Second-Order Sensitivity Analysis of Uncollided Particle Contributions to Radiation Detector Responses

Abstract

-This work presents an application of Cacuci's Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to the simplified Boltzmann equation that models the transport of uncollided particles through a medium to compute efficiently and exactly all of the first-and second-order derivatives (sensitivities) of a detector's response with respect to the system's isotopic number densities, microscopic cross sections, source emission rates, and detector response function. The off-the-shelf PARTISN multigroup discrete ordinates code is employed to solve the equations underlying the 2nd-ASAM. The accuracy of the results produced using PARTISN is verified by using the results of three test configurations: (1) a homogeneous sphere, for which the response is the exactly known total uncollided leakage, (2) a multiregion two-dimensional (r-z) cylinder, and (3) a two-region sphere for which the response is a reaction rate. For the homogeneous sphere, results for the total leakage as well as for the respective first-and second-order sensitivities are in excellent agreement with the exact benchmark values. For the nonanalytic problems, the results obtained by applying the 2nd-ASAM to compute sensitivities are in excellent agreement with central-difference estimates. The efficiency of the 2nd-ASAM is underscored by the fact that, for the cylinder, only 12 adjoint PARTISN computations were required by the 2nd-ASAM to compute all of the benchmark's 18 first-order sensitivities and 224 second-order sensitivities, in contrast to the 877 PARTISN calculations needed to compute the respective sensitivities using central finite differences, and this number does not include the additional calculations that were required to find appropriate values of the perturbations to use for the central differences.Keywords -Second-order adjoint sensitivity analysis, particle and radiation transport, response variance and skewness.Note -Some figures may be in color only in the electronic version.

Application of the Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology to Compute First- and Second-Order Sensitivities of Flux Functionals in a Multiplying System with Source

“…The present work extends significantly the results presented in Ref. 17 by applying the 2nd-ASAM to the neutron transport equation that models a multiplying subcritical system comprising a nonfission neutron source. Section II of this work recalls succinctly the Boltzmann transport equation describing the transport of neutrons within a finite multiplying medium with an internal nonfission source, defining this physical system's parameters and responses.…”

“…In particular, the results in Ref. 17 show that only 12 largescale adjoint particle transport computations were required by using the 2nd-ASAM to compute all of the detector's response to the flux of uncollided particles for the 18 first-order sensitivities and 224 second-order sensitivities, in contrast to the 877 large-scale forward particle transport calculations needed to compute the respective sensitivities using central finite differences, and this number did not include the additional calculations that were required to find appropriate values of the perturbations to use for the central differences.…”

“…(22) in conjunction with the boundary conditions given in Eq. (17) to construct the following 1st-LASS for the first-level adjoint function ψ 1 ð Þ r; Ω ð Þ:…”

“…(127) can be computed only after having obtained the solution δφ r; Ω; E ð Þof the 1st-LFSS defined in Eqs. (16) and (17). To avoid the need for solving the 1st-LFSS, the indirect-effect term defined in Eq.…”

Illustrative Application of the 2<sup>nd</sup>-Order Adjoint Sensitivity Analysis Methodology to a Paradigm Linear Evolution/Transmission Model: Point-Detector Response

“…The application of the general second-order adjoint sensitivity analysis metho-dology presented in [1] is illustrated in this work by means of a simple mathematical model which expresses a conservation law of the model's state function. This paradigm model is representative of transmission of particles and/or radiation through materials [2] [3], chemical kinetics processes [4] [5], radioactive decay modeled by the Bateman equation, etc.…”