1, ENDF/BVII.1 and established with the CAB model. The discrepancies slightly decrease when the temperature increases. In the CAB model, the use of diffusion instead of free gas for molecular translation allows to better reproduce the experimental data measured below 1 meV. Detailed comparisons with experimental data are given in reference [23].
In the thermal energy range, the magnitude of the differences does not change over the temperature range between 6 °C to 80 °C. The discrepancies between JEFF3.1.1 and the CAB model reach ≃5% at 25.3 meV. In the case of ENDF/BVII.1, the discrepancies remain smaller or equal to 2.5%.
Fig. 8 Comparison of the total crosssections of H in H_{2}O calculated at 10 °C and 80 °C. The discrepancy bands of ±2.5% and ±5% are shown to appreciate the discrepancy between the models. 
4.2 Reactivity excess as a function of temperature
The interpretation of the three MISTRAL configurations was performed with the JEFF3.1.1 library. As shown in the flowchart of Figure 3, we have only replaced the thermal scattering laws of H in H_{2}O by those calculated with the processing scheme presented in Sections 3.2 and 3.3. Results reported in Table 4 represent the differences in reactivity Δρ(T), where Δ indicates the deviation from the experimental values. Contributions of the experimental uncertainties and those coming from the MonteCarlo calculations are taken into account separately. For each configuration the statistical uncertainty due to the MonteCarlo calculations is close to ±2 pcm. The experimental uncertainties account for uncertainties that mainly come from the kinetic parameters, the measurements of the doubling time and of the boron concentration. The final experimental uncertainty ranges from ±10 pcm to ±25 pcm. In the present work, the contribution of the kinetic parameters to the final uncertainty is small because we have used β_{eff} values which have been measured during the MISTRAL1 and MISTRAL2 programs. Results reported in reference [32] are 788 ± 12 pcm for MISTRAL1 (UOX core) and 370 ± 6 pcm for MISTRAL2 (MOX core). Technological uncertainties are not included. For the EOLE facility, the magnitude of such uncertainties is close to ±200 pcm.
The top plot of Figure 9 shows the Δρ(T) values obtained for the MISTRAL1 experiment as a function of the temperature. Using the JEFF3.1.1 library, we observe a slight overestimation of the core reactivity (+192 pcm at 20 °C). Compared to JEFF3.1.1, the thermal scattering laws of ENDF/BVII.1 and those from the CAB model increase the calculated reactivity by respectively +65 pcm and +100 pcm on average.
The middle and bottom plots of Figure 9 show the Δρ(T) values obtained for the MISTRAL2 and MISTRAL3 experiments. Using the JEFF3.1.1 library, the MISTRAL2 core reactivity is overestimated by +732 pcm at 20 °C. As for MISTRAL1, the calculated reactivity increases when the thermal scattering laws of ENDF/BVII.1 and those from the CAB model are used. For MISTRAL2, the mean differences are +80 pcm and +180 pcm respectively. Similar trends are obtained for MISTRAL3 (+60 pcm and +140 pcm). Larger differences are reached because MOX fuel calculations are more sensitive to the thermal scattering laws of hydrogen in light water. One of the relevant results is the sizeable overestimation of the experimental reactivity by the calculations. It reaches Δρ ≃ 900 pcm when the thermal scattering laws calculated with the CAB model are used. Such an integral trend could be attribuated to the americium crosssections. New experimental works seem to indicate that the thermal capture crosssection and the capture resonance integral of ^{241}Am could be overestimated in JEFF3.1.1 by 15% and 20%, respectively [33]. A new ^{241}Am evaluation was included in the latest version of the JEFF library (JEFF3.2) for improving the calculations of the k_{eff} values [34]. In the case of the MISTRAL programs, this new ^{241}Am evaluation provides improved Δρ values at room temperature ranging from 200 pcm to 300 pcm [35].
Each component of the frequency spectrum (Eq. (10)) was investigated in order to understand the origin of the increase of the reactivity (few tens of pcm) when the ENDF/BVII.1 and CAB models are used. This increase can be explained by two competitive behaviors, which can be observed in Figure 8. The increase of the calculated reactivity is mainly connected to the decrease of the total crosssection between 0.01 eV and 1 eV, which is partially compensated by an increase of the total crosssection between 0.001 eV and 0.01 eV. Above 0.01 eV, the total crosssection is driven by the continuous part of the frequency spectrum ρ_{c}(β) (Fig. 1), indicating that the intermolecular vibrations have a major contribution in the transport of neutrons in the moderator.
To obtain a curve of the excess reactivity as a function of the temperature in units of degree Celsius, results in pcm were fitted with an empirical function. Linear [4], quadratic [19] or cubic [16] polynomials are used in the literature:(12)where the coefficients a_{i} are free parameters. In the present work, a systematic study is reported as a function of the degree of the polynomial (n = 1, 2, 3). The fitting algorithm of the CONRAD code was used [36]. A Chisquare test provides a measure of the goodnessoffit. It was used to select the optimal degree of the polynomial. Table 5 reports the final Chisquare values provided by the CONRAD code for n = 1, 2, 3. The originality of our work is the simultaneous analysis of the Δρ(T) values calculated for the MISTRAL2 and MISTRAL3 programs. The calculations were performed by introducing a free normalization factor which does not change the shape of Δρ as a function of the temperature. This approach aims to provide a global trend for the MOX configurations.
For MISTRAL1, a quadratic polynomial (n = 2) gives a rather good description of the Δρ(T) results. A different trend is observed for MOX fuel. A simple linear fit (n = 1) of the MISTRAL2 and MISTRAL3 results provides better Chisquare values than a cubic polynomial. Best fit curves are reported in Figure 9. The corresponding polynomial coefficients a_{i} are given in Table 6. The quoted uncertainties account for the statistical uncertainties in order to quantify the contribution of the MonteCarlo calculations only. The propagation of the experimental uncertainties was already addressed in the frame of the previous analysis performed with the deterministic code APOLLO2 [16,17].
Differences in reactivity Δρ = C − E (in pcm) obtained with the thermal scattering laws of JEFF3.1.1, ENDF/BVII.1 and calculated with the CAB model for the MISTRAL1, MISTRAL2 and MISTRAL3 configurations. The statistical uncertainty due to the MonteCarlo calculations is ±2 pcm.
Fig. 9 Differences in reactivity Δρ(T) obtained with the thermal scattering laws of JEFF3.1.1, ENDF/BVII.1 and calculated with the CAB model for the MISTRAL1, MISTRAL2 and MISTRAL3 configurations. The solid lines represent the best fit curves calculated with the CONRAD code [36]. 
4.3 Calculation errors on the reactivity temperature coefficient.
The temperature effect on the reactivity can be expressed by the temperature coefficient α_{iso}, defined as the change in reactivity due to a change in temperature. The deviation from the experimental values is given by the derivative of equation (12) with respect to the temperature:(13)where Δα_{cor} is a correction factor introduced to account for the thermal expansion of the materials. Such a correction was applied in the previous interpretation of the MISTRAL1 experiment with the deterministic code APOLLO2 [16]. In the present work, we decided to use a similar strategy for a better comparison of the MISTRAL1 results.
Temperature variation produces a thermal expansion of the fuel pellet, clad and grid that will have an impact on the effective multiplication factor. The lattice pitch will increase with the temperature, modifying the moderation ratio. Consequently, the contribution of the resonance absorption will decrease and the resonance escape probability will increase. The aluminum overclad will have an opposite effect because its objective is to remove moderator, compensating the increase in the moderation ratio. UOX and MOX oxides have a lower thermal expansion coefficient than aluminum. They are characterized by a volume change of the order of 0.3% between 5 °C and 80 °C, which has a slight impact on the resonance absorption. For MISTRAL1, the thermal expansion of the materials was calculated from a linear fit based on four temperatures (6 °C, 20 °C, 40 °C and 80 °C). The deduced correction is:
The present result is twice as large as the correction found in the previous interpretations performed with the APOLLO2 code. Unfortunately, no obvious explanations were found for understanding the differences. For MISTRAL2 and MISTRAL3, the calculated reactivity includes the thermal dilatation effects and no correction is needed (Δα_{cor} = 0).
The calculation errors on RTC are summarized in Table 7. Results are averaged over broad temperature intervals between the temperature T_{1} and T_{2}:(14)
The present work provides the first interpretation of the RTC errors with the MonteCarlo code TRIPOLI4^{®}. Previous calculations were performed with the deterministic code APOLLO2. For the UOX configuration (MISTRAL1), we observe differences of about 0.8 pcm/°C between the APOLLO2 and TRIPOLI4^{®} results. The origin of such a systematic bias is hard to explain, especially since a better agreement is achieved for the MOX configurations (MISTRAL2 and MISTRAL3). However, results averaged over the broad temperature range [10 °C–80 °C] remain consistent with those reported in reference [17] and still confirm that the calculation errors are lower (UOX core) or nearly equal (MOX core) to the target accuracy of 1 pcm/°C:
The comparison of the TRIPOLI4^{®} results indicates that the thermal scattering laws of H in H_{2}O of the JEFF3.1.1 and ENDF/BVII.1 libraries provide similar Δα_{iso}(T) values. For the three MISTRAL configurations, no substantial differences can be observed between the calculation errors at low and high temperature, where spectral effects and water density effects dominate, respectively.
For the UOX configuration, an improvement is achieved above 40 °C with the CAB model. As shown in Figure 10, this improvement reaches +0.6 pcm/°C at 80 °C. As a result, the calculation errors on RTC over the broad temperature range [10 °C–80 °C] are close to zero when the thermal scattering law of H in H_{2}O from the CAB model is used:
For the MOX cores, the CAB model for H in H_{2}O leads to a substantial improvement of the calculation errors on RTC. Over the temperature range of interest, we obtain:
Although the magnitude of the improvement, close to +0.3 pcm/°C, is similar to the experimental uncertainty, it remains significant compared to the statistical uncertainty of +0.02 pcm/°C coming from the MonteCarlo calculations. The observed differences between the CAB model and the other libraries are not due to a statistical bias. They point out a positive impact of the CAB model on the RTC calculations for MOX fuel.
The results reported for UOX fuel confirm the conclusions reported in reference [19] concerning the nonnegligible contribution of the thermal scattering crosssection of Hydrogen on RTC calculations. Through the interpretation of the MISTRAL1 configuration, we observe the impact of the thermal scattering laws of H in H_{2}O on the water density effects, increasing with the temperature. Such effects are of the same order of magnitude as the experimental uncertainties and their contributions to the calculation errors Δα_{iso} are similar to other nuclear data, such as the shape of the thermal crosssections of the fissile isotopes [37,38].
Summary of the calculation errors Δα_{iso} (in pcm/°C) for the MISTRAL experiments obtained with the MonteCarlo code TRIPOLI4^{®}. Our results are compared with those obtained with the deterministic code APOLLO2 [17]. The experimental uncertainties are also given in pcm/°C. The contribution of the statistical uncertainties due to the MonteCarlo calculations (±0.02 pcm/°C) is negligible.
Fig. 10 Calculation errors on RTC as a function of the temperature for the MISTRAL1 experiments. The uncertainty bands account for the statistical uncertainty of the MonteCarlo calculations. 
5 Conclusions
A 3D model of the EOLE reactor by using the TRIPOLI4^{®} MonteCarlo code was used for the first time to achieve the interpretation of the RTC experiments performed in the MISTRAL1, MISTRAL2 and MISTRAL3 configurations as a function of the temperature. This approach has not only confirmed previous results established with the deterministic code APOLLO2 but also provided new integral trends in relation with the thermal scattering laws of hydrogen bound to H_{2}O.
The comparison of the excess of reactivity calculated with three different sets of thermal scattering laws (JEFF3.1.1, ENDF/BVII.1 and CAB model) shows the impact of the intermolecular vibration modes of the water molecule in the neutron transport. The decrease of the translational mode in favor to the rotational mode leads to an increase of the calculated reactivity that can reach +180 pcm when the CAB model is used for the interpretation of the MISTRAL2 configuration (MOX fuel).
In the whole temperature range of interest for this work [10 °C–80 °C], the calculation error on RTC for a standard UOX lattice is close to −0.4 pcm/°C when the JEFF3.1.1 library is used. Such a bias vanishes and becomes closer to zero (Δα_{iso} =−0.02 pcm/°C) when the thermal scattering laws are replaced by those generated with the CAB model. This result indicates that the spectral component of the error in the RTC as well as the water expansion effects are well described. For MOX fuel configurations, the calculation error on RTC is of the order of −1.0 pcm/°C by using the JEFF3.1.1 library. A similar trend is reached when the thermal scattering laws are replaced by those of the ENDF/BVII.1 library. Our MonteCarlo calculations show a slight reduction of the bias with the thermal scattering laws coming from the CAB model. The calculation error on RTC becomes closer to −0.7 pcm/°C. Such an improvement (+0.3 pcm/°C) is of the same order of magnitude as the uncertainty of the ^{239}Pu thermal crosssection shapes.
Results obtained with the CAB model aim at demonstrating the interest of using Molecular Dynamic simulations for producing reliable thermal scattering laws of hydrogen bound in light water. In cold operating conditions at atmospheric pressure, Molecular Dynamic simulations seem to provide better S(α, β) tables at temperatures where the change of water phase becomes relevant.
Acknowledgments
Thanks are addressed to Olivier Litaize and Yannick Peneliau, from the Nuclear Data group of CEA Cadarache, for the valuable discussions and their relevant advices during the interpretation of the MISTRAL programs with the MonteCarlo code TRIPOLI4^{®}.
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Cite this article as: Juan Pablo Scotta, Gilles Noguere, David Bernard, Jose Ignacio Marquez Damian, Alain Santamarina, Impact of the thermal scattering law of H in H_{2}O on the isothermal temperature reactivity coefficients for UOX and MOX fuel lattices in cold operating conditions, EPJ Nuclear Sci. Technol. 2, 28 (2016)
All Tables
Excess of reactivity calculated with the TRIPOLI4^{®} code for the MISTRAL1 configuration at 20 °C. The differences C_{i} − C_{1} are calculated by using the result obtained with the official T4 library as reference.
Differences in reactivity Δρ = C − E (in pcm) obtained with the thermal scattering laws of JEFF3.1.1, ENDF/BVII.1 and calculated with the CAB model for the MISTRAL1, MISTRAL2 and MISTRAL3 configurations. The statistical uncertainty due to the MonteCarlo calculations is ±2 pcm.
Summary of the calculation errors Δα_{iso} (in pcm/°C) for the MISTRAL experiments obtained with the MonteCarlo code TRIPOLI4^{®}. Our results are compared with those obtained with the deterministic code APOLLO2 [17]. The experimental uncertainties are also given in pcm/°C. The contribution of the statistical uncertainties due to the MonteCarlo calculations (±0.02 pcm/°C) is negligible.
All Figures
Fig. 1 Comparison of the continuous and discrete frequency spectrum for H in H_{2}O at 294 K. 

In the text 
Fig. 2 Radial crosssections of the MISTRAL1 (750 UOX fuel pins), MISTRAL2 (1572 MOX fuel pins) and MISTRAL3 (1388 MOX fuel pins) cores. For MISTRAL2, the given core is the configuration at 20 °C. 

In the text 
Fig. 3 Flowchart of the calculation scheme used to produce S(α, β) tables for the TRIPOLI4^{®} code [20]. The processing of the S(α, β) tables from the three TSL data files of interest for this work is performed with the NJOY code [25]. The crosssections of the JEFF3.1.1 library is used for the neutron transport and only S(α, β) of light water are replaced by taking the needed information from alternatively the JEFF3.1.1, ENDF/BVII.1 and CAB libraries. The CADTOOL package [31] provides an easytouse interface for automated sequential processing schemes. 

In the text 
Fig. 4 Comparison of the ^{1}H and H in H_{2}O total crosssections calculated at T = 300 K. The thermal energy cutoff is equal to 4.95 eV. Then, TRIPOLI4^{®} uses the Sampling of the Velocity of the Target nucleus (SVT) up to T_{max} = 400k_{B}T. Beyond this energy, the static Assymptotic Kernel (AK) approximation is applied. 

In the text 
Fig. 5 Interpolation of the model parameters established by Mattes [28] between 6 °C to 80 °C. The meaning of each parameter is given in Table 2. 

In the text 
Fig. 6 Continuous frequency spectra for H in H_{2}O for the broad temperature mesh of the JEFF1.1.1 library (top plot) and interpolated over a fine temperature mesh (bottom plot). 

In the text 
Fig. 7 Total crosssection of H in H_{2}O calculated with the broad temperature mesh of the JEFF1.1.1 library (20 °C, 50 °C and 100 °C) and interpolated over a fine temperature mesh (from 6 °C to 80 °C). 

In the text 
Fig. 8 Comparison of the total crosssections of H in H_{2}O calculated at 10 °C and 80 °C. The discrepancy bands of ±2.5% and ±5% are shown to appreciate the discrepancy between the models. 

In the text 
Fig. 9 Differences in reactivity Δρ(T) obtained with the thermal scattering laws of JEFF3.1.1, ENDF/BVII.1 and calculated with the CAB model for the MISTRAL1, MISTRAL2 and MISTRAL3 configurations. The solid lines represent the best fit curves calculated with the CONRAD code [36]. 

In the text 
Fig. 10 Calculation errors on RTC as a function of the temperature for the MISTRAL1 experiments. The uncertainty bands account for the statistical uncertainty of the MonteCarlo calculations. 

In the text 