ON THE IMPORTANCE OF SECOND-ORDER RESPONSE SENSITIVITIES TO NUCLEAR DATA IN REACTOR PHYSICS UNCERTAINTY ANALYSIS

This invited keynote presentation compares the relative importance of 1 st -order versus 2 nd order sensitivities of the leakage response of an OECD/NEA benchmark (polyethylene-reflected plutonium sphere) to the nuclear data characterizing this benchmark. The imprecisely known parameters underlying the neutron transport computational model for this benchmark include 180 group-averaged total microscopic cross sections, 21600 group-averaged scattering microscopic cross sections, 60 parameters describing the fission process, 30 parameters describing the fission spectrum, 10 parameters describing the system’s sources, and 6 isotopic number densities. Thus, this benchmark comprises 21886 1st-order sensitivities of the leakage response with respect to the model parameters, and 478,996,996 2nd-order sensitivities, of which 239,509,441 are distinct. The exact deterministic computation of all of these 1st- and 2nd-order sensitivities was made possible by the application of the Second-Order Adjoint Sensitivity Analysis Methodology (2 nd -ASAM) developed by Cacuci. Thousands (out of the 32 400 elements) of the 2 nd -order sensitivities of the leakage response with respect to the total cross sections turned out to be significantly larger than the largest corresponding 1st-order sensitivities, contrary to some previously held beliefs in the reactor physics community. Hence, it will be shown that neglecting the 2nd-order sensitivities to total cross sections would cause very large non-conservative errors by under-reporting the response’s variance and expected value. The 2nd-order sensitivities also cause the response distribution to be skewed towards positive values relative to the expected value, which, in turn, is significantly larger than the computed value of the leakage response. The result presented in this paper also underscore the need for obtaining reliable cross section covariance data, which are not available at this time.


INTRODUCTION
Several important subcritical fundamental physics benchmarks within the Nuclear Energy Agency (NEA) International Criticality Safety Benchmark Evaluation Project (ICSBEP) Handbook [1] use as particle source a 4.5 kg alpha-phase plutonium sphere colloquially known as the "BeRP ball," which was originally constructed at Los Alamos National Laboratory in 1980 [2] for conducting experiments aimed at estimating the reactivity worth of beryllium reflectors (hence the acronym "BeRP" for "berylliumreflected plutonium"). Subsequent experiments have used the BeRP ball as the source for other subcritical experiments (accepted in the ICSBEP book as fundamental physics benchmarks) using tungsten, nickel or polyethylene reflectors around the BeRP ball. In their computational evaluation of neutron multiplicity measurements for the "polyethylene-reflected BeRP" ball, Miller et al. [3] showed that the computational results disagreed significantly with the corresponding measurements of neutron multiplicity and concluded that "only a subtle variation in the value of the average number of neutrons produced per fission" for 239Pu was able to improve the simulations of the plutonium sphere…" However, the "polyethylene-reflected BeRP ball" benchmark contains many imprecisely known nuclear data parameters, in addition to those that were incidentally investigated in [3]. Therefore, the sensitivities to all of the benchmark's nuclear data would need to be quantified in order to assess the uncertainties in the computed responses that are, in turn, compared to measurements. For this purpose, Cacuci's 2 nd -ASAM for subcritical systems [4] has been applied to compute all of the 1 st -and 2 nd -order sensitivities of the leakage response of this benchmark with respect to the nuclear data that characterizes this benchmark. This work summarizes the most significant results obtained for the 1 st -and 2 nd -order sensitivities of the "polyethylene-reflected BeRP ball" benchmark's leakage response with respect to the benchmark's groupaveraged isotopic total cross sections, highlighting the finding that many of the 2 nd -order sensitivities are much larger than the corresponding 1 st -order sensitivities. The 2 nd -order sensitivities shift the leakage response's expected value significantly away from the computed value of the leakage response, and cause asymmetries in the response distribution. Furthermore, the effects of the 2 nd -order leakage response sensitivities with respect to the isotopic total cross sections on the leakage responses expected value, variance, and skewness, are much larger than the corresponding effects stemming from the 1st-order sensitivities.

LARGEST FIRST-AND SECOND-ORDER SENSITIVITIES OF THE PERP BENCHMARK'S LEAKAGE WITH RESPECT TO PERP'S NUCLEAR DATA
The "polyethylene-reflected plutonium sphere," which will henceforth be abbreviated as "PERP," contains two materials designated as "material 1" and "material 2". Material 1 (core) contains the following isotopes: 239 Pu, 240 Pu, 69 Ga, 71 Ga. Material 2 (reflector) contains the following isotopes: C and 1 H. The neutron flux distribution within the PERP benchmark is computed by using the PARTISN [5] multigroup discrete ordinates particle transport code. The specific computations in this work were performed using a P3 Legendre expansion of the scattering cross section, an angular quadrature of S256, and a fine-mesh spacing of 0.005 cm (comprising 759 meshes for the plutonium sphere of radius of 3.794 cm, and 762 meshes for the polyethylene shell of thickness of 3.81 cm). The PARTISN [5] computations used the MENDF71X [6] 618-group cross section data collapsed to 30 G = energy groups. The details of these computations are presented in [7]. Since the total leakage is physically more meaningful than count rates (because it does not depend on the detector configuration), the total leakage of the PERP benchmark, which will be denoted as ( ) L α , will be considered as the response of interest for sensitivity analysis. The leakage response displayed the largest 1 st -and 2 nd -order sensitivities with respect to the group-averaged total microscopic cross sections for 1 H, as presented in Tables 1 and 2 (see Ref. 7 for complete results for all isotopes).

UNCERTAINTY QUANTIFICATION
Correlations among the group total cross sections are not available for the PERP benchmark. Two extreme situations can be considered, as follows: (i) all cross sections are uncorrelated, which will be considered in Section 3.1 below; (ii) all cross sections are fully correlated, which will be considered in Section 3.2 below.

Uncorrelated Total Microscopic Cross Sections
Considering only the contributions from the group-averaged uncorrelated total microscopic cross sections, the expected value of the leakage response has the expression where the superscript "(2,U)" indicates "2 nd -order, uncorrelated" cross sections, the subscript "t" indicates contributions solely from the group-averaged total microscopic cross sections, and where: , s denotes the standard deviation associated with the group-averaged total microscopic cross sections , addition to being uncorrelated, the total microscopic cross sections are also normally-distributed, which will be indicated using the superscript "(U,N)" then the following expressions hold: (i) the variance, , of the leakage response, L , due to the variances of uncorrelated and normally distributed microscopic total cross sections is defined and hence the skewness 1 γ of the leakage response would vanish and the response distribution would by default be assumed to be Gaussian.

Fully Correlated Total Microscopic Cross Sections
The effects of correlations among the group-averaged microscopic total cross sections, and hence the impact of the second-order mixed sensitivities of the leakage response to these cross sections, can also be assessed in the extreme case of fully correlated cross sections. Thus, if the group-averaged microscopic total cross sections were fully correlated (denoted by using the superscript "FC", the expectation value of the leakage response would be given by the expression denotes the contributions from both the unmixed and mixed 2 nd -order sensitivities when the total cross sections parameters are fully correlated, and is given by the following expression: The additional 2 nd -order contributions to the expectation value of the leakage response when the total cross sections are fully correlated is denoted , where the superscript "MSC" denotes "mixed second-order correlated," and is computed using the following expression: The contributions stemming from the mixed 2 nd -order sensitivities when the total cross sections are fully correlated and normally distributed will be denoted as [ ] ( , ) , where the superscript "(MSC,N)" indicates "Mixed Second-order sensitivities, fully Correlated Normally distributed parameters." These The effects of the first-and second-order sensitivities on the response moments (expected value, variance and skewness) can be highlighted by considering uniform values for the standard deviations of the group-averaged isotopic total microscopic cross sections and using these values together with the respective sensitivities. The results thus obtained are presented in Table 4, for illustrative uniform relative standard deviations for the parameters of 5% and 10%, respectively, for the standard deviations , g t i s .  As indicated in Table 4, the trends displayed by the results for a uniform standard deviation of 5% for the group isotopic microscopic cross sections are amplified significantly when this uniform standard deviation is increased to 10%. Thus, the 2 nd -order contribution   Table 4 Table 4 also indicate the effect of the correlations induced by the mixed 2 nd -order sensitivities is to render the leakage response distribution to be more symmetrical about the response's expected value.

CONCLUSIONS
Even when the group-averaged microscopic total cross sections are uncorrelated, the results in Table 4 indicate that the importance of the 2 nd -order sensitivities relative to the importance of the 1 st -order ones increases as the parameters uncertainties increase. The effects of the 2 nd -order sensitivities are to increase the expected value of the response versus the computed response value, which shifts to positive values the distribution of the leakage response in parameters space. Also, the contributions of the 2 nd -order sensitivities to the response's variance overtake the contributions of the 1 st -order sensitivities to the response's variance already for relatively small (ca. 5%) parameter standard deviations. These effects are rapidly amplified when the parameters are less precisely known. In particular, it has been shown that, for a uniform standard deviation of 10%, the 2 nd -order sensitivities contribute 72% of the expected value for the leakage response The results presented in Table 4 also indicate that the mixed 2 nd -order sensitivities play a very significant role in determining the moments of the leakage response distribution for correlated cross sections. The importance of the mixed 2 nd -order sensitivities increases as the relative standard deviations for the cross sections increase. For fully correlated cross sections, for example, neglecting the 2 nd -order sensitivities would cause an error as large as 2000% in the expected value of the leakage response, and up to 6000% in the variance of the leakage response. Of course, neither the fully uncorrelated nor the fully correlated illustrative examples presented in this work realistically describe the actual physical situations regarding the total microscopic cross section parameters. The fully uncorrelated case underestimates reality while the fully correlated case overestimates it. In reality, total cross sections are partially correlated, so reality falls in between the fully uncorrelated and fully correlated cases. However, correlations among cross sections are scantly available at this time in the evaluated nuclear data files. The purpose of this work is to draw attention to the fact that 2 nd -order sensitivities are important, and their effects must be assessed for each physical system under consideration. In particular for the PERP benchmark, it has been shown that the 2 nd -order sensitivities are even more important than the 1 st -order ones. While the effects of the 2 nd -order sensitivities may be less marked for other reactor physics systems, the point is that they are not always negligible, as they have been considered in hitherto in the published literature. Furthermore, the effects of the mixed 2 nd -order sensitivities, as illustrated for the PERP benchmark in this work, underscore the need for future experimental research aimed at obtaining values for the correlations that might exist among the total cross sections, which are unavailable at this time.
The complete 2 nd -order sensitivity analysis of the PERP benchmark can be found in References 7 -12.