Third Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: II. Computed Sensitivities

This work presents the results of the exact computation of (180) 3 = 5,832,000 third-order mixed sensitivities of the leakage response of a polyethy-lene-reflected plutonium (PERP) experimental benchmark with respect to the benchmark’s 180 microscopic total cross sections. This computation was made possible by applying the Third-Order Adjoint Sensitivity Analysis Methodology developed by Cacuci. The numerical results obtained in this work revealed that many of the 3 rd -order sensitivities are significantly larger than their corresponding 1 st - and 2 nd -order ones, which is contrary to the widely held belief that higher-order sensitivities are all much smaller and hence less important than the first-order ones, for reactor physics systems. In particular, the largest 3 rd -order relative sensitivity is the mixed sensitivity of the PERP leakage response with respect to the lowest energy-group (30) total cross sections of 1 H (“isotope 6”) and 239 Pu (“isotope 1”). These two isotopes are shown in this work to be the two most important parameters affecting the PERP benchmark’s leakage response. By comparison, the largest 1 st -order sensitivity is that of the PERP leakage response with respect to the lowest energy-group total cross section of

Methodology developed by Cacuci. The numerical results obtained in this work revealed that many of the 3 rd -order sensitivities are significantly larger than their corresponding 1 st -and 2 nd -order ones, which is contrary to the widely held belief that higher-order sensitivities are all much smaller and hence less important than the first-order ones, for reactor physics systems. In particular, the largest 3 rd -order relative sensitivity is the mixed sensitivity respect to the lowest energy-group (30) total cross sections of 1 H ("isotope 6") and 239 Pu ("isotope 1"). These two isotopes are shown in this work to be the two most important parameters affecting the PERP benchmark's leakage response. By comparison, the largest 1 st -order sensitivity is that of the PERP leakage response with respect to the lowest energy-group total cross section of isotope 1 H, having the value ( ) ( ) . The 3 rd -order sensitivity analysis presented in this work is the first ever such analysis in the field of reactor physics. The consequences of the results presented in this work on the uncertainty analysis of the PERP benchmark's leakage response will be presented in a subsequent work.

Introduction
The accompanying Part I [1] has reported the exact mathematical expressions of the 3 rd -order sensitivities of the leakage response of the OECD/NEA subcritical polyethylene-reflected plutonium (acronym: PERP) metal fundamental physics benchmark [2] with respect to the benchmark's group-averaged microscopic total cross sections. The exact mathematical expressions of these 3 rd -order sensitivities were derived by applying the general Third-Order Adjoint Sensitivity Analysis Methodology conceived by Cacuci [3]. This work will present the numerical results obtained by using the formulas derived in [1] for the (180) 3 third-order mixed sensitivities of the PERP's leakage response with respect to the benchmark's 180 group-averaged microscopic total cross sections. The numerical results obtained for the 3 rd -order relative sensitivities are then compared with the corresponding 1 st -and 2 nd -order ones, which have been computed and reported in [4] [5]. The magnitudes and distributions of the 3 rd -order mixed relative sensitivities will be illustrated in 3D plots. This work is organized as follows: Section 2 reports the numerical results for the 180 third-order "unmixed" sensitivities of the PERP's leakage response with respect to the microscopic total cross sections, comparing them with the corresponding 1st-and 2nd-order sensitivities. Section 3 presents the numerical results for the mixed third-order sensitivities, highlighting the magnitudes and distributions for the largest of these. Section 4 summarizes and highlights the significance of the pioneering results obtained in this work.

Numerical Results for Third-Order Unmixed Sensitivities of the PERP Leakage Response to Total Cross Sections
The characteristics of the OECD/NEA polyethylene-reflected plutonium (acronym: PERP) metal sphere benchmark for subcritical neutron and gamma measurements have been detailed in Part I [1] and in [4] [5]. The mathematical expression of the 3 rd -order mixed sensitivities ( ) 3 , , , 1, , of the PERP leakage response with respect to the group-averaged microscopic total cross sections has been derived in the accompanying Part I [1] and is reproduced in the Appendix, for convenient reference. In this Section, the computed values of the 3 rd -order unmixed relative sensitivities, i.e., ( ) ( ) ( )( ) , , 1, , 30, 1, , 6 g g g g g g t j t j t j t j t j t j L L S g j σ σ σ σ σ σ ∂ ∂ ∂ = =    . The term "unmixed" denotes the sensitivity of the PERP leakage response with respect to the same parameter. The numerical values for the 1 st -order and 2 nd -order unmixed relative sensitivities of PERP leakage response with respect to the benchmark's total cross sections have been computed and documented in [4].
Tables 1-6 present side-by-side comparisons of the unmixed sensitivities of order 1-3 for each of the six isotopes contained in the PERP benchmark. Table 1 presents the comparison of 1 st -order, 2 nd -order and 3 rd -order unmixed relative sensitivities for isotope 1 ( 239 Pu), for all energy groups 1, , 30 g =  . This comparison indicates that, for the same energy group, the absolute values of the 3 rd -order relative sensitivities are generally much larger than the corresponding values of both the 1 st -and 2 nd -order sensitivities. Specifically, for the energy groups 6, , 26 g =  and 30 g = , the values of the 3 rd -order relative sensitivities are around 1.6 -6.6 times of the corresponding values of the 2nd-order sensitivities, and are larger than the corresponding values of the 1 st -order sensitivities by factors ranging from 2.0 to 29.7 times. The largest values (shown in bold in the table) for the 1 st -order, 2 nd -order and 3 rd -order relative sensitivities all occur for Table 1. Comparison of the 1st-order, 2nd-order and 3rd-order unmixed relative sensi-   (  )   1  2  3  ,2  ,2  ,2  ,2  ,2  ,2   ,  , ,  , , , 1, ,30 for isotope 2 ( 240 Pu). Table 3. Comparison of the 1st-order, 2nd-order and 3rd-order unmixed relative sensi-     The results presented in Tables 2-4 indicate that all the 1 st -, 2 nd -and 3 rd -order unmixed relative sensitivities for the isotopes 2, 3 and 4 (namely, 240 Pu, 69 Ga and 71 Ga), respectively, are very small (i.e., in the order of 10 −2 and less). For the same energy group of each isotope, the value of the 1 st -order relative sensitivity is generally the largest, followed by the 2nd-order sensitivity, while the 3 rd -order sensitivity is generally the smallest. For instance, for the same energy group of isotopes 69 Ga and 71 Ga, respectively, the values of the 1 st -order relative sensitivities are ca. 2-3 orders of magnitudes greater than the corresponding values of the 2 nd -order sensitivities, and ca. 4-5 orders of magnitudes greater than the corresponding values of the 3 rd -order ones. Also, all of the 1 st -and 3 rd -order unmixed relative sensitivities that are presented in Tables 2-4 are negative, while all the 2 nd -order unmixed relative sensitivities are positive.
As shown in Table 5, the values of the 1 st -, 2 nd -and 3 rd -order unmixed relative sensitivities for isotope 5 (C) are mostly of the order of 10 −1 or 10 −2 (or less) for all energy groups, except for the lowest energy group (g = 30). For each energy group of g = 1 … 29, the 1 st -order relative sensitivities are the largest, followed by the 2 nd -order and the 3 rd -order sensitivities. Specifically, for these groups, the absolute values of the 1 st -order relative sensitivities are ca. one order of magnitude greater than that of the corresponding 2 nd -order sensitivities, while the 2 nd -order sensitivities are generally 1 to 3 times greater than the corresponding 3 rd -order ones. However, for the lowest group (g = 30), the 3 rd -order relative sensitivity ( ) ( ) the absolute value of the 1 st -order relative sensitivity ( ) ( ) As shown in Table 6, many of the relative sensitivities for isotope 6 ( 1 H) have absolute values greater than 1.0, including 6 first-order sensitivities, 16 second-order unmixed sensitivities, and 18 third-order unmixed sensitivities.
For energy groups g = 1 … 11, the 1 st -order sensitivities are slightly larger (in absolute values) than the corresponding 2 nd -and 3 rd -order ones, but all are small.
For energy groups g = 12 … 29, all of the 3 rd -order unmixed relative sensitivities have absolute values greater than 1.0 and are significantly larger than the corresponding 1 st -and 2 nd -order ones. Depending on the specific energy group, the absolute values of the 3 rd -order relative sensitivity are ca. 2 to 6 times larger than the corresponding 2 nd -order ones and ca. 3 to 27 times larger than the values of the corresponding 1 st -order sensitivities. The largest absolute values for all the 1 st -, 2 nd -and 3 rd -order sensitivities are highlighted in bold digits in Table 6; they occur for the lowest-energy group (g = 30), where the 3 rd -order sensitivity  Tables 2-5, the 3 rd -order relative sensitivities are all smaller than both the 1 st -order and 2 nd -order relative sensitivities for isotopes 2, 3, 4, and 5 (with one exception, for isotope 5 at g = 30). However, the results that are presented in Table 1 and Table 6 indicate that for isotope 1 ( 239 Pu) and isotope 6 ( 1 H), the 3 rd -order relative sensitivities are generally larger than the corresponding 1 st -order and 2 nd -order relative sensitivities. For isotope 1 ( 239 Pu), the largest 3 rd -order unmixed sensitivity is about 4 times larger than the corresponding 2 nd -order relative sensitivity, and about 20 times larger than that of the corresponding 1 st -order one. Notably for isotope 6 ( 1 H), the largest 3 rd -order unmixed sensitivity is about 70 times larger than the corresponding 2 nd -order relative sensitivity and is ca. 3000 times larger than the corresponding 1 st -order one. The results presented in Tables 1-6 indicate that the 1 st -order and 3 rd -order unmixed relative sensitivities are negative while the 2 nd -order unmixed relative sensitivities are all positive, for all the isotopes contained in the PERP sphere. The results in Tables 1-6 indicate that largest 1 st -, 2 nd -and 3 rd -order sensitivities, and hence the most important consequential effects for the PERP benchmark's leakage response, arise from the microscopic total cross sections of isotopes 1 H and 239 Pu. American Journal of Computational Mathematics As indicated in [4], all the numerical values for the 1 st -order and unmixed 2 nd -order relative sensitivities in Tables 1-6 have been independently verified with the results being obtained from the central-difference estimates using forward PARTISN [6] computations in which the isotopic total cross section for each group was perturbed by small amounts, as needed for the respective finite-difference formulas. Similar verifications of the values obtained by using the 3rd-LASS were also performed in this work for 3 rd -order unmixed relative sensitivities, which were alternatively computed using central-difference methods in conjunction with forward PARTISN [6] computations in which the isotopic total cross section for each group was perturbed by small amounts. The result of the verification of the largest sensitivities for isotope 239 Pu, which occurs in group 12, is presented in Table 7. Furthermore, Table 8 shows the verification of largest overall unmixed sensitivities of the PERP benchmark's leakage response to the microscopic total cross section, which occurs in group 30 of isotope 1 H. The results in both Table 7 and Table 8 provide confidence that the numerical solution of the 3 rd -LASS is not only efficient but also very accurate.

S
Min. value = −4.58 × 10 −5 at g = g' = 13, g" = 30 , , , , , , , , , , , , The submatrices which comprise components having absolute values greater than 1.0, as shown in Tables 9-14, will be discussed in detail in sub-Sections 3.1-3.16, below.  appear on the 3D-plot's respective axes. The absolute values of these 2027 elements are illustrated by spheres, where the size and color of each sphere indicate visually the magnitude of the respective relative sensitivity. In addition, Figure 1 also shows the projection (colored in grey) of the 3 rd -order mixed relative sensitivities onto the bottom plane, i.e., the g−g' plane. As shown in Figure 1 Table 15. Notably, all these 4 elements are located on the diagonal of this three-dimensional submatrix, with 12 g g g ′ ′′ = = = , 13 g g g ′ ′′ = = = , 14 g g g ′ ′′ = = = , and 16 g g g ′ ′′ = = = , respectively. The largest value among these sensitivities is attained by the 3rd-order mixed relative sensitivity     , which involves the 12 th energy group of the total cross section for isotope 1 ( 239 Pu) and the 30 th energy group of the total cross section for isotope 5 (C).  involves the 12 th energy group of the total cross section for isotope 1 ( 239 Pu) and the 30 th energy group of the total cross section for isotope 6 ( 1 H).  The magnitudes and distribution of these 817 components are illustrated in Figure 5. Due to the wide spread of the values of these components, Figure 5 employs a logarithm scale (instead of the linear scale employed in Figures 1-4) for scaling the size of the spheres as well as for the legend of the colors. As shown in Figure 5  ; the largest component involves the 30 th energy group of the microscopic total cross sections for isotopes 240 Pu, C and C, respectively.  270.08 , ,

Third-Order Relative Sensitivities
, which involves the 30 th energy group of the microscopic total cross section for isotopes 240 Pu, C and 1 H.  energy group of the total cross section for isotope 4 ( 71 Ga), isotope 6 ( 1 H) and isotope 6 ( 1 H). The second largest 3 rd -order mixed relative sensitivity is  Figure 12 shows the distribution of the 16 components of this submatrix which have absolute values greater than 1.0. As shown in the figure, all of these 16 components are located on the three edges of the cube. The largest 3 rd -order mixed relative sensitivity is attained by ( ) (  )   3  30  30  30  ,5  ,5  ,5 17. , , 45

Third-Order Relative Sensitivities
, which occurs at the vertex of the cube, corresponding to the 30 th energy group of the microscopic total cross section for isotope C.   Figure 14, most of the large components are located in the region confined by , , 12, ,30 g g g ′ ′′ =  ; in particular, most of the components that have absolute values greater than 100.0 are located on the three edges of the cube. The largest 3 rd -order mixed relative sensitivity in this submatrix is attained by ( ) ( ) S σ σ σ ′ ′′ = = = = − × , which involves the 30 th energy group of the microscopic total cross section for isotope 1 H. As illustrated in Figure 15, most of these large components (having absolute values greater than 1.0) are located in the region bordered by , , 12, ,30 g g g ′ ′′ =  , while most of the components that have absolute values greater than 1000.0 are located on the three edges of the cube.

Summary and Conclusions
This work has presented the numerical computation of the (180) 3 third-order  mixed sensitivities   ,   3  ,   , , ,  6; , ,  1  , 1 , ,30 , , g g g t j t k t l j k l L g g g σ σ σ ′ ′′ ′ ′′ ∂ ∂ ∂ ∂ = =   of the PERP benchmark's total leakage response with respect to the benchmark's 180 microscopic total cross sections. The largest magnitudes attained by the 1 st -, 2 ndand 3 rd -order relative sensitivities of the PERP benchmark's leakage response with respect to the microscopic total cross sections, are summarized in Table 16, underscoring the finding that the total number of 3 rd -order relative sensitivities that have large values (greater than 1.0) is significantly higher than the number of large 1 st -and 2nd-order sensitivities. Table 16 indicates that the absolute value of the largest 3 rd -order relative sensitivity is ca. 437 times larger than the largest 2 nd -order sensitivity and is ca. 20,000 times larger than the largest 1 st -order sensitivity. All of the largest 1 st -, 2 nd -and 3 rd -order sensitivities involve the microscopic total cross section for the lowest (30 th ) energy group of isotope 1 H (i.e., 30 ,6 t σ ). The largest unmixed 3 rd -order sensitivity is also with respect to 30 ,6 t σ , namely  Table 6. However, the largest overall 3 rd -order sensitivity is the mixed 3 rd -order sensitivity The following conclusions can be drawn from the results reported in this work: 1) For all isotopes contained in the PERP benchmark, all of the 1 st -order and 3 rd -order unmixed relative sensitivities of the PERP's leakage response to the PERP's microscopic total cross sections are negative, while all the unmixed 2 nd -order ones are positive.
2) The properties of the unmixed sensitivities for the isotopes contained in the PERP benchmark have been discussed in Section 2, in conjunction with the numerical results presented in Tables 1-6 and will therefore not be repeated here.
3) The number of 3 rd -order mixed relative sensitivities that have large values (and are therefore important) is far greater than the number of important 2 ndand 1 st -order sensitivities. All of the important 3 rd -order mixed relative sensitivities have negative values.  6) The 3 rd -order sensitivity analysis presented in this work is the first ever such analysis in reactor physics. The consequences of the results presented in this work on the uncertainty analysis of the PERP benchmark's leakage response will be presented in the accompanying Part III [7].