Speaker
Description
Nuclear data evaluation principally relies on adjustments of nuclear parameters on differential experiment data. To add another source of information, one may use integral experiments data. Usually, this process is applied on multi-group cross sections, with the help of deterministic transport code. However, nuclear data obtained are often biased toward the integral experiment studied, or the calculation scheme used. To obtain the best adjustment, and not deteriorate nuclear data physical correlations, one has to work directly with nuclear parameters. Such an assimilation needs to account for parameters influence not only on cross sections, but also angular (AD) and energy (ED) distributions. Most of all, to exclude all bias from multi-group discretization, continuous energy data has to be used.
Previous works [a] managed to extend the assimilation process to nuclear parameters, and even by using continuous energy [b]. However, these were restricted to cross-sections influence, and continuous energy calculations limited to specific nuclear parameters (resonance parameters). A recent coupling [c] between the nuclear reaction code CONRAD [d] and the stochastic code TRIPOLI-4 \textregistered \ [e] allows calculating full sensitivities of integral experiment reactivity to nuclear parameters, with contributions from all nuclear data, without energy and models restrictions. The CONRAD code is used to compute sensitivities of nuclear data to nuclear parameters, and TRIPOLI-4 \textregistered \ computes reactivity sensitivities to nuclear data. This second part uses perturbation theory, implemented here with the Iterated Fission Probability method. All these contributions are used to compute full sensitivity $S$ of reactivity $\rho$ to nuclear parameter $\Gamma$:
$S^\rho_\Gamma = \int_E \left[\frac{\partial \rho}{\partial \sigma(E)} \frac{\partial \sigma(E)}{\partial \Gamma} + \frac{\partial \rho}{\partial AD(E)} \frac{\partial AD(E)}{\partial \Gamma} + \frac{\partial \rho}{\partial ED(E)} \frac{\partial ED(E)}{\partial \Gamma}\right]\text{d}E.$
These sensitivities calculations are a main component of the assimilation process, and this work seeks to study the impact of a complete assimilation, where sensitivities are calculated by taking account of all the nuclear data, compared to a more classic multi-group cross section adjustment. For this work, the assimilation is achieved with Bayesian inference, solved by a Newton method (iterative Generalized Least Square (GLS)). Such a method needs information about parameters values $\vec{x}$, correlations between nuclear parameters ${M}_{x}^{\text{prior}}$, the expected integral experiments data $\vec{y}$, sensitivities $S$ of this data to nuclear parameters, correlations between the integral experiments ${M}_{y}$, and data estimation $\vec{t}$ obtained with the values of the studied parameters, so that, for each iteration $n$:
$M_{x}^{n} = [{M_{x}^{\text{prior}}}^{-1} + S_{n-1}^\text{T} M_{y}^{-1} S_{n-1}]^{-1},$
and:
$\vec{x}_n = \vec{x}_{n-1} + M_x^n [S^\text{T}_{n-1} M_y^{-1}(\vec{y}-\vec{t}_{n-1}) + M_x^{\text{prior}^{-1}} (\vec{x}_{n-1}-\vec{x}^\text{prior})],$
with $x^{\text{prior}}$ the initial nuclear parameters.
This work focus on the ICSPBEP benchmark Jezebel [f]: We used its reactivity to adjust the plutonium-239 Optical Model Potential (OMP) parameters from Morillon-Romain [g] parametrization, and fission-related parameters. From an initial plutonium-239 nuclear parameter evaluation, obtained after adjusting total and fission cross section over two sets of differential data, we realized two assimilations of Jezebel's reactivity: The first one uses the complete assimilation, with complete sensitivities, and all the nuclear data are updated at each iteration. The second assimilation is achieved by only working with cross section, to be the closest to the classic adjustment.
The expected multiplication factor ($k_\text{eff}$) expected from Jezebel is 1.00000 $\pm$ 200 pcm. The $k_\text{eff}$ obtained after a Monte-Carlo transport simulation, using the initial plutonium-239 evaluation, was 0.97634 $\pm$ 1 pcm. For the complete assimilation, the fitted parameters give a $k_\text{eff}$ of 0.99833 $\pm$ 1 pcm, and $k_\text{eff}$ = 0.99882 $\pm$ 1 pcm for the cross-section only adjustment. Both methods manage to fit Jezebel expected reactivity. Thus, the difference of 49 pcm highlights the importance of fitting nuclear parameters with full sensitivities. Regarding parameters correlations, presented in the figure below, we see an initial anticorrelation between OMP and fission parameters. Correlations matrix after assimilations are similar (small numerical differences), with a trend to decorrelate OMP parameters.
[Correlation matrix available here in case of indico error: https://imagizer.imageshack.com/v2/xq90/922/m5slFm.png]2
We are now attempting to compare the complete assimilation method to a determinist multi-group cross section adjustment, and to include neutron spectrum parameters in the adjustment, to fully characterize the impact of complete assimilation. The results, to be presented in the full paper, will be based on a more realistic initial plutonium-239 evaluation (the preliminary differential adjustment realized here was intentionally coarse, to accelerate computation time).
[a] C. De Saint Jean et al. “Evaluation of Cross Section Uncertainties Using Physical Constraints: Focus on Integral Experiments.” Nuclear Data Sheets, Volume 123, 2015, Pages 178-184, ISSN 0090-3752.
[b] V. Sobes et al. “Resonance Parameter Adjustment Basedon Integral Experiments.” Nuclear Science and Engineering,volume 183(3), pp. 347–355(2016).
[c] Elias Vandermeersch et al. “Two examples of recent advances in sensitivity calculations.” EPJ Nuclear Sci. Technol., 7 (2021) 13.
[d] P. Archier et al. “CONRAD Evaluation Code: Development Status and Perspectives.” Nuclear Data Sheets,volume 118, pp. 488–490 (2014)\newline
[e] E. Brun et al. “TRIPOLI-4® , CEA, EDF and AREVA reference Monte Carlo code.”Annals of Nuclear Energy,volume 82, pp. 151–160 (2015). Joint International Conference onSupercomputing in Nuclear Applications and Monte Carlo 2013, SNA + MC 2013.
[f] OECD-NEA. “International Handbook of Evaluated Criticality Safety Benchmark Experi-ments.“ Springer, City and Country (2018).
[g] B. Morillon and P. Romain. “Dispersive and global spherical optical model with a localenergy approximation for the scattering of neutrons by nuclei from1 keV to 200 MeV.” PhysRev C,volume 70, p. 014601 (2004).