^{1}

^{1}

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 polyethylene-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
^{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
^{nd}-order sensitivity is
^{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.

The accompanying Part I [^{rd}-order sensitivities of the leakage response of the OECD/NEA subcritical polyethylene-reflected plutonium (acronym: PERP) metal fundamental physics benchmark [^{rd}-order sensitivities were derived by applying the general Third-Order Adjoint Sensitivity Analysis Methodology conceived by Cacuci [^{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 [^{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.

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 [^{rd}-order mixed sensitivities ∂ 3 L ( α ) / ∂ t j ∂ t k ∂ t l , j , k , l = 1 , ⋯ , J σ t of the PERP leakage response with respect to the group-averaged microscopic total cross sections has been derived in the accompanying Part I [^{rd}-order unmixed relative sensitivities, i.e., S ( 3 ) ( σ t , j g , σ t , j g , σ t , j g ) ≜ ( ∂ 3 L / ∂ σ t , j g ∂ σ t , j g ∂ σ t , j g ) ( σ t , j g σ t , j g σ t , j g / L ) , g = 1 , ⋯ , 30 , j = 1 , ⋯ , 6 , are compared to the corresponding values of the 1^{st}-order relative sensitivities S ( 1 ) ( σ t , j g ) ≜ ( ∂ L / ∂ σ t , j g ) ( σ t , j g / L ) for g = 1 , ⋯ , 30 , j = 1 , ⋯ , 6 , as well as to the corresponding values of the 2^{nd}-order unmixed relative sensitivities S ( 2 ) ( σ t , j g , σ t , j g ) ≜ ( ∂ 2 L / ∂ σ t , j g ∂ σ t , j g ) ( σ t , j g σ t , j g / L ) , g = 1 , ⋯ , 30 , j = 1 , ⋯ , 6 . 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 [

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.

^{st}-order, 2^{nd}-order and 3^{rd}-order unmixed relative sensitivities for isotope 1 (^{239}Pu), for all energy groups g = 1 , ⋯ , 30 . 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 g = 6 , ⋯ , 26 and g = 30 , 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

g | 1st-Order | 2nd-Order | 3rd-Order | g | 1st-Order | 2nd-Order | 3rd-Order |
---|---|---|---|---|---|---|---|

1 | −0.0003 | 0.0003 | −0.0003 | 16 | −0.779 | 3.487 | −23.10 |

2 | −0.0007 | 0.0005 | −0.0005 | 17 | −0.364 | 1.578 | −10.07 |

3 | −0.0019 | 0.0015 | −0.0015 | 18 | −0.227 | 0.995 | −6.428 |

4 | −0.009 | 0.007 | −0.008 | 19 | −0.181 | 0.789 | −5.063 |

5 | −0.046 | 0.043 | −0.054 | 20 | −0.155 | 0.601 | −3.431 |

6 | −0.135 | 0.162 | −0.267 | 21 | −0.137 | 0.479 | −2.480 |

7 | −0.790 | 1.987 | −7.294 | 22 | −0.099 | 0.297 | −1.313 |

8 | −0.726 | 1.768 | −6.270 | 23 | −0.081 | 0.205 | −0.777 |

9 | −0.843 | 2.205 | −8.454 | 24 | −0.051 | 0.123 | −0.438 |

10 | −0.845 | 2.177 | −8.247 | 25 | −0.060 | 0.138 | −0.473 |

11 | −0.775 | 1.879 | −6.691 | 26 | −0.063 | 0.158 | −0.581 |

12 | −1.320 | 4.586 | −23.71 | 27 | −0.017 | 0.022 | −0.039 |

13 | −1.154 | 4.039 | −20.96 | 28 | −0.003 | 0.002 | −0.0017 |

14 | −0.952 | 3.435 | −18.29 | 29 | −0.035 | 0.072 | −0.226 |

15 | −0.690 | 2.487 | −13.18 | 30 | −0.461 | 1.353 | −5.980 |

g | 1st-Order | 2nd-Order | 3rd-Order | g | 1st-Order | 2nd-Order | 3rd-Order |
---|---|---|---|---|---|---|---|

1 | −2.060 × 10^{−5} | 1.052 × 10^{−6} | −6.857 × 10^{−8} | 16 | −4.864 × 10^{−2} | 1.052 × 10^{−6} | −5.606 × 10^{−3} |

2 | −4.117 × 10^{−5} | 2.089 × 10^{−6} | −1.358 × 10^{−7} | 17 | −2.236 × 10^{−2} | 2.089 × 10^{−6} | −2.328 × 10^{−3} |

3 | −1.192 × 10^{−4} | 6.055 × 10^{−6} | −3.948 × 10^{−7} | 18 | −1.358 × 10^{−2} | 6.055 × 10^{−6} | −1.383 × 10^{−3} |

4 | −5.638 × 10^{−4} | 2.947 × 10^{−5} | −1.994 × 10^{−6} | 19 | −1.021 × 10^{−2} | 2.947 × 10^{−5} | −9.170 × 10^{−4} |

5 | −2.894 × 10^{−3} | 1.730 × 10^{−4} | −1.370 × 10^{−5} | 20 | −8.914 × 10^{−3} | 1.730 × 10^{−4} | −6.590 × 10^{−4} |

6 | −8.513 × 10^{−3} | 6.485 × 10^{−4} | −6.744 × 10^{−5} | 21 | −6.716 × 10^{−3} | 6.485 × 10^{−4} | −2.947 × 10^{−4} |

7 | −4.958 × 10^{−2} | 7.836 × 10^{−3} | −1.806 × 10^{−3} | 22 | −4.676 × 10^{−3} | 7.836 × 10^{−3} | −1.364 × 10^{−4} |

8 | −4.574 × 10^{−2} | 7.026 × 10^{−3} | −1.571 × 10^{−3} | 23 | −7.458 × 10^{−3} | 7.026 × 10^{−3} | −6.187 × 10^{−4} |

9 | −5.318 × 10^{−2} | 8.769 × 10^{−3} | −2.120 × 10^{−3} | 24 | −4.371 × 10^{−3} | 8.769 × 10^{−3} | −2.703 × 10^{−4} |

10 | −5.345 × 10^{−2} | 8.711 × 10^{−3} | −2.087 × 10^{−3} | 25 | −8.131 × 10^{−4} | 8.711 × 10^{−3} | −1.170 × 10^{−6} |

11 | −4.909 × 10^{−2} | 7.547 × 10^{−3} | −1.703 × 10^{−3} | 26 | −9.171 × 10^{−4} | 7.547 × 10^{−3} | −1.776 × 10^{−6} |

12 | −8.364 × 10^{−2} | 1.842 × 10^{−2} | −6.032 × 10^{−3} | 27 | −1.862 × 10^{−2} | 1.842 × 10^{−2} | −4.965 × 10^{−2} |

13 | −7.145 × 10^{−2} | 1.548 × 10^{−2} | −4.974 × 10^{−3} | 28 | −9.671 × 10^{−3} | 1.548 × 10^{−2} | −3.722 × 10^{−2} |

14 | −5.953 × 10^{−2} | 1.342 × 10^{−2} | −4.466 × 10^{−3} | 29 | −1.364 × 10^{−4} | 1.342 × 10^{−2} | −1.385 × 10^{−8} |

15 | −4.267 × 10^{−2} | 9.506 × 10^{−3} | −3.114 × 10^{−3} | 30 | −7.909 × 10^{−3} | 9.506 × 10^{−3} | −3.016 × 10^{−5} |

g | 1st-Order | 2nd-Order | 3rd-Order | g | 1st-Order | 2nd-Order | 3rd-Order |
---|---|---|---|---|---|---|---|

1 | −9.214 × 10^{−7} | 2.104 × 10^{−9} | −6.132 × 10^{−12} | 16 | −2.551 × 10^{−3} | 3.733 × 10^{−5} | −8.089 × 10^{−7} |

2 | −1.974 × 10^{−6} | 4.804 × 10^{−9} | −1.497 × 10^{−11} | 17 | −1.262 × 10^{−3} | 1.893 × 10^{−5} | −4.187 × 10^{−7} |

3 | −6.012 × 10^{−6} | 1.541 × 10^{−8} | −5.068 × 10^{−11} | 18 | −8.411 × 10^{−4} | 1.371 × 10^{−5} | −3.289 × 10^{−7} |

4 | −3.036 × 10^{−5} | 8.545 × 10^{−8} | −3.114 × 10^{−10} | 19 | −8.605 × 10^{−4} | 1.790 × 10^{−5} | −5.485 × 10^{−7} |

5 | −1.587 × 10^{−4} | 5.204 × 10^{−7} | −2.260 × 10^{−9} | 20 | −6.458 × 10^{−4} | 1.050 × 10^{−5} | −2.506 × 10^{−7} |

6 | −4.353 × 10^{−4} | 1.696 × 10^{−6} | −9.018 × 10^{−9} | 21 | −3.919 × 10^{−4} | 3.949 × 10^{−6} | −5.856 × 10^{−8} |

7 | −2.107 × 10^{−3} | 1.415 × 10^{−5} | −1.386 × 10^{−7} | 22 | −1.489 × 10^{−4} | 6.668 × 10^{−7} | −4.408 × 10^{−9} |

8 | −1.717 × 10^{−3} | 9.897 × 10^{−6} | −8.307 × 10^{−8} | 23 | −1.104 × 10^{−4} | 3.859 × 10^{−7} | −2.008 × 10^{−9} |

9 | −1.912 × 10^{−3} | 1.133 × 10^{−5} | −9.845 × 10^{−8} | 24 | −3.199 × 10^{−5} | 4.778 × 10^{−8} | −1.059 × 10^{−10} |

10 | −1.956 × 10^{−3} | 1.166 × 10^{−5} | −1.022 × 10^{−7} | 25 | −1.726 × 10^{−5} | 1.136 × 10^{−8} | −1.118 × 10^{−11} |

11 | −1.943 × 10^{−3} | 1.182 × 10^{−5} | −1.055 × 10^{−7} | 26 | −5.147 × 10^{−5} | 1.046 × 10^{−7} | −3.139 × 10^{−10} |

12 | −3.756 × 10^{−3} | 3.714 × 10^{−5} | −5.464 × 10^{−7} | 27 | −2.586 × 10^{−5} | 4.825 × 10^{−8} | −1.331 × 10^{−10} |

13 | −3.522 × 10^{−3} | 3.762 × 10^{−5} | −5.957 × 10^{−7} | 28 | −8.496 × 10^{−7} | 1.192 × 10^{−10} | −2.523 × 10^{−14} |

14 | −2.987 × 10^{−3} | 3.371 × 10^{−5} | −5.624 × 10^{−7} | 29 | −6.754 × 10^{−7} | 2.747 × 10^{−11} | −1.682 × 10^{−15} |

15 | −2.182 × 10^{−3} | 2.485 × 10^{−5} | −4.163 × 10^{−7} | 30 | −2.542 × 10^{−5} | 4.111 × 10^{−9} | −1.002 × 10^{−12} |

g | 1st-Order | 2nd-Order | 3rd-Order | g | 1st-Order | 2nd-Order | 3rd-Order |
---|---|---|---|---|---|---|---|

1 | −6.266 × 10^{−7} | 9.730 × 10^{−10} | −1.928 × 10^{−12} | 16 | −1.662 × 10^{−3} | 1.585 × 10^{−5} | −2.237 × 10^{−7} |

2 | −1.345 × 10^{−6} | 2.230 × 10^{−9} | −4.734 × 10^{−12} | 17 | −8.176 × 10^{−4} | 7.950 × 10^{−6} | −1.139 × 10^{−7} |

3 | −4.103 × 10^{−6} | 7.176 × 10^{−9} | −1.611 × 10^{−11} | 18 | −5.318 × 10^{−4} | 5.479 × 10^{−6} | −8.310 × 10^{−8} |

4 | −2.069 × 10^{−5} | 3.967 × 10^{−8} | −9.849 × 10^{−11} | 19 | −4.939 × 10^{−4} | 5.898 × 10^{−6} | −1.037 × 10^{−7} |

5 | −1.072 × 10^{−4} | 2.374 × 10^{−7} | −6.966 × 10^{−10} | 20 | −3.976 × 10^{−4} | 3.979 × 10^{−6} | −5.847 × 10^{−8} |

6 | −2.906 × 10^{−4} | 7.557 × 10^{−7} | −2.683 × 10^{−9} | 21 | −2.344 × 10^{−4} | 1.413 × 10^{−6} | −1.253 × 10^{−8} |

7 | −1.397 × 10^{−3} | 6.218 × 10^{−6} | −4.037 × 10^{−8} | 22 | −2.170 × 10^{−3} | 1.416 × 10^{−4} | −1.364 × 10^{−5} |

8 | −1.149 × 10^{−3} | 4.436 × 10^{−6} | −2.492 × 10^{−8} | 23 | −1.337 × 10^{−4} | 5.659 × 10^{−7} | −3.568 × 10^{−9} |

9 | −1.295 × 10^{−3} | 5.202 × 10^{−6} | −3.063 × 10^{−8} | 24 | −1.322 × 10^{−5} | 8.156 × 10^{−9} | −7.470 × 10^{−12} |

10 | −1.327 × 10^{−3} | 5.368 × 10^{−6} | −3.192 × 10^{−8} | 25 | −7.518 × 10^{−6} | 2.154 × 10^{−9} | −9.232 × 10^{−13} |

11 | −1.318 × 10^{−3} | 5.439 × 10^{−6} | −3.296 × 10^{−8} | 26 | −2.313 × 10^{−5} | 2.112 × 10^{−8} | −2.848 × 10^{−11} |

12 | −2.549 × 10^{−3} | 1.710 × 10^{−5} | −1.707 × 10^{−7} | 27 | −1.201 × 10^{−5} | 1.041 × 10^{−8} | −1.333 × 10^{−11} |

13 | −2.375 × 10^{−3} | 1.711 × 10^{−5} | −1.828 × 10^{−7} | 28 | −4.131 × 10^{−7} | 2.818 × 10^{−11} | −2.901 × 10^{−15} |

14 | −2.005 × 10^{−3} | 1.521 × 10^{−5} | −1.705 × 10^{−7} | 29 | −3.512 × 10^{−7} | 7.429 × 10^{−12} | −2.365 × 10^{−16} |

15 | −1.481 × 10^{−3} | 1.145 × 10^{−5} | −1.302 × 10^{−7} | 30 | −1.665 × 10^{−5} | 1.764 × 10^{−9} | −2.815 × 10^{−13} |

g | 1st-Order | 2nd-Order | 3rd-Order | g | 1st-Order | 2nd-Order | 3rd-Order |
---|---|---|---|---|---|---|---|

1 | −9.992 × 10^{−6} | 1.066 × 10^{−6} | 1.038 × 10^{−7} | 16 | −2.074 × 10^{−1} | 1.415 × 10^{−1} | −1.429 × 10^{−1} |

2 | −2.017 × 10^{−5} | 2.185 × 10^{−6} | 4.236 × 10^{−8} | 17 | −1.665 × 10^{−1} | 9.779 × 10^{−2} | −8.554 × 10^{−2} |

3 | −6.373 × 10^{−5} | 7.901 × 10^{−6} | −3.833 × 10^{−7} | 18 | −1.439 × 10^{−1} | 7.678 × 10^{−2} | −6.114 × 10^{−2} |

4 | −2.996 × 10^{−4} | 3.873 × 10^{−5} | −3.872 × 10^{−6} | 19 | −1.310 × 10^{−1} | 6.625 × 10^{−2} | −5.004 × 10^{−2} |

5 | −1.597 × 10^{−3} | 2.359 × 10^{−4} | −3.370 × 10^{−5} | 20 | −1.212 × 10^{−1} | 5.905 × 10^{−2} | −4.297 × 10^{−2} |

6 | −4.403 × 10^{−3} | 6.521 × 10^{−4} | −1.175 × 10^{−4} | 21 | −1.129 × 10^{−1} | 5.347 × 10^{−2} | −3.780 × 10^{−2} |

7 | −3.698 × 10^{−2} | 9.376 × 10^{−3} | −3.113 × 10^{−3} | 22 | −1.036 × 10^{−1} | 4.747 × 10^{−2} | −3.247 × 10^{−2} |

8 | −4.631 × 10^{−2} | 1.447 × 10^{−2} | −5.744 × 10^{−3} | 23 | −9.589 × 10^{−2} | 4.280 × 10^{−2} | −2.851 × 10^{−2} |

9 | −4.502 × 10^{−2} | 1.114 × 10^{−2} | −3.553 × 10^{−3} | 24 | −8.693 × 10^{−2} | 3.756 × 10^{−2} | −2.422 × 10^{−2} |

10 | −5.135 × 10^{−2} | 1.368 × 10^{−2} | −4.754 × 10^{−3} | 25 | −8.213 × 10^{−2} | 3.496 × 10^{−2} | −2.220 × 10^{−2} |

11 | −5.645 × 10^{−2} | 1.633 × 10^{−2} | −6.262 × 10^{−3} | 26 | −7.550 × 10^{−2} | 3.142 × 10^{−2} | −1.949 × 10^{−2} |

12 | −1.345 × 10^{−1} | 6.055 × 10^{−2} | −3.799 × 10^{−2} | 27 | −6.727 × 10^{−2} | 2.701 × 10^{−2} | −1.617 × 10^{−2} |

13 | −1.529 × 10^{−1} | 8.249 × 10^{−2} | −6.342 × 10^{−2} | 28 | −6.224 × 10^{−2} | 2.437 × 10^{−2} | −1.422 × 10^{−2} |

14 | −1.504 × 10^{−1} | 8.573 × 10^{−2} | −7.064 × 10^{−2} | 29 | −5.995 × 10^{−2} | 2.298 × 10^{−2} | −1.312 × 10^{−2} |

15 | −1.299 × 10^{−1} | 6.928 × 10^{−2} | −5.391 × 10^{−2} | 30 | −7.847 × 10^{−1} | 3.016 × 10^{0} | −1.745 × 10^{1} |

g | 1st-Order | 2nd-Order | 3rd-Order | g | 1st-Order | 2nd-Order | 3rd-Order |
---|---|---|---|---|---|---|---|

1 | −8.471 × 10^{−6} | 7.636 × 10^{−7} | 6.322 × 10^{−8} | 16 | −1.164 × 10^{0} | 4.460 × 10^{0} | −2.530 × 10^{1} |

2 | −2.060 × 10^{−5} | 2.280 × 10^{−6} | 4.516 × 10^{−8} | 17 | −1.173 × 10^{0} | 4.853 × 10^{0} | −2.991 × 10^{1} |

3 | −6.810 × 10^{−5} | 9.021 × 10^{−6} | −4.677 × 10^{−7} | 18 | −1.141 × 10^{0} | 4.828 × 10^{0} | −3.049 × 10^{1} |

4 | −3.932 × 10^{−4} | 6.673 × 10^{−5} | −8.758 × 10^{−6} | 19 | −1.094 × 10^{0} | 4.619 × 10^{0} | −2.913 × 10^{1} |

5 | −2.449 × 10^{−3} | 5.549 × 10^{−4} | −1.216 × 10^{−4} | 20 | −1.033 × 10^{0} | 4.284 × 10^{0} | −2.655 × 10^{1} |

6 | −9.342 × 10^{−3} | 2.935 × 10^{−3} | −1.123 × 10^{−3} | 21 | −9.692 × 10^{0} | 3.937 × 10^{0} | −2.388 × 10^{1} |

7 | −7.589 × 10^{−2} | 3.949 × 10^{−2} | −2.690 × 10^{−2} | 22 | −8.917 × 10^{−1} | 3.515 × 10^{0} | −2.069 × 10^{1} |

8 | −9.115 × 10^{−2} | 5.604 × 10^{−2} | −4.380 × 10^{−2} | 23 | −8.262 × 10^{−1} | 3.177 × 10^{0} | −1.823 × 10^{1} |

9 | −1.358 × 10^{−1} | 1.014 × 10^{−1} | −9.758 × 10^{−2} | 24 | −7.495 × 10^{−1} | 2.792 × 10^{0} | −1.552 × 10^{1} |

10 | −1.659 × 10^{−1} | 1.428 × 10^{−1} | −1.604 × 10^{−1} | 25 | −7.087 × 10^{−1} | 2.604 × 10^{0} | −1.427 × 10^{1} |

11 | −1.899 × 10^{−1} | 1.849 × 10^{−1} | −2.385 × 10^{−1} | 26 | −6.529 × 10^{−1} | 2.349 × 10^{0} | −1.260 × 10^{1} |

12 | −4.446 × 10^{−1} | 6.620 × 10^{−1} | −1.373 × 10^{0} | 27 | −5.845 × 10^{−1} | 2.039 × 10^{0} | −1.061 × 10^{1} |

13 | −5.266 × 10^{−1} | 9.782 × 10^{−1} | −2.590 × 10^{0} | 28 | −5.474 × 10^{−1} | 1.885 × 10^{0} | −9.678 × 10^{0} |

14 | −5.772 × 10^{−1} | 1.262 × 10^{0} | −3.991 × 10^{0} | 29 | −5.439 × 10^{−1} | 1.891 × 10^{0} | −9.800 × 10^{0} |

15 | −5.820 × 10^{−1} | 1.391 × 10^{0} | −4.581 × 10^{0} | 30 | −9.366 × 10^{0} | 4.296 × 10^{2} | −2.966 × 10^{4} |

the 12^{th} energy group, i.e., S ( 1 ) ( σ t , 1 g = 12 ) = − 1.32 , S ( 2 ) ( σ t , 1 g = 12 , σ t , 1 g = 12 ) = 4.586 and S ( 3 ) ( σ t , 1 g = 12 , σ t , 1 g = 12 , σ t , 1 g = 12 ) = − 23.71 . It is noteworthy that all of the 1^{st}-order and 3^{rd}-order unmixed relative sensitivities are negative, while all the unmixed 2^{nd}-order ones are positive. A negative value for the 1^{st}-order sensitivity signifies that an increase in the microscopic total cross section σ t , 1 g will cause a decrease in the leakage L.

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 ^{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 S ( 3 ) ( σ t , 5 g = 30 , σ t , 5 g = 30 , σ t , 5 g = 30 ) = − 17.45 has the largest absolute value, followed by the 2^{nd}-order unmixed relative sensitivity S ( 2 ) ( σ t , 5 g = 30 , σ t , 5 g = 30 ) = 3.01 ; the absolute value of the 1^{st}-order relative sensitivity S ( 1 ) ( σ t , 5 g = 30 ) = − 0.782 is the smallest.

As shown in ^{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 ^{rd}-order sensitivity S ( 3 ) ( σ t , 6 g = 30 , σ t , 6 g = 30 , σ t , 6 g = 30 ) = − 2.966 × 10 4 attains a very large absolute value; the 2^{nd}-order sensitivity S ( 2 ) ( σ t , 6 g = 30 , σ t , 6 g = 30 ) = 4.283 × 10 2 is also the largest among the 2^{nd}-order ones, while the 1^{st}-order sensitivity S ( 1 ) ( σ t , 6 g = 30 ) = − 9.338 is only slightly smaller (in absolute value) than S ( 1 ) ( σ t , 6 g = 21 ) = − 9.692 .

As indicated by the results presented in 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 ^{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.

As indicated in [^{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 [^{rd}-order unmixed relative sensitivities, which were alternatively computed using central-difference methods in conjunction with forward PARTISN [^{239}Pu, which occurs in group 12, is presented in ^{1}H. The results in both ^{rd}-LASS is not only efficient but also very accurate.

The matrix of 3^{rd}-order mixed relative sensitivities

S ( 3 ) ( σ t , j g , σ t , k g ′ , σ t , l g ″ ) ≜ ( ∂ 3 L / ∂ σ t , j g ∂ σ t , k g ′ ∂ σ t , l g ″ ) ( σ t , j g σ t , k g ′ σ t , l g ″ / L ) , j , k , l = 1 , ⋯ , 6 ; g , g ′ , g ″ = 1 , ⋯ , 30 has dimensions J σ t × J σ t × J σ t ( = 180 × 180 × 180 ) . To facilitate the presentation of its elements, the matrix S ( 3 ) ( σ t , j g , σ t , k g ′ , σ t , l g ″ ) has been partitioned into I × I × I = 6 × 6 × 6 = 216 submatrices, each of dimensions G × G × G = 30 × 30 × 30 . Because of symmetry of the 3^{rd}-order sensitivities, only 56 of these 216 submatrices are district. For example, the values of the corresponding components of the following submatrices are identical: S ( 3 ) ( σ t , j = 1 g , σ t , k = 2 g ′ , σ t , l = 3 g ″ ) , S ( 3 ) ( σ t , j = 1 g , σ t , k = 3 g ′ , σ t , l = 2 g ″ ) , S ( 3 ) ( σ t , j = 2 g , σ t , k = 1 g ′ , σ t , l = 3 g ″ ) , S ( 3 ) ( σ t , j = 2 g , σ t , k = 3 g ′ , σ t , l = 1 g ″ ) , S ( 3 ) ( σ t , j = 3 g , σ t , k = 1 g ′ , σ t , l = 2 g ″ ) and S ( 3 ) ( σ t , j = 3 g , σ t , k = 2 g ′ , σ t , l = 1 g ″ ) . Therefore, only the summary of the main features of each distinct submatrix are presented in Tables 9-14, in the following form: for a submatrix that comprises elements with relative sensitivities having absolute values greater than 1.0, the total number of such elements are counted and shown in the table. Otherwise, if the relative sensitivities of all elements of a submatrix have values lying in the interval (−1.0, 1.0), only the element having the largest absolute value in the submatrix, together with the phase-space coordinates of that element, are listed in the respective Table.

For the submatrices which comprise components with absolute values less than 1.0, as shown in Tables 9-14, most of the largest absolute values are of the order of 10^{−2} or smaller, particularly the mixed 3^{rd}-order relative sensitivities involving the microscopic total cross sections of isotopes ^{240}Pu, ^{69}Ga, and ^{71}Ga.

Unmixed Relative Sensitivities | 1st-Order | 2nd-Order | 3rd-Order |
---|---|---|---|

Computed results using adjoint functions (from | −1.319 | 4.586 | −23.78 |

Results using central-difference method | −1.320 | 4.649 | −24.25 |

Unmixed Relative Sensitivities | 1st-Order | 2nd-Order | 3rd-Order |
---|---|---|---|

Computed results using adjoint functions (from | −9.366 | 429.6 | −2.966 × 10^{4} |

Results using central-difference method | −9.369 | 430.6 | −3.017 × 10^{4} |

Isotopes | l = 1 (^{239}Pu) | l = 2 (^{240}Pu) | l = 3 (^{69}Ga) | l = 4 (^{71}Ga) | l = 5 (C) | l = 6 (^{1}H) |
---|---|---|---|---|---|---|

k = 1 (^{239}Pu) | S ( 3 ) ( σ t , 1 g , σ t , 1 g ′ , σ t , 1 g ″ ) 2027 elements with absolute values > 1.0 | S ( 3 ) ( σ t , 1 g , σ t , 1 g ′ , σ t , 2 g ″ ) 4 elements with absolute values > 1.0 | S ( 3 ) ( σ t , 1 g , σ t , 1 g ′ , σ t , 3 g ″ ) Min. value = −7.56 × 10^{−2} at g = g' = g'' = 16 | S ( 3 ) ( σ t , 1 g , σ t , 1 g ′ , σ t , 4 g ″ ) Min. value = −4.92 × 10^{−2} at g = g' = g" = 16 | S ( 3 ) ( σ t , 1 g , σ t , 1 g ′ , σ t , 5 g ″ ) 188 elements with absolute values >1.0 | S ( 3 ) ( σ t , 1 g , σ t , 1 g ′ , σ t , 6 g ″ ) 3090 elements with absolute values >1.0 |

k = 2 (^{240}Pu) | S ( 3 ) ( σ t , 1 g , σ t , 2 g ′ , σ t , 2 g ″ ) Min. value = −9.52 × 10^{−2} at g = g' = g" = 12 | S ( 3 ) ( σ t , 1 g , σ t , 2 g ′ , σ t , 3 g ″ ) Min. value = −4.71 × 10^{−3} at g = g' = g" = 16 | S ( 3 ) ( σ t , 1 g , σ t , 2 g ′ , σ t , 4 g ″ ) Min. value = −3.07 × 10^{−3} at g = g' = g" = 16 | S ( 3 ) ( σ t , 1 g , σ t , 2 g ′ , σ t , 5 g ″ ) Min. value = −5.25 × 10^{−1} at g = g' = 12, g" = 30 | S ( 3 ) ( σ t , 1 g , σ t , 2 g ′ , σ t , 6 g ″ ) 120 elements with absolute values > 1.0 | |

k = 3 (^{69}Ga) | S ( 3 ) ( σ t , 1 g , σ t , 3 g ′ , σ t , 3 g ″ ) Min. value = −2.47 × 10^{−4} at g = g' = g" = 16 | S ( 3 ) ( σ t , 1 g , σ t , 3 g ′ , σ t , 4 g ″ ) Min. value = −1.61 × 10^{−4} at g = g' = g" = 16 | S ( 3 ) ( σ t , 1 g , σ t , 3 g ′ , σ t , 5 g ″ ) Min. value = −2.30 × 10^{−2} at g = g' = 12, g" = 30 | S ( 3 ) ( σ t , 1 g , σ t , 3 g ′ , σ t , 6 g ″ ) Min. value = −2.82 × 10^{−1} at g = g' = 12, g" = 30 | ||

k = 4 (^{71}Ga) | S ( 3 ) ( σ t , 1 g , σ t , 4 g ′ , σ t , 4 g ″ ) Min. value = −6.25 × 10^{−4} at g = g' = g" = 22 | S ( 3 ) ( σ t , 1 g , σ t , 4 g ′ , σ t , 5 g ″ ) Min. value = −1.60 × 10^{−2} at g = g' = 12, g" = 30 | S ( 3 ) ( σ t , 1 g , σ t , 4 g ′ , σ t , 6 g ″ ) Min. value = −1.90 × 10^{−1} at g = g' = 12, g" = 30 | |||

k = 5 (C) | S ( 3 ) ( σ t , 1 g , σ t , 5 g ′ , σ t , 5 g ″ ) 817 elements with absolute value > 1.0 | S ( 3 ) ( σ t , 1 g , σ t , 5 g ′ , σ t , 6 g ″ ) 2013 elements with absolute values > 1.0 | ||||

k = 6 (^{1}H) | S ( 3 ) ( σ t , 1 g , σ t , 6 g ′ , σ t , 6 g ″ ) 7998 elements with absolute values > 1.0 |

Isotopes | l = 2 (^{240}Pu) | l = 3 (^{69}Ga) | l = 4 (^{71}Ga) | l = 5 (C) | l = 6 (^{1}H) |
---|---|---|---|---|---|

k = 2 (^{240}Pu) | S ( 3 ) ( σ t , 2 g , σ t , 2 g ′ , σ t , 2 g ″ ) Min. value = −4.96 × 10^{−2} at g = g' = g" = 27 | S ( 3 ) ( σ t , 2 g , σ t , 2 g ′ , σ t , 3 g ″ ) Min. value = −2.94 × 10^{−4} at g = g' = g" = 16 | S ( 3 ) ( σ t , 2 g , σ t , 2 g ′ , σ t , 4 g ″ ) Min. value = −1.92 × 10^{−4} at g = g' = g" = 16 | S ( 3 ) ( σ t , 2 g , σ t , 2 g ′ , σ t , 5 g ″ ) Min. value = −3.66 × 10^{−2} at g = g' = 27, g" = 30 | S ( 3 ) ( σ t , 2 g , σ t , 2 g ′ , σ t , 6 g ″ ) Min. value = −4.37 × 10^{−1} at g = g' = 27, g" = 30 |

k = 3 (^{69}Ga) | S ( 3 ) ( σ t , 2 g , σ t , 3 g ′ , σ t , 3 g ″ ) Min. value = −1.54 × 10^{−5} at g = g' = g" = 16 | S ( 3 ) ( σ t , 2 g , σ t , 3 g ′ , σ t , 4 g ″ ) Min. value = −1.01 × 10^{−5} at g = g' = g" = 16 | S ( 3 ) ( σ t , 2 g , σ t , 3 g ′ , σ t , 5 g ″ ) Min. value = −1.50 × 10^{−3} at g = g' = 12, g" = 30 | S ( 3 ) ( σ t , 2 g , σ t , 3 g ′ , σ t , 6 g ″ ) Min. value = −1.78 × 10^{−2} at g = g' = 12, g" = 30 | |

k = 4 (^{71}Ga) | S ( 3 ) ( σ t , 2 g , σ t , 4 g ′ , σ t , 4 g ″ ) Min. value = −2.94 × 10^{−5} at g = g' = g" = 22 | S ( 3 ) ( σ t , 2 g , σ t , 4 g ′ , σ t , 5 g ″ ) Min. value = −1.01 × 10^{−3} at g = g' = 12, g" = 30 | S ( 3 ) ( σ t , 2 g , σ t , 4 g ′ , σ t , 6 g ″ ) Min. value = −1.21 × 10^{−2} at g = g' = 12, g" = 30 | ||

k = 5 (C) | S ( 3 ) ( σ t , 2 g , σ t , 5 g ′ , σ t , 5 g ″ ) 75 elements with absolute value > 1.0 | S ( 3 ) ( σ t , 2 g , σ t , 5 g ′ , σ t , 6 g ″ ) 452 elements with absolute values > 1.0 | |||

k = 6 (^{1}H) | S ( 3 ) ( σ t , 2 g , σ t , 6 g ′ , σ t , 6 g ″ ) 1621 elements with absolute values > 1.0 |

Isotopes | l = 3 (^{69}Ga) | l = 4 (^{71}Ga) | l = 5 (C) | l = 6 (^{1}H) |
---|---|---|---|---|

k = 3 (^{69}Ga) | S ( 3 ) ( σ t , 3 g , σ t , 3 g ′ , σ t , 3 g ″ ) Min. value = −8.09 × 10^{−7} at g = g' = g" = 16 | S ( 3 ) ( σ t , 3 g , σ t , 3 g ′ , σ t , 4 g ″ ) Min. value = −5.27 × 10^{−7} at g = g' =g" = 16 | S ( 3 ) ( σ t , 3 g , σ t , 3 g ′ , σ t , 5 g ″ ) Min. value = −6.80 × 10^{−5} at g = g' = 13, g" = 30 | S ( 3 ) ( σ t , 3 g , σ t , 3 g ′ , σ t , 6 g ″ ) Min. value = −8.11 × 10^{−4} at g = g' = 13, g" = 30 |

k = 4 (^{71}Ga) | S ( 3 ) ( σ t , 3 g , σ t , 4 g ′ , σ t , 4 g ″ ) Min. value = −1.10 × 10^{−6} at g = 12, g' = g" = 22 | S ( 3 ) ( σ t , 3 g , σ t , 4 g ′ , σ t , 5 g ″ ) Min. value = −4.58 × 10^{−5} at g = g' = 13, g" = 30 | S ( 3 ) ( σ t , 3 g , σ t , 4 g ′ , σ t , 6 g ″ ) Min. value = −5.47 × 10^{−4} at g = g' = 13, g" = 30 | |

k = 5 (C) | S ( 3 ) ( σ t , 3 g , σ t , 5 g ′ , σ t , 5 g ″ ) Min. value = −7.27 × 10^{−2} at g = g' = g" = 30 | S ( 3 ) ( σ t , 3 g , σ t , 5 g ′ , σ t , 6 g ″ ) Min. value = −8.68 × 10^{−1} at g = g' = g" = 30 | ||

k = 6 (^{1}H) | S ( 3 ) ( σ t , 3 g , σ t , 6 g ′ , σ t , 6 g ″ ) 15 elements with absolute values > 1.0 |

Isotopes | l = 4 (^{71}Ga) | l = 5 (C) | l = 6 (^{1}H) |
---|---|---|---|

k = 4 (^{71}Ga) | S ( 3 ) ( σ t , 4 g , σ t , 4 g ′ , σ t , 4 g ″ ) Min. value = −1.36 × 10^{−5} at g = 12, g' = g" = 22 | S ( 3 ) ( σ t , 4 g , σ t , 4 g ′ , σ t , 5 g ″ ) Min. value = −2.11 × 10^{−4} at g = g' = 22, g" = 30 | S ( 3 ) ( σ t , 4 g , σ t , 4 g ′ , σ t , 6 g ″ ) Min. value = −2.52 × 10^{−3} at g = g' = 22, g" = 30 |

k = 5 (C) | S ( 3 ) ( σ t , 4 g , σ t , 5 g ′ , σ t , 5 g ″ ) Min. value = −4.76 × 10^{−2} at g = g' = g" = 30 | S ( 3 ) ( σ t , 4 g , σ t , 5 g ′ , σ t , 6 g ″ ) Min. value = −5.69 × 10^{−1} at g = g' = g" = 30 | |

k = 6 (^{1}H) | S ( 3 ) ( σ t , 4 g , σ t , 6 g ′ , σ t , 6 g ″ ) 11 elements with absolute values > 1.0 |

Isotopes | l = 5 (C) | l = 6 (^{1}H) |
---|---|---|

k = 5 (C) | S ( 3 ) ( σ t , 5 g , σ t , 5 g ′ , σ t , 5 g ″ ) 16 elements with absolute values > 1.0 | S ( 3 ) ( σ t , 5 g , σ t , 5 g ′ , σ t , 6 g ″ ) 179 elements with absolute values > 1.0 |

k = 6 (^{1}H) | S ( 3 ) ( σ t , 5 g , σ t , 6 g ′ , σ t , 6 g ″ ) 1563 elements with absolute values > 1.0 |

Isotopes | l = 6 (^{1}H) |
---|---|

k = 6 (^{1}H) | S ( 3 ) ( σ t , 6 g , σ t , 6 g ′ , σ t , 6 g ″ ) 6999 elements with absolute values > 1.0 |

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.

The values of the 3^{rd}-order relative sensitivities of the leakage response with respect to the microscopic total cross sections for isotope 1 (^{239}Pu) are all negative. As summarized in ^{rd}-order mixed relative sensitivities onto the bottom plane, i.e., the g−g' plane. As shown in ^{rd}-order relative sensitivity in the submatrix S ( 3 ) ( σ t , j = 1 g , σ t , k = 1 g ′ , σ t , l = 1 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 is S ( 3 ) ( σ t , 1 g = 12 , σ t , 1 g ′ = 12 , σ t , 1 g ″ = 12 ) = − 23.71 , which occurs for the 12^{th} energy group of the total cross section for isotope 1 (^{239}Pu).

The values of the 3^{rd}-order mixed relative sensitivities belonging to the submatrix S ( 3 ) ( σ t , j = 1 g , σ t , k = 1 g ′ , σ t , l = 2 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 are all negative. In this submatrix, only 4 elements have absolute values greater than 1.0, as listed in

Components | Values |
---|---|

S ( 3 ) ( σ t , 1 g = 12 , σ t , 1 g ′ = 12 , σ t , 2 g ″ = 12 ) | −1.50 |

S ( 3 ) ( σ t , 1 g = 13 , σ t , 1 g ′ = 13 , σ t , 2 g ″ = 13 ) | −1.30 |

S ( 3 ) ( σ t , 1 g = 14 , σ t , 1 g ′ = 14 , σ t , 2 g ″ = 14 ) | −1.14 |

S ( 3 ) ( σ t , 1 g = 16 , σ t , 1 g ′ = 16 , σ t , 2 g ″ = 16 ) | −1.44 |

The submatrix S ( 3 ) ( σ t , j = 1 g , σ t , k = 1 g ′ , σ t , l = 5 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3rd-order mixed relative sensitivities of the leakage response with respect to the total cross sections of isotope 1 (^{239}Pu) and isotope 5 (C). Among the total 27,000 components in the submatrix, the majority of them (namely, 25,259 components) have negative sensitivities, with values between −8.3 and 0; only few components (e.g., 1741 components) have positive sensitivities, which are very small having values in the order of 10^{−4} or less. The submatrix S ( 3 ) ( σ t , j = 1 g , σ t , k = 1 g ′ , σ t , l = 5 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises only 188 components that have absolute values greater than 1.0; these components are depicted in ^{rd}-order relative sensitivity in the submatrix S ( 3 ) ( σ t , j = 1 g , σ t , k = 1 g ′ , σ t , l = 5 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 is S ( 3 ) ( σ t , 1 g = 12 , σ t , 1 g ′ = 12 , σ t , 5 g ″ = 30 ) = − 8.29 , 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).

The submatrix S ( 3 ) ( σ t , j = 1 g , σ t , k = 1 g ′ , σ t , l = 6 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3^{rd}-order mixed relative sensitivities of the leakage response with respect to the microscopic total cross sections of isotope 1 (^{239}Pu) and isotope 6 (^{1}H). The majority (i.e., 25,259 out of the total 27,000) of the components of this submatrix have negative values. The remaining 1741 components have positive values, which are very small (of the order of 10^{−3} and smaller). The matrix S ( 3 ) ( σ t , j = 1 g , σ t , k = 1 g ′ , σ t , l = 6 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 includes 3090 components that have absolute values greater than 1.0. These 3090 components are depicted in

(ca. 5% of 3090) have absolute values between 10.0 and 50.0; and 3) 12 components have absolute values between 50.0 and 99.0. As illustrated in ^{rd}-order mixed relative sensitivity is S ( 3 ) ( σ t , 1 g = 12 , σ t , 1 g ′ = 12 , σ t , 6 g ″ = 30 ) = − 98.98 , 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 6 (^{1}H).

The submatrix S ( 3 ) ( σ t , j = 1 g , σ t , k = 2 g ′ , σ t , l = 6 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3^{rd}-order mixed relative sensitivities of the leakage response with respect to the microscopic total cross sections of isotope 1 (^{239}Pu), isotope 2 (^{240}Pu) and isotope 6 (^{1}H). As was the case for the submatrices investigated in the foregoing (cf. sections 3.3 and 3.4), 25,259 of the 27,000 components in the submatrix have negative values; the remaining 1741 components have very small positive values, of the order of 10^{−4} or less. ^{rd}-order relative sensitivity in the submatrix is S ( 3 ) ( σ t , 1 g = 12 , σ t , 2 g ′ = 12 , σ t , 6 g ″ = 30 ) = − 6.27 , which involves the 12^{th} energy group of the total cross section for ^{239}Pu, ^{240}Pu and the 30^{th} energy group of the total cross section for ^{1}H.

The submatrix S ( 3 ) ( σ t , j = 1 g , σ t , k = 5 g ′ , σ t , l = 5 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3rd-order mixed relative sensitivities of the leakage response with respect to the microscopic total cross sections of isotopes ^{239}Pu, C and C. In this submatrix, 23,881 (out of the total 27,000) components have negative values, while the remaining 3119 components have positive values. All the positive values are very small (of the order of 10^{−4} or less) whereas the negative values are widely spread between 0.0 and −1319.5. Among the negative components, 817 components have 3^{rd}-order mixed relative sensitivities with absolute values greater than 1.0. The magnitudes and distribution of these 817 components are illustrated in ^{th} energy group of the microscopic total cross section for isotope 1 (^{239}Pu), isotope 5 (C) and isotope 5 (C).

The submatrix S ( 3 ) ( σ t , j = 1 g , σ t , k = 6 g ′ , σ t , l = 6 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3^{rd}-order mixed relative sensitivities of the leakage response with respect to the microscopic total cross sections of isotope 1 (^{239}Pu), isotope 6 (^{1}H) and isotope 6 (^{1}H). As in the cases analyzed in the foregoing, most (namely: 23,881 out of 27,000) components of this submatrix have negative values spread in the wide range from 0.0 to −1.9 × 10^{5}. Only a relatively small number (namely 3119 components out of 27,000) have positive values; these positive values are very small, of the order of 10^{−3} or less. The submatrix #Math_214#, comprises 7998 components which have absolute values greater than 1.0, which are depicted in ^{5}. Due to the very wide range of values, these 3^{rd}-order relative sensitivities are plotted on a log scale in ^{rd}-order mixed relative sensitivity is S ( 3 ) ( σ t , 1 g = 30 , σ t , 6 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 1.88 × 10 5 , which involves the 30^{th} energy groups of the total cross section for isotope 1 (^{239}Pu), isotope 6 (^{1}H) and isotope 6 (^{1}H).

The submatrix S ( 3 ) ( σ t , j = 2 g , σ t , k = 5 g ′ , σ t , l = 5 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3^{rd}-order mixed relative sensitivities of the leakage response with respect to the microscopic total cross sections of isotopes ^{240}Pu, C and C. As was the case for

the submatrices discussed in the foregoing, the values of the 3^{rd}-order mixed relative sensitivities for the majority (namely, 23,881 out of 27,000) of the components of this submatrix are negative, having values from 0.0 to −22.7, whereas the other 3119 components have very small positive values (of the order of 10^{−5} or less). ^{th} energy group of the microscopic total cross sections for isotopes ^{240}Pu, C and C, respectively.

The submatrix S ( 3 ) ( σ t , j = 2 g , σ t , k = 5 g ′ , σ t , l = 6 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3^{rd}-order mixed relative sensitivities of the leakage response with respect to the microscopic total cross sections of isotopes ^{240}Pu, C and ^{1}H. Similarly, among the 27,000 components in this submatrix, the majority (namely: 23,881 of 27,000) of these components have negative values for the 3rd-order mixed relative sensitivities, having values in the range from 0.0 to −271.0; the other 3119 components have very small positive values (of the order of 10^{−4} or less).

(ca. 82%) have absolute values between 1.0 and 5.0; 16 components have values in the range 5.0 to 10.0; and 63 components have values between 10.0 and 50.0; only 3 components have absolute values larger than 50.0. The largest 3^{rd}-order mixed relative sensitivities in this submatrix is S ( 3 ) ( σ t , 2 g = 30 , σ t , 5 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 270.08 , which involves the 30^{th} energy group of the microscopic total cross section for isotopes ^{240}Pu, C and ^{1}H.

The submatrix S ( 3 ) ( σ t , j = 2 g , σ t , k = 6 g ′ , σ t , l = 6 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 , comprises the 3^{rd}-order mixed relative sensitivities of the leakage response with respect to the microscopic total cross sections of isotopes ^{240}Pu, ^{1}H and ^{1}H. As was the case with the submatrices analyzed in the foregoing, 23,881 elements of this submatrix have negative values and only 3119 components have positives values; the positive values are very small (order of 10^{−4} or less) whereas the negative values range from 0.0 to −3.2 × 10^{3}. Using a logarithmic scale, ^{rd}-order mixed relative sensitivity in the submatrix S ( 3 ) ( σ t , j = 2 g , σ t , k = 6 g ′ , σ t , l = 6 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 , is S ( 3 ) ( σ t , 2 g = 30 , σ t , 6 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 3.22 × 10 3 , which involves the 30^{th} energy group of the total cross section for isotope 2 (^{240}Pu), isotope 6 (^{1}H) and isotope 6 (^{1}H). The next largest 3^{rd}-order mixed relative sensitivity is S ( 3 ) ( σ t , 2 g = 28 , σ t , 6 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 2.43 × 10 3 .

The submatrix S ( 3 ) ( σ t , j = 3 g , σ t , k = 6 g ′ , σ t , l = 6 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3^{rd}-order mixed relative sensitivities of the leakage response with respect to the microscopic total cross sections of isotopes ^{69}Ga, ^{1}H and ^{1}H. Likewise, among the 27,000 components in this submatrix, 23,881 components have negative values varying between −10.4 to 0.0; the rest 3119 components have very small positive values (of the order of 10^{−5} or less). ^{rd}-order mixed relative sensitivity is S ( 3 ) ( σ t , 3 g = 30 , σ t , 6 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 10.36 , which involves the 30^{th} energy group of the total cross section for isotope 3 (^{69}Ga), isotope 6 (^{1}H) and isotope 6 (^{1}H).

The submatrix S ( 3 ) ( σ t , j = 4 g , σ t , k = 6 g ′ , σ t , l = 6 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3^{rd}-order mixed relative sensitivities of the leakage response with respect to the microscopic total cross sections of isotopes ^{71}Ga, ^{1}H and ^{1}H. Of the 27,000 components of this submatrix, 23,881 have negative values varying between 0.0 and −6.79; the remaining 3119 components have very small positive values (of the order of 10^{−5} or less). ^{rd}-order mixed relative sensitivity is S ( 3 ) ( σ t , 4 g = 30 , σ t , 6 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 6.79 , which involves the 30^{th}

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 S ( 3 ) ( σ t , 4 g = 22 , σ t , 6 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 6.63 .

The submatrix S ( 3 ) ( σ t , j = 5 g , σ t , k = 5 g ′ , σ t , l = 5 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3^{rd}-order mixed relative sensitivities of the leakage response with respect to the microscopic total cross sections of isotope C. This submatrix is symmetric about the principal diagonal defined by g = g ′ = g ″ = 0 and g = g ′ = g ″ = 30 . The majority (i.e., 23,207 out of 27,000) of this submatrix’s components have negative values (varying between −17.45 and 0.0) while the remaining 3793 components have very small positive values (of the order of 10^{−5} or less). ^{rd}-order mixed relative sensitivity is attained by S ( 3 ) ( σ t , 5 g = 30 , σ t , 5 g ′ = 30 , σ t , 5 g ″ = 30 ) = − 17.45 , which occurs at the vertex of the cube, corresponding to the 30^{th} energy group of the microscopic total cross section for isotope C.

The submatrix S ( 3 ) ( σ t , j = 5 g , σ t , k = 5 g ′ , σ t , l = 6 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3^{rd}-order mixed relative sensitivities of the leakage response with respect to the microscopic total cross sections of isotopes C, C and ^{1}H. For this submatrix, 23,207 out of the 27,000 components have negative values, and the other 3739 components have positive values. The positive values are negligibly small (of the order of 10^{−4} or less) whereas the negative values vary from −208.2 to 0.0. Among the 23,207 negative values, 179 of them have absolute values greater than 1.0, and are depicted in ^{th} energy group of the total cross section for isotopes C, C and ^{1}H.

The submatrix S ( 3 ) ( σ t , j = 5 g , σ t , k = 6 g ′ , σ t , l = 6 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3^{rd}-order mixed relative sensitivities of the leakage response with respect to the

microscopic total cross sections of isotopes C, ^{1}H and ^{1}H. The majority (i.e., 23,207 out of 27,000) of the components of this submatrix have negative values; the remaining 3793 components have positives values. Again, the positive values are very small, of the order of 10^{−4} or less; whereas the negative values span a wide range between −2.5 × 10^{3} and 0.0. Among the negative sensitivities, 1563 components have absolute values larger than 1.0, as illustrated in ^{rd}-order mixed relative sensitivity in this submatrix is attained by S ( 3 ) ( σ t , 5 g = 30 , σ t , 6 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 2.49 × 10 3 , involving the 30^{th} energy group of the total cross section for isotopes C, ^{1}H and ^{1}H. The next largest 3^{rd}-order mixed relative sensitivity is S ( 3 ) ( σ t , 5 g = 16 , σ t , 6 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 2.21 × 10 2 .

Lastly, the submatrix S ( 3 ) ( σ t , j = 6 g , σ t , k = 6 g ′ , σ t , l = 6 g ″ ) , g , g ′ , g ″ = 1 , ⋯ , 30 comprises the 3^{rd}-order mixed relative sensitivities of the leakage response with respect to the microscopic total cross sections of isotope 6 (^{1}H). Among the 27,000 components in this submatrix, 23,207 of them have negative values for the 3rd-order mixed relative sensitivities, which vary in the wide range from −2.97 × 10^{4} to 0.0; the other 3793 components have negligibly small positive values, of the order of 10^{−4} or less. There are 6999 components having absolute values greater than 1.0. ^{rd}-order mixed relative sensitivity S ( 3 ) ( σ t , 6 g = 30 , σ t , 6 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 2.97 × 10 4 , which involves the 30^{th} energy group of the microscopic total cross section for isotope ^{1}H. As illustrated in

This work has presented the numerical computation of the (180)^{3} third-order mixed sensitivities ∂ 3 L / ∂ σ t , j g ∂ σ t , k g ′ ∂ σ t , l g ″ , j , k , l = 1 , ⋯ , 6 ; g , g ′ , g ″ = 1 , ⋯ , 30 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^{nd}- and 3^{rd}-order relative sensitivities of the PERP benchmark’s leakage response with respect to the microscopic total cross sections, are summarized in ^{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.

^{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., σ t , 6 30 ). The largest unmixed 3^{rd}-order sensitivity is also with respect to σ t , 6 30 , namely S ( 3 ) ( σ t , 6 g = 30 , σ t , 6 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 2.966 × 10 4 , as presented in ^{rd}-order sensitivity is the mixed 3^{rd}-order sensitivity S ( 3 ) ( σ t , 1 g = 30 , σ t , 6 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 1.88 × 10 5 , which also involves the microscopic total cross section for the 30^{th} energy group of isotope ^{239}Pu (i.e., σ t , 1 g = 30 ).

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^{nd}- and 1^{st}-order sensitivities. All of the important 3^{rd}-order mixed relative sensitivities have negative values.

S ( 1 ) ( σ t ) | S ( 2 ) ( σ t , σ t ) | S ( 3 ) ( σ t , σ t , σ t ) | |
---|---|---|---|

Number of elements with absolute values between 1.0 and 10.0 | 8 | 665 | 45,970 |

Number of elements with absolute values between 10.0 and 100.0 | 0 | 54 | 11,861 |

Number of elements with absolute values > 100.0 | 0 | 1 | 1199 |

Largest relative sensitivity | S ( 1 ) ( σ t , 6 30 ) = − 9.366 | S ( 2 ) ( σ t , 6 30 , σ t , 6 30 ) = 429.6 | S ( 3 ) ( σ t , 1 30 , σ t , 6 30 , σ t , 6 30 ) = − 1.88 × 10 5 |

4) Most of the 3^{rd}-order mixed relative sensitivities that involve the microscopic total cross sections of isotopes ^{240}Pu, ^{69}Ga, and ^{71}Ga have values of the order of 10^{−2} or less. However, many 3^{rd}-order mixed relative sensitivities involving the microscopic total cross sections of isotopes ^{239}Pu, C, and ^{1}H have large values.

5) The largest 1^{st}-, 2^{nd}- and 3^{rd}-order sensitivities are S ( 1 ) ( σ t , 6 30 ) = − 9.366 , S ( 2 ) ( σ t , 6 30 , σ t , 6 30 ) = 429.6 and S ( 3 ) ( σ t , 1 g = 30 , σ t , 6 g ′ = 30 , σ t , 6 g ″ = 30 ) = − 1.88 × 10 5 , respectively. Thus, the largest 1^{st}-, 2^{nd}-order sensitivities are all with respect to the microscopic total cross section for the 30^{th} energy group of isotope ^{1}H (i.e., σ t , 6 30 ), and the largest 3^{rd}-order sensitivity is with respect to the microscopic total cross section for the 30^{th} energy group of isotopes ^{239}Pu and ^{1}H (i.e., σ t , 1 g = 30 and σ t , 6 30 ). All in all, the microscopic total cross section of isotopes ^{1}H and ^{239}Pu are the two most important parameters affecting the PERP benchmark’s leakage response, since they are involved in all of the large 2^{nd}- and 3^{rd}-order sensitivities.

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 [

The authors declare no conflicts of interest regarding the publication of this paper.

Fang, R.X. and Cacuci, D.G. (2020) Third Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: II. Computed Sensitivities. American Journal of Computational Mathematics, 10, 529-558. https://doi.org/10.4236/ajcm.2020.104030

The mathematical expression of the 3^{rd}-order mixed sensitivities ∂ 3 L ( α ) / ∂ t j ∂ t k ∂ t l , j , k , l = 1 , ⋯ , J σ t of the PERP leakage response with respect to the group-averaged microscopic total cross sections has been derived in the accompanying Part I [

∂ 3 L ( α ) ∂ t j ∂ t k ∂ t l = − { 〈 ψ 1 ( 3 ) , g ( j , k ; r , Ω ) , φ g ( r , Ω ) 〉 ( 1 ) + 〈 ψ 2 ( 3 ) , g ( j , k ; r , Ω ) , ψ ( 1 ) , g ( r , Ω ) 〉 ( 1 ) + 〈 ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , ψ 1 ( 2 ) , g ( j ; r , Ω ) 〉 ( 1 ) + 〈 ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , ψ 2 ( 2 ) , g ( j ; r , Ω ) 〉 ( 1 ) } δ g l g N i l , m l , for j , k , l = 1 , ⋯ , J σ t , (1)

where

〈 ψ 1 ( 3 ) , g ( j , k ; r , Ω ) , φ g ( r , Ω ) 〉 ( 1 ) = ∑ g = 1 G ∫ d V ∫ 4 π d Ω ψ 1 ( 3 ) , g ( j , k ; r , Ω ) φ g ( r , Ω ) , (2)

〈 ψ 2 ( 3 ) , g ( j , k ; r , Ω ) , ψ ( 1 ) , g ( r , Ω ) 〉 ( 1 ) = ∑ g = 1 G ∫ d V ∫ 4 π d Ω ψ 2 ( 3 ) , g ( j , k ; r , Ω ) ψ ( 1 ) , g ( r , Ω ) , (3)

〈 ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , ψ 1 ( 2 ) , g ( j ; r , Ω ) 〉 ( 1 ) = ∑ g = 1 G ∫ d V ∫ 4 π d Ω ψ 3 ( 3 ) , g ( j , k ; r , Ω ) ψ 1 ( 2 ) , g ( j ; r , Ω ) , (4)

〈 ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , ψ 2 ( 2 ) , g ( j ; r , Ω ) 〉 ( 1 ) = ∑ g = 1 G ∫ d V ∫ 4 π d Ω ψ 4 ( 3 ) , g ( j , k ; r , Ω ) ψ 2 ( 2 ) , g ( j ; r , Ω ) . (5)

The forward multigroup neutron fluxes φ g ( r , Ω ) are the solutions [

B g ( α ) φ g ( r , Ω ) = Q g ( r ) , g = 1 , ⋯ , G , (6)

φ g ( r d , Ω ) = 0 , Ω ⋅ n < 0 , g = 1 , ⋯ , G , (7)

where r d is the radius of the PERP sphere, and where

#Math_317# (8)

Q g ( r ) ≜ ∑ k = 1 N f λ k N k , 1 F k S F ν k S F ( 2 π a k 3 b k e − a k b k 4 ) ∫ E g + 1 E g d E e − E / a k sinh b k E . (9)

The multigroup adjoint fluxes ψ ( 1 ) , g ( r , Ω ) are the solutions of the following 1st-Level Adjoint Sensitivity System (1^{st}-LASS), which has been solved in [

A g ( α ) ψ ( 1 ) , g ( r , Ω ) = Ω ⋅ n δ ( r − r d ) , g = 1 , ⋯ , G , (10)

ψ ( 1 ) , g ( r d , Ω ) = 0 , Ω ⋅ n > 0 , g = 1 , ⋯ , G . (11)

The 2^{nd}-level adjoint fluxes ψ 1 ( 2 ) , g ( j ; r , Ω ) and ψ 2 ( 2 ) , g ( j ; r , Ω ) , j = 1 , ⋯ , J σ t ; g = 1 , ⋯ , G are the solutions of the following 2^{nd}-Level Adjoint Sensitivity System (2^{nd}-LASS), which have been solved in [

A g ( α ) ψ 1 ( 2 ) , g ( j ; r , Ω ) = − δ g j g N i j , m j ψ ( 1 ) , g ( r , Ω ) , j = 1 , ⋯ , J σ t ; g = 1 , ⋯ , G , (12)

ψ 1 , j ( 2 ) , g ( r d , Ω ) = 0 , Ω ⋅ n > 0 ; j = 1 , ⋯ , J σ t ; g = 1 , ⋯ , G , (13)

#Math_326# (14)

ψ 2 , j ( 2 ) , g ( r d , Ω ) = 0 , Ω ⋅ n < 0 ; j = 1 , ⋯ , J σ t ; g = 1 , ⋯ , G . (15)

The 3^{rd}-level adjoint fluxes ψ 1 ( 3 ) , g ( j , k ; r , Ω ) , ψ 2 ( 3 ) , g ( j , k ; r , Ω ) , ψ 3 ( 3 ) , g ( j , k ; r , Ω ) and ψ 4 ( 3 ) , g ( j , k ; r , Ω ) are the solutions of the following 3^{rd}-Level Adjoint Sensitivity System (3^{rd}-LASS) [

A g ( α ) ψ 4 ( 3 ) , g ( j , k ; r , Ω ) = − ψ ( 1 ) , g ( r , Ω ) δ g k g N i k , m k , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , (16)

ψ 4 ( 3 ) , g ( j , k ; r d , Ω ) = 0 , Ω ⋅ n > 0 , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , (17)

A g ( α ) ψ 1 ( 3 ) , g ( j , k ; r , Ω ) = − [ ψ 1 ( 2 ) , g ( j ; r , Ω ) δ g k g N i k , m k + ψ 4 ( 3 ) , g ( j , k ; r , Ω ) δ g j g N i j , m j ] , (18)

ψ 1 ( 3 ) , g ( j , k ; r d , Ω , E ) = 0 , Ω ⋅ n > 0 , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , (19)

B g ( α ) ψ 3 ( 3 ) , g ( j , k ; r , Ω ) = − φ g ( r , Ω ) δ g k g N i k , m k , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , (20)

ψ 3 ( 3 ) , g ( j , k ; r d , Ω ) = 0 , Ω ⋅ n < 0 , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , (21)

B g ( α ) ψ 2 ( 3 ) , g ( j , k ; r , Ω ) = − [ ψ 2 ( 2 ) , g ( j ; r , Ω ) δ g k g N i k , m k + ψ 3 ( 3 ) , g ( j , k ; r , Ω ) δ g j g N i j , m j ] , (22)

ψ 2 ( 3 ) , g ( j , k ; r d , Ω ) = 0 , Ω ⋅ n < 0 , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j . (23)

The 3^{rd}-level adjoint fluxes ψ 3 ( 3 ) , g ( j , k ; r , Ω ) and ψ 4 ( 3 ) , g ( j , k ; r , Ω ) are solved first by running the PARTISN-code [^{nd}-level adjoint fluxes ψ 1 ( 2 ) , g ( j ; r , Ω ) and ψ 2 ( 2 ) , g ( j ; r , Ω ) to compute the remaining 3^{rd}-level adjoint fluxes ψ 1 ( 3 ) , g ( j , k ; r , Ω ) and ψ 2 ( 3 ) , g ( j , k ; r , Ω ) , respectively. Similarly, ψ 1 ( 3 ) , g ( j , k ; r , Ω ) are computed by running PARTISN [^{rd}-order mixed sensitivities, ∂ 3 L ( α ) / ∂ t j ∂ t k ∂ t l , j , k , l = 1 , ⋯ , J σ t are computed by using these fluxes in Equation (1).

Th components t j , j = 1 , ⋯ , J σ t are defined as follows [

[ t 1 , ⋯ , t J σ t ] † ≜ [ σ t , i = 1 1 , σ t , i = 1 2 , ⋯ , σ t , i = 1 G , ⋯ , σ t , i g , ⋯ , σ t , i = I 1 , ⋯ , σ t , i = I G ] † , i = 1 , ⋯ , I = 6 ; g = 1 , ⋯ , G = 30 ; J σ t = I × G = 180. (24)

In Equation (24), the dagger denotes “transposition,” σ t , i g denotes the microscopic total cross section for isotope i and energy group g, and J σ t denotes the total number of microscopic total cross sections. It is convenient to write the 3^{rd}-order absolute sensitivities of the PERP’s leakage response with respect to the microscopic total cross sections as follows: ∂ 3 L / ∂ σ t , j g ∂ σ t , k g ′ ∂ σ t , l g ″ , j , k , l = 1 , ⋯ , I ; g , g ′ , g ″ = 1 , ⋯ , G , for the I = 6 isotopes and G = 30 energy groups of the PERP benchmark. The matrix ∂ 3 L / ∂ σ t , j g ∂ σ t , k g ′ ∂ σ t , l g ″ , j , k , l = 1 , ⋯ , I ; g , g ′ , g ″ = 1 , ⋯ , G of 3^{rd}-order absolute sensitivities has dimensions J σ t × J σ t × J σ t ( = 180 × 180 × 180 ) , where J σ t = G × I = 30 × 6 . The matrix of 3^{rd}-order relative sensitivities, denoted as S ( 3 ) ( σ t , j g , σ t , k g ′ , σ t , l g ″ ) , is defined as follows:

#Math_366# (25)