Assimilation of integral experiments on high-energy nuclear parameters

. Current assimilation of integral experiments often consists in adjusting multi-group cross sections with feedbacks from critical reference benchmarks. In order to maintain the constraints coming from nuclear models, we present here a method to achieve assimilation of integral experiment on nuclear parameters, from which nuclear data are evaluated. This method, based on Bayesian inference, uses continuous energy reactivity sensitivities to nuclear parameters, throughout all the nuclear data types (cross section, angular distribution, energy distribution, fission multiplicity and spectrum). This improvement was made possible by coupling a stochastic transport code and a nuclear data evaluation code. The study of a test case – the assimilation of Jezebel ICSBEP benchmark on a plutonium-239 toy evaluation – shows that angular and energy distributions have a non-negligible impact on the assimilation process and results.


Introduction
Nuclear data evaluation relies, in part, on adjustments of nuclear parameters on differential experiment data. To add another source of information, one can use the data from integral experiments. Usually, this process is applied on multi-group cross sections, using deterministic transport code. Unfortunately, this may deteriorate physical constraints coming from the nuclear models, used to evaluate nuclear data. To get the best adjustment, and maintain the physical correlations of nuclear data, one has to work directly with nuclear parameters: Thus, we can take into account the influence of parameters not only on cross sections, but also on angular (AD) and energy (ED) distributions, when these latter are obtained from the same set of parameters. Above all, to exclude multi-group discretization bias, continuous energy data should be used. Previous works have successfully extended the assimilation process to nuclear parameters, and even using continuous energy [1,2]. However, these were limited to the influence of cross-sections, and continuous energy calculations were limited to specific nuclear parameters (resonance parameters). A recent coupling [3] between the nuclear reaction code CONRAD [4] and a development version of the stochastic code TRIPOLI-4 ® [5] allows calculating full sensitivities of integral experiment reactivity to nuclear parameters, with contributions from all nuclear data, without energy and models restrictions [6,7]. With such tool, we are now able to adjust nuclear parameters and phenomenological models on integral experiment results, and especially here, criticality experiments reactivity. * e-mail: elias.vandermeersch@cea.fr

Complete feedback on nuclear parameters
The assimilation of criticality assimilation may be achieved with Bayesian inference, where one updates prior knowledge of nuclear parameters with results from integral experiments. With the help of the CONRAD code, such problem is here linearized and solved by iterations of a Newton method. Stochastic methods can be used to solve this type of problem and avoid linearization, but with a high computational cost, particularly for the adjustment of nuclear parameters. Here, using information about initial nuclear parameters values ⃗ x prior , their correlations M prior x , the expected integral experiments reactivities ⃗ y, the reactivities sensitivities S to nuclear parameters, correlations between the integral experiments M y , and reactivity estimation ⃗ t, one tries to find a set of parameters ⃗ x that minimizes χ: (1) For each iteration, the nuclear parameters values, their correlations, the sensitivities and the estimated reactivity are updated.
Reactivity estimation ⃗ t is obtained here by a stochastic transport simulation.
In the same time, CONRAD/TRIPOLI-4 ® weak coupling compute reactivity sensitivity to nuclear parameters: Indeed, the sensitivity S of the reactivity ρ to the nuclear parameter Γ may be expressed as: with σ, AD and ED the cross sections, angular and energy distributions, and r the neutron induced nuclear reactions. For prompt neutron multiplicity ν p and neutron spectrum nuclear parameters χ p : One should note that only σ, AD and ED share the same models, meaning that Γ, λ and T are completely separated and non interfering nuclear parameters. The CON-RAD code computes nuclear data sensitivities to nuclear parameters, and TRIPOLI-4 ® reactivity sensitivities to nuclear data, using perturbation theory with the Iterated Fission Probability method [8]. 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 into account all the nuclear data.

Application case on a plutonium-239 evaluation
This work focuses on the ICSPBEP benchmark Jezebel [9], a bare sphere of plutonium and gallium alloy, producing a fast neutron spectrum. A plutonium-239 toy evaluation is generated with CONRAD, from a set of nuclear parameters and their associated models. As CONRAD  cannot produce delayed neutron data, these have been extracted from JEFF-3.1.1 [10], and will be excluded during the adjustment. Regarding prompt neutron multiplicity, a simple linear model has been implemented, using two parameters: the interception (thermal value) and the slope of the linear curve. Such a model underestimates the prompt neutron plutonium-239 multiplicity compared to JEFF-3.1.1 evaluation, as shown in figure 1. Finally, Optical Model Potential (OMP) parameters, from Morillon-Romain [11] parameterization, have been fitted on Poenitz differential experiments, in order to obtain some initial correlations.
We focused on the nine most effective parameters, estimated here with a variation of the standard regression coefficient (SRC), defined by:  Figure 3: Studied parameters' initial correlation matrix.
With x the nuclear parameter of interest and δx its standard deviation. Based on the results presented in figure 2, it appears that Jezebel assimilation will mostly have an impact on Optical Model, prompt fission spectrum parameters, and thermal value of the prompt neutron multiplicity. We added the slope of the multiplicity to the set of parameters studied, to allow more flexibility on the simplest model of our evaluation. All the parameters selected, their value and their associated nuclear data are recalled in table 1. The initial correlation of the parameters is depicted in figure 3: at this step, there is no link between models: For example, fission spectrum parameters are not correlated to Optical Model or Fission parameters. Regarding multiplicity, parameters are, for now, considered uncorrelated.
The first assimilation of Jezebel's reactivity was conducted using full sensitivities (through cross section, an-gular distribution (AD) and energy distribution (ED), and all nuclear data are updated at each iteration. Results are presented in table 2, and the final parameter correlations in figure 4.
The expected multiplication factor (k eff ) for Jezebel is 1.00000 ± 110 pcm [9]. Focusing on the first case, the assimilation manages to fit the parameters so that the Jezebel simulated k eff reaches the expected value. Most of the changes concern prompt spectrum fission spectrum, as expected from Jezebel characteristics, and because of the parameters large uncertainties. Correlations are altered, for instance an anti-correlation between our multiplicity parameters appears. Regarding links between models, they appear mostly for the parameters highly modified by the assimilation.
This assimilation leads to a probable over-estimate prompt neutrons multiplicity, particularly in the thermal energy range (in comparison to JEFF-3.1.1 value, cf. figure 1). The assimilation manages to find a balance that fits our experiment, however, some constraints such as our linear multiplicity model may lead to a non-realistic evaluation. Such problem may be corrected by using a proper multiplicity model, fitted on differential experiments, a stochastic solving of the Bayesian problem, and assimilate multiples experiments in the same process.

Angular and energetic distribution impact
In the second case, feedbacks from prompt multiplicity and fission spectrum are used and updated at each iteration. However, for fission and optical model parameters (used for σ, AD, ED), only cross sections feedback is used and updated. This means that angular and energy distributions remain unchanged during the process. Once again, the assimilation process manages to find a balance, with a k eff within the expected experimental  range. The correlation matrix, depicted at the right of figure 4, is very similar to the one obtained with a full assimilation. Interestingly, by removing some freedom of action on the Optical Model and Fission parameters, we mainly influenced the parameters of other models: As in our evaluation, flexibility is primarily offered by prompt multiplicity and fission spectrum parameters, assimilation compensates for the lack of freedom on these parameters. Such example illustrates how the current method, based only on cross section adjustment, may wrongly fit some parameters, by removing degrees of freedom in the assimilation process.

Conclusion
This weak coupling between a nuclear data and a stochastic transport code allowed to assimilate reactivity to nuclear parameters, with feedback through all nuclear data and using continuous energy. A simple test case, with the Jezebel benchmark, shows the impact of neglecting angular and energy distributions. Further work is underway to achieve a strong coupling, in order to avoid formatting nuclear data and facilitate the assimilation process. Such coupling will also ease the development of more complex assimilation methods, such as those based on the Generalized Perturbation Theory, in order to achieve assimilation of reaction rates for example. At the same time, CONRAD evaluation capabilities will be extended, in order to work on more complete evaluation data.