Second-Order Analysis: Effects of Total Cross Sections

Abstract

This chapter presents the application of the 2nd-CASAM-L to compute efficiently and exactly the second-order sensitivities of the PERP model’s leakage response to the model’s group-averaged total microscopic cross sections. It is found that among the total of 32400 = 180 × 180 second-order sensitivities, 720 of these elements have relative sensitivities greater than 1.0, and many of the second-order sensitivities are much larger than the corresponding first-order ones. The largest second-order sensitivities involve the total cross sections of ²³⁹Pu and ¹H. The overall largest element is the unmixed second-order relative sensitivity ${S}^{(2)}\left({\sigma}_{t,6}^{30},{\sigma}_{t,6}^{30}\right)=429.6$, which occurs in the lowest-energy group for ¹H. Neglecting the second-order sensitivities would cause an erroneous reporting of the response’s expected value and also a very large nonconservative error by underreporting of the response variance. For example, if the parameters were uncorrelated and had a uniform standard deviation of 10%, neglecting second- (and higher-) order sensitivities would cause a nonconservative error by underreporting of the response variance by a factor of 947%. If the cross sections were fully correlated, neglecting the second-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. In all cases, neglecting the second-order sensitivities would erroneously predict a Gaussian distribution in parameter space (for the PERP leakage response) centered about the computed value of the leakage response. In reality, the second-order sensitivities cause the leakage distribution in parameter space to be skewed toward positive values relative to the expected value, which, in turn, is significantly shifted to much larger positive values than the computed leakage value. The effects of the second-order sensitivities underscore the need for obtaining reliable data for correlations that might exist among the total cross sections; such data is unavailable at this time.