Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: III. Effects of Imprecisely Known Microscopic Fission Cross Sections and Average Number of Neutrons per Fission

The Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) is applied to compute the first-order and second-order sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) experimental system with respect to the following nuclear data: Group-averaged isotopic microscopic fission cross sections, mixed fission/total, fission/scattering cross sections, average number of neutrons per fission (ν), mixed ν/total cross sections, ν/scattering cross sections, and ν/fission cross sections. The numerical results obtained indicate that the 1st-order relative sensitivities for these nuclear data are smaller than the 1st-order sensitivities of the PERP leakage response with respect to the total cross sections but are larger than those with respect to the scattering cross sections. The vast majority of the 2nd-order unmixed sensitivities are smaller than the corresponding 1st-order ones, but several 2nd-order mixed relative sensitivities are larger than the 1st-order ones. In particular, several 2nd-order sensitivities for 239Pu are significantly larger than the corresponding 1st-order ones. It is also shown that the effects of the 2nd-order sensitivities of the PERP benchmark’s leakage response with respect to the benchmark’s parameters underlying the average number of neutrons per fission, ν, on the moments (expected value, variance, and skewness) of the PERP benchmark’s leakage response distribution are negligible by comparison to the corresponding effects (on the response distribution) stemming from uncertainties in the total cross sections, but are larger than the corresponding effects (on the response distribution) stemming from uncertainties in the fission and scattering cross sections.


Introduction
This work, designated as "Part III," continues the presentations of results, commenced in Part I [1] and set forth in Part II [2], produced within the ongoing second-order comprehensive sensitivity analysis to nuclear data of the polyethylene-reflected plutonium (PERP) metal sphere benchmark described The physical system considered in this work is the same polyethylene-reflected plutonium (acronym: PERP) metal sphere benchmark [3] as was considered in the 2nd-order sensitivity and uncertainty analyses performed for the group-averaged total microscopic cross sections [1] and the group-averaged scattering cross sections [2], respectively. As in [1,2], the neutron flux is computed by solving numerically the neutron transport equation using the PARTISN [4] multigroup discrete ordinates transport code. For the PERP benchmark under consideration, PARTISN [4] solves the following multi-group approximation of the neutron transport equation with a spontaneous fission source provided by the code SOURCES4C [6]: B g (α)ϕ g (r, Ω) = Q g (r), g = 1, . . . , G, ϕ g (r d , Ω) = 0, Ω · n < 0, g = 1, . . . , G where r d denotes the external radius of the PERP benchmark, and where B g (α)ϕ g (r, Ω) Ω·∇ϕ g (r, Ω) + Σ g t (r) ϕ g (r, Ω) − G g =1 4π Σ g →g s (r, Ω → Ω) ϕ g (r, Ω )dΩ − χ g (r) In Equation (1), the vector α denotes the "vector of imprecisely known model parameters", which has been defined in [1] as α σ t ; σ s ; σ f ; ν; p; q; N † , having the vector-components σ t , σ s , σ f , ν, p, q and N which comprise the various model parameters for the microscopic total cross sections, scattering cross sections, fission cross sections, average number of neutrons per fission, fission spectra, sources, and isotopic number densities, respectively.

First-Order Sensitivities ∂L(α α α)/∂σ σ σ f
The first-order sensitivity of the PERP leakage response to the group-averaged microscopic fission cross sections, which will be denoted as ∂L(α)/∂ f j f =σ f , comprises two types of contributions.
The first type of contributions, which will be denoted as ∂L(α)/∂ f j (1) f =σ f , arises from quantities that involve the macroscopic fission cross sections directly, while the second type of contributions stems indirectly, through the macroscopic total cross sections, which comprise the fission cross sections in their definitions. The contributions ∂L(α)/∂ f j (1) f =σ f are computed using the following particular forms of Equations (152), (156) and (157) in [5]. For convenient referencing, the corresponding equations from [5] used in this work were reproduced in Appendix B.
The second type of contributions, which will be denoted as ∂L(α)/∂ f j (2) f =σ f , includes the contributions stemming from the total cross sections, since the total cross sections comprises the fission cross sections. The contributions are computed using Equation (150) in [5] in conjunction with the , to obtain: Adding Equations (6) and (10) yields the following expression: V dV 4π dΩ ψ (1),g (r, Ω)ϕ g (r, Ω) ∂Σ t g ∂ f j , f or j = 1, . . . , J σ f .
The numerical values of the 1st-order relative sensitivities, S (1) σ ∂L/∂σ g f ,i σ g f ,i /L , i = 1, 2; g = 1, . . . , 30, of the leakage response with respect to the fission microscopic cross sections for the six isotopes contained in the PERP benchmark will be presented in Section 2.3, below, in tables that will also include comparisons with the numerical values of the corresponding 2nd-order unmixed relative sensitivities S (2)
The parameters f j and f m 2 in Equations (17), (18) and (20) correspond to the fission cross sections, and are therefore denoted as f j ≡ σ g j f ,i j and f m 2 ≡ σ g m 2 f ,i m 2 , respectively, where the subscripts i m 2 and g m 2 refer to the isotope and energy groups associated with the parameter f m 2 , respectively, and where the index m 2 is defined in Equation (17). Noting that and inserting the results obtained in Equation (12) and in Equations (22)-(25) into Equations (18), (20) and (17) reduces the latter equation to the following expression: where U (2),g 1,j;0 (r) 4π dΩ u (2),g 1,j (r, Ω),
Additional contributions stem from Equation (160) in [5], in conjunction with the relation , which takes on the following particular form: Inserting the results obtained in Equations (23) and (24) into Equation (41), and performing the respective angular integrations, yields the following simplified expression for Equation (41): where the flux moments ξ (2),g m 2 1, j;0 (r) and ξ (2),g 2, j;0 (r) are defined as follows: Further contributions stem from Equation (177) in [5] in conjunction with the relations , as follows: Inserting the results obtained in Equation (37) into Equation (45) reduces the latter to the following expression: Collecting the partial contributions obtained in Equations (26), (38), (42) and (46), yields the following result: (2),g m 2 1,j (r, Ω)ψ (1),g m 2 (r, Ω) + u (2),g m 2 2,j (r, Ω)ϕ g m 2 (r, Ω) , The 2nd-order absolute sensitivities of the leakage response with respect to the fission cross sections, i.e., ∂ 2 L/∂σ g f ,i ∂σ g f ,k , i, k = 1, . . . , N f ; g, g = 1, . . . , G, for the N f = 2 fissionable isotopes and G = 30 energy groups of the PERP benchmark are computed using Equation (47). The (Hessian) matrix For convenient comparisons, the numerical results presented in this section are presented in unit-less values of the relative sensitivities that correspond f ,k and are defined as follows: The numerical results obtained for the matrix S (2) σ g f ,i , σ g f ,k , i, k = 1, 2; g, g = 1, . . . , 30, have been partitioned into N f × N f = 4 submatrices, each of dimensions G × G(= 30 × 30); the summary of the main features of each submatrix is presented in Table 1. The results for the submatrices are presented in the following form: when a submatrix comprises elements with relative sensitivities with 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 is listed in Table 1, together with the phase-space coordinates of that element. The submatrix S (2) Table 1 comprises components with absolute values greater than 1.0; it will therefore be discussed in detail in subsequent sub-sections of this section.
Max. value = 6.97 × 10 −2 at g = 12, g = 12 i = 2 ( 240 Pu) Max. value = 6.97 × 10 −2 at g = 12, g = 12 Max. value = 3.60 × 10 −3 at g = 12, g = 12 The 2nd-order mixed sensitivities ∂ 2 L(α)/∂σ f ∂σ f are mostly positive. Among all . . . , 30, a total of 3508 out of 3600 elements have positive values, and most of them are very small, as indicated in Table 1. However, among all the J σ f × J σ f (= 60 × 60) elements, 11 of them have very large relative sensitivities, with values greater than 1.0, as noted in the table. All of these larger sensitivities reside in the sub-matrix S (2) σ g f ,1 , σ g f ,1 , and relate to the fission cross sections in isotope 239 Pu. The overall maximum relative sensitivity is S (2) σ 12 f ,1 , σ 12 f ,1 = 1.348. Additional details about the sub-matrix S (2) σ g f ,1 , σ g f ,1 are provided in the following section. Also noted in Table 1 are the results that all of the mixed 2nd-order relative sensitivities involving the fission cross sections of isotope 240 Pu (i.e., σ g f ,2 ) have absolute values smaller than 1.0. The elements with the maximum absolute value in each of the respective submatrices relate to the fission cross sections for the 12th energy group of isotopes 239 Pu and 240 Pu.
The 2nd-order unmixed sensitivities S (2) . . . , 30, which are the elements on the diagonal of the matrix S (2) σ g f ,i , σ g f ,k , i, k = 1, 2; g, g = 1, . . . , 30, can be directly compared to the values of the 1st-order relative sensitivities . . . , 30, for the leakage response with respect to the fission cross section parameters. These comparisons are presented in Tables 2 and 3 for the two fissionable isotopes contained in the PERP benchmark. Table 2 compares the 1st-order to the 2nd-order relative sensitivities for isotope 1 ( 239 Pu). This comparison indicates that the values of the 2nd-order sensitivities are comparable to, and generally smaller than, the corresponding values of the 1st-order sensitivities for the same energy group, except for the 12th energy group, where the 2nd-order relative sensitivity is larger. The largest values (shown in bold in the table) for the 1st-order and 2nd-order relative sensitivities both related to the 12th energy group of isotope 239 Pu. It is noteworthy that all of the 1st-order relative sensitivities are positive, signifying that an increase in σ g f ,1 will cause an increase in L.
Comparing the corresponding results in Table 2 in this work with Table 5 of Part I [1] and Table 6 of Part II [2] reveals that the absolute values of the 1st-order relative sensitivities with respect to the fission cross sections are significantly smaller than the corresponding 1st-order relative sensitivities with respect to the total cross sections, but they are approximately one order of magnitude larger than the corresponding 1st-order relative sensitivities with respect to the 0th-order self-scattering cross sections for isotope 239 Pu. Likewise, the absolute values of the 2nd-order unmixed relative sensitivities with respect to the fission cross sections are approximately 50-90% smaller than the corresponding 2nd-order unmixed relative sensitivities to the total cross sections, but they are approximately one to two orders of magnitudes larger than the corresponding 2nd-order unmixed relative sensitivities for the 0th-order self-scattering cross sections for isotope 239 Pu.  Table 3 presents the results for the 1st-order and 2nd-order unmixed relative sensitivities for isotope 2 ( 240 Pu). These results show that the values for both the 1st-and 2nd-order relative sensitivities are all very small, and the absolute values of the 2nd-order unmixed relative sensitivities are at least one order of magnitude smaller than the corresponding values of the 1st-order ones for all energy groups. The largest 1st-order relative sensitivity is S (1) σ 12 f ,i=2 = 4.568 × 10 −2 , and the largest 2nd-order unmixed relative sensitivity is S (2) σ 12 f ,i=2 , σ 12 f ,k=2 = 3.602 × 10 −3 ; both occur for the 12th energy group of the fission cross section of 240 Pu.

Second-Order Relative Sensitivities
, σ g f ,k=1 , g, g = 1, . . . , 30 Figure 1 depicts the 2nd-order mixed relative sensitivity results obtained for S (2) f ,k=1 /L , g, g = 1, . . . , 30, for the leakage response with respect to the fission cross sections of 239 Pu. This matrix is symmetrical with respect to its principal diagonal. As shown in Figure 1, the largest 2nd-order relative sensitivities are concentrated in the energy region confined by the energy groups g = 7, . . . , 14 and g = 7, . . . , 14. The numerical values of these elements are presented in Table 4. Shown in bold in this Table are 11 sensitivities, all involving the 12th energy group of the fission cross sections σ f ,k=1 of 239 Pu, which have values greater than 1.0. The largest value among these sensitivities is attained by the relative 2nd-order unmixed sensitivity Figure 1 also shows that the majority (877 out of 900) of the elements of S (2) , σ g f ,k=1 have positive 2nd-order relative sensitivities. The remaining 23 elements are located mostly on the diagonal of S (2) , σ g f ,k=1 and have negative values, as presented in Table 2, above.

Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Parameters Underlying the Benchmark's Fission and Total Cross Sections
This Section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities ∂ 2 L(α)/∂σ f ∂σ t , of the leakage response with respect to the group-averaged fission and total microscopic cross sections of all isotopes of the PERP benchmark. As has been shown by Cacuci [5], these mixed sensitivities can be computed using two distinct expressions, involving distinct 2nd-level adjoint systems and corresponding adjoint functions, by considering either the computation of ∂ 2 L(α)/∂σ f ∂σ t or the computation of ∂ 2 L(α)/∂σ t ∂σ f . These two distinct paths for computing the 2nd-order sensitivities with respect to group-averaged fission and total microscopic cross sections will be presented in Section 3.1 and, respectively, Section 3.2. The end results produced by these two distinct paths must be identical to one another, thus providing a mutual "solution verification" that the respective computations were performed correctly. Moreover, the computational time for these two distinct paths can be much different, and one of them provides the best computational speed, as will be further illustrated by the numerical results presented in Section 3.3.
The matrix ∂ 2 L/∂ f j ∂t m 2 , j = 1, . . . , J σ f ; m 2 = 1, . . . , J σ f has dimensions J σ f × J σt (= 60 × 180); corresponding to this matrix is the matrix denoted as S (2) σ g f ,i , σ g t,k of 2nd-order relative sensitivities, which is defined as follows: To facilitate the presentation and interpretation of the numerical results, the J σ f × J σt (= 60 × 180) matrix S (2) σ g f ,i , σ g t,k has been partitioned into N f × I = 2 × 6 submatrices, each of dimensions G × G = 30 × 30. The summary of the main features of each of these submatrices is presented in Table 5. Table 5. Summary presentation of the matrix S (2) 35 elements with absolute values > 1.0 Min. value = −1.67 × 10 −1 at g = 12, g = 12 Min. value = −7.48 × 10 −3 at g = 12, g = 12 Min. value = −5.08 × 10 −3 at g = 12, g = 12 1 element with absolute value > 1.0 48 elements with absolute values > 1.0 Min. value = −1.36 × 10 −1 at g = 12, g = 12 Min. value = −8.62 × 10 −3 at g = 12, g = 12 Min. value = −3.87 × 10 −4 at g = 12, g = 12 Min. value = −2.63 × 10 −4 at g = 12, g = 12 Min. value = −6.04 × 10 −2 at g = 12, g = 30 Min. value = −7.21 × 10 −1 at g = 12, g = 30 Most of the values of the J σ f × J σt (= 10, 800) elements in the matrix S (2) σ g f ,i , σ g t,k , i = 1, 2; k = 1, . . . , 6; g, g = 1, . . . , 30 are very small, and 10,704 elements out of 10,800 elements have negative values. The results in Table 5 indicate that, when the 2nd-order mixed relative sensitivities involve the fission cross sections of the isotope 240 Pu or the total cross sections of isotopes 240 Pu, 69 Ga and 71 Ga, their absolute values are all smaller than 1.0, and the element with the most negative value in each of the submatrices always relates to the fission cross sections for the 12th energy group and the total cross sections for either the 12th energy group or the 30th energy group of the isotopes. There are 84 elements with large relative sensitivities, having values greater than 1.0, as indicated in Table 5. Those large sensitivities reside in the submatrices S (2) 5 and S (2) 6 , respectively. All of these 84 large sensitivities involve the fission cross sections of isotope 239 Pu, and the total cross sections of isotopes 239 Pu, C and 1 H. Of the sensitivities summarized in Table 5, the single largest relative value is S (2) σ 12 f ,1 , σ 30 t,6 = −13.92. The results obtained for the 2nd-order mixed relative sensitivity of the leakage response with respect to the fission microscopic cross sections of 239 Pu and to the total microscopic cross sections of 239 Pu, denoted as S (2) /L , g, g = 1, . . . , 30, are summarized in Table 6 and depicted in Figure 2. Almost all, namely 894 out of 900, elements in this submatrix have negative 2nd-order relative sensitivities; only 6 elements have small positive values. As shown in Figure 2, there are some large 2nd-order mixed relative sensitivities concentrated in the energy region confined by the energy groups g = 7, . . . , 14 and g = 7, . . . , 16. The actual numerical values of these large elements are presented in Table 6, which comprises 35 elements having values greater than 1.0, as shown in bold in this table. The largest absolute value in this submatrix is attained by the relative 2nd-order mixed sensitivity S (2) t,k=1 = −2.630, which involves the 12th energy group for both the fission and total cross sections of isotope 239 Pu. Table 6. Components of S (2)  The absolute values of the mixed sensitivities in row g = 12 are the largest among all g = 1, . . . , 30 rows, including rows not presented in Table 6. In other words, the absolute value of mixed relative sensitivities involving the fission cross section parameter σ g=12 f ,i=1 are always the largest among all groups g = 1, . . . , 30. Similarly, the values of the mixed sensitivities in group g = 12 are the most negative among all groups g = 1, . . . , 30, with one exception for the sensitivity value located in groups g = 13 and g = 12 which is less negative than the value located in groups g = 13 and g = 13.

Second-Order Relative Sensitivities
The matrix S (2) /L , comprising the 2nd-order sensitivities of the leakage response with respect to the fission cross sections of isotope 1 ( 239 Pu) and the total cross sections of isotope 5 (C), contains a single large element that has an absolute value greater than 1.0, namely S (2) σ 12 f ,1 , σ 30 t,5 = −1.167.

Second-Order Relative Sensitivities
The submatrix S (2) /L , comprising the 2nd-order relative sensitivities of the leakage response with respect to the fission microscopic cross sections of isotope 1 ( 239 Pu) and the total microscopic cross sections of isotope 6 ( 1 H), is illustrated in Figure 3. The submatrix S (2) 6 , g, g = 1, . . . , 30 includes 48 elements that have absolute values greater than 1.0, as specified, in bold, in Table 7; 35 out of these 48 elements are located in the energy phase-space confined by the energy groups g = 7, . . . , 13 and g = 16, . . . , 25. The other 13 elements are located in energy groups g = 6, . . . , 30 and g = 30. The largest negative value is displayed by the 2nd-order relative sensitivity of the leakage response with respect to the 12th energy group of the fission cross section for 239 Pu and the 30th energy group of the total cross section for 1 H,  As shown in Table 7, the values of the 2nd-order mixed sensitivities involving the fission cross section parameter σ g=12 f ,i=1 , in energy group g = 12, are the most negative among all energy groups g = 1, . . . , 30. In addition to the sensitivities presented in Table 7, the following 2nd-order mixed relative sensitivities of the leakage response with respect to the fission microscopic cross sections of 239 Pu and the total microscopic cross sections of 1 H have absolute values greater than 1.0: S (2) σ 12 f ,1 , σ 23 t,6 = −1.213,

Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Parameters Underlying the Benchmark's Fission and Scattering Cross Sections
This Section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities ∂ 2 L(α)/∂σ f ∂σ s , of the leakage response with respect to the group-averaged fission and scattering microscopic cross sections of all isotopes of the PERP benchmark. Similarly, the 2nd-order mixed sensitivities ∂ 2 L(α)/∂σ f ∂σ s can also be computed using the alternative expressions for ∂ 2 L(α)/∂σ s ∂σ f . These two distinct paths for computing the 2nd-order sensitivities with respect to group-averaged fission and scattering microscopic cross sections will be presented in Section 4.1 and, respectively, Section 4.2 as follows. As will be discussed in detail in Section 4.3, below, the pathway for computing ∂ 2 L(α)/∂σ f ∂σ s turns out to be 60 times more efficient than the pathway for computing ∂ 2 L(α)/∂σ s ∂σ f .

Computing the Second-Order Sensitivities
The equations needed for deriving the expressions of the 2nd-order sensitivities ∂ 2 L/∂ f j ∂s m 2 , j = 1, . . . , J σ f ; m 2 = 1, . . . , J σs will differ from each other depending on whether the parameter s m 2 corresponds to the 0th-order (l = 0) scattering cross sections or to the higher-order (l ≥ 1) scattering cross sections, since, as shown in Equation (A3) of Appendix A, the 0th-order scattering cross sections contribute to the total cross sections, while the higher-order ones do not. Therefore, the zeroth order scattering cross sections must be considered separately from the higher order scattering cross sections. As described in [1] and Appendix A, the total number of zeroth-order (l = 0) scattering cross section comprise in σ s is denoted as J σs,l=0 , where J σs,l=0 = G × G × I; and the total number of higher order (i.e., l ≥ 1) scattering cross sections comprised in σ s is denoted as J σs,l≥1 , where J σs,l≥1 = G × G × I × ISCT, with J σs,l=0 + J σs,l≥1 = J σs . There are two distinct cases, as follows: (1) where the quantities f j refer to the parameters underlying the fission microscopic cross sections, while the quantities s m 2 refer to the parameters underlying the 0th-order scattering microscopic cross sections; and (2) where the quantities f j refer to the parameters underlying the fission microscopic cross sections, and the quantities s m 2 refer to the parameters underlying the l th -order (l ≥ 1) scattering microscopic cross sections.
The equations needed for deriving the expression of the 2nd-order mixed sensitivities are obtained by particularizing Equations (158), (159), (177) and (178) in [5] to the PERP benchmark, where Equation (178) provides the contributions arising directly form the respective fission and scattering cross sections, while Equations (158), (159) and (177) provide contributions arising indirectly through the total cross sections, since both the fission cross sections and the 0th-order scattering cross sections are part of the total cross sections. The expression obtained by particularizing Equation (178) in [5] to the PERP benchmark yields: In Equation (67), the parameters indexed by f j correspond to the fission cross sections, which means that f j ≡ σ g j f ,i j , while the parameters indexed by s m 2 correspond to the 0th-order scattering cross where the subscripts i m 2 , l m 2 , g m 2 and g m 2 refer to the isotope, order of Legendre expansion, and energy groups associated with the parameter s m 2 , respectively. Noting that inserting the results obtained in Equations (68) and (69) into Equation (67), using the addition theorem for spherical harmonics in one-dimensional geometry, performing the respective angular integrations, and finally setting l m 2 = 0 in the resulting expression yields the following simplified expression for Equation (67): where the 0th-order moments ϕ (r), U (2),g m 2 1, j;0 (r) and U (2),g m 2 2, j;0 (r) have been defined previously in Equations (15), (16), (27) and (28), respectively.
Using Equation (158) in [5] in conjunction with the relations yields the following contributions: where the adjoint functions ψ Noting that, and inserting the results obtained in Equations (72) and (73) into Equation (71), yields the following simplified expression for Equation (71): Additional contributions stem from Equation (159) in [5], which takes on the following particular form: Inserting the results obtained in Equations (68) and (69) into Equation (75), using the addition theorem for spherical harmonics in one-dimensional geometry and performing the respective angular integrations, and setting l m 2 = 0, yields the following simplified expression for Equation (75): where the 0th-order moments ξ (2),g m 2 1,j;0 (r) and ξ (2),g m 2 2, j;0 (r) have been defined previously in Equations (43) and (44), respectively.
Using Equation (177) in [5] in conjunction with the relation ∂Σ t g ∂t m 2 yields the final set of contributions, namely: Replacing the result obtained in Equation (73) into Equation (77) yields the following simplified expression: Collecting the partial contributions obtained in Equations (70), (74), (76) and (78), yields the following result: For computing the 2nd-order sensitivities , the parameters f j ≡ σ is obtained by particularizing Equations (159) and (178) in [5] to the PERP benchmark, which yields, where the first two terms arise from Equation (178) while the last two terms arise from Equations (159). Inserting the results obtained in Equations (68) and (69) into Equation (80), using the addition theorem for spherical harmonics in one-dimensional geometry and performing the respective angular integrations, yields the following expression: The results to be computed using the expressions for ∂ 2 L(α)/∂σ f ∂σ s obtained in Equations (79) and (81) can be verified, because of the symmetry of the mixed 2nd-order sensitivities, by obtaining the expressions for ∂ 2 L(α)/∂σ s ∂σ f , which also requires separate consideration of the zeroth-order scattering cross sections. The two cases involved are as follows: (1) where the quantities s j refer to the parameters underlying the 0th-order scattering cross sections, while the quantities f m 2 refer to the parameters underlying the fission cross sections; (2) where the quantities s j refer to the parameters underlying the l th -order (l ≥ 1) scattering cross sections, and the quantities f m 2 refer to the parameters underlying the fission cross sections.

Second-Order Sensitivities
The equations needed for deriving the expression of the 2nd-order mixed sensitivities are obtained by particularizing Equations (158), (160), (167) and (169) in [5] to the PERP benchmark. The expression obtained by particularizing Equation (169) in [5] to the PERP benchmark is as follows: where the 2nd-level adjoint functions, θ  51) and (52) of Part II [2], which are reproduced below for convenient reference: (90) In Equation (88), the parameters indexed by s j correspond to the 0th-order scattering cross sections, so that s j ≡ σ g j →g j s,l j =0,i j , while the parameters indexed by f m 2 correspond to the fission cross sections, so Inserting the results obtained in Equations (23) and (24) into Equation (88), yields the following simplified expression for Equation (88): where the 0th-order moments Θ (2),g m 2 1, j;0 (r) and Θ (2),g 2, j;0 (r) are defined as follows: Using Equation (158) in [5] in conjunction with the relations yields the following contributions: where the 2nd-level adjoint functions, ψ  [2], which are reproduced below for convenient reference: Noting that and inserting the results obtained in Equations (101) and (37) into Equation (96), yields the following simplified expression for Equation (96): Additional contributions stem from Equation (160) in [5], which takes on the following particular form: Inserting the results obtained in Equations (23) and (24) into Equation (103), yields the following simplified expression for Equation (103): Using Equation (167) in [5] in conjunction with the relation ∂Σ t g yields the following contributions: Inserting the results obtained in Equation (37) into Equation (105), yields the following simplified expression: Collecting the partial contributions obtained in Equations (93), (102), (104) and (106), yields the following result:
The dimensions of the matrix s,l,k , is defined as follows: To facilitate the presentation and interpretation of the numerical results, the J σ f × J σs (= 60 × 21, 600) matrix S (2) Table 8 presents the results for 2nd-order relative sensitivities of the leakage response with respect to the fission cross sections and the 0th-order scattering cross sections for all isotopes, All of these 2nd-order relative sensitivities are smaller than 1.0. The value of the largest element of each of the respective sub-matrices is presented in Table 8, together with the phase-space coordinates of the respective element. For the 2nd-order mixed sensitivities with respect to the 0th-order scattering cross sections, the values can be positive or negative, but there are more positive values than negative ones.
For example, the submatrix S (2) σ g f , 1 , σ g →h s,l= 0,1 , having dimensions G × (G · G) = 30 × 900, comprises 7577 positive elements, 2563 negative elements, while the remaining elements are zero. The largest absolute values of the mixed 2nd-order sensitivities all involve the fission cross sections for the 12th energy group of isotopes 239 Pu or 240 Pu, and (mostly) the 0th-order self-scattering cross sections in the 12th energy group for isotopes 239 Pu, 240 Pu, 69 Ga, 71 Ga andC, or (occasionally) the 0th-order out-scattering cross section σ 16→17 s,l= 0,k=6 for isotope 1 H. All of the largest elements in the respective sub-matrix are positive, and the vast majority of them are very small. The overall largest element in the matrix S (2) s,l= 0,1 = 3.03 × 10 −1 . Table 8. Summary presentation of the matrix S (2) Table 9 summarizes the results for the 2nd-order mixed relative sensitivities of the leakage response with respect to the fission cross sections and the 1st-order scattering cross sections for all isotopes,  Table 9, the largest absolute values of the mixed 2nd-order sensitivities all involve the fission cross sections σ g=12 f ,i , i = 1, 2 for the 12th energy group of isotopes 239 Pu or 240 Pu, and either the 1st-order self-scattering cross sections σ 7→7 s,l=1,k , k = 1, . . . , 4 in the 7th energy group for isotopes 239 Pu, 240 Pu, 69 Ga and 71 Ga, or the 1st-order self-scattering cross sections σ 12→12 s,l=1,k , k = 5, 6 in the 12th energy group for isotopes C and 1 H. All of the largest (in absolute value) elements are negative, and the vast majority of them are very small. The overall most negative element in the matrix S (2) s,l= 1,1 = −1.70 × 10 −1 .     Table 10, all of the largest absolute values of the mixed 2nd-order sensitivities involve the fission cross sections σ g=12 f ,i , i = 1, 2 for the 12th energy group of isotopes 239 Pu or 240 Pu, and involve either the 2nd-order self-scattering cross sections σ 12→12 s,l=2,i=6 in the 12th energy group for isotope 1 H, or the 2nd-order self-scattering cross sections σ 7→7 s,l=2,k , k = 1, . . . , 5 in the 7th energy group for isotopes 239 Pu, 240 Pu, 69 Ga, 71 Ga and C. As shown in Table 10, all of the largest elements in the respective sub-matrix are positive, and the vast majority of them are very small. The overall largest element in the matrix S (2) s,l= 2,1 = 1.02 × 10 −2 .     Table 11, the mixed 2nd-order sensitivities having the largest absolute values involve the fission cross sections σ g=12 f ,i , i = 1, 2 for the 12th energy group of isotopes 239 Pu or 240 Pu, and either the 3rd-order self-scattering cross sections σ 12→12 s,l=3,i=6 for the 12th energy group for isotope 1 H or the 3rd-order self-scattering cross sections σ 7→7 s,l=3,k , k = 1, . . . , 5 for the 7th energy group for isotopes 239 Pu, 240 Pu, 69 Ga, 71 Ga andC. The overall largest (in absolute value) element of the matrix S (2) s,l= 3,k=6 = −1.25 × 10 −2 . Table 11. Summary presentation of the matrix S (2) The results in Tables 9-11 indicate that the largest mixed second-order relative sensitivities in matrices S (2)  s,l=3,k frequently involve the self-scattering cross sections in the 7th-energy group namely, σ 7→7 s,l,k , l = 1, 2, 3; k = 1, . . . , 4, which is likely due to the fact that, for isotope 239 Pu, the scattering cross section σ 7→7 s,l,k=1 , l = 1, 2, 3 has the largest value among all scattering cross sections σ g →h s,l,k=1 , g , h = 1, . . . , 30, for l = 1, 2, 3. Figure 4 shows the energy-group structure of the fission spectrum for isotope 239 Pu, highlighting that most of the spectrum is concentrated in the energy region g = 7, . . . , 14, with the largest portion contained in group 12. It is therefore not surprising that most of the large mixed 2nd-order relative sensitivities of ∂ 2 L(α)/∂σ f ∂σ f , ∂ 2 L(α)/∂σ f ∂σ t , ∂ 2 L(α)/∂σ f ∂σ s and ∂ 2 L(α)/∂ν∂σ f are concentrated in the energy region g = 7, . . . , 14 of the fission cross sections of 239 Pu. In particular, the 1st-and 2nd -order sensitivities of leakage response to the fission cross sections of 239 Pu are both related to the 12th energy group, which is expected since energy-group 12 contains the largest portion of the fission spectrum of 239 Pu.
When the parameters f j correspond to the average number of neutrons per fission, i.e., f j ≡ ν g j i j , the following relation holds: Inserting Equation (112) into Equation (111) yields the following simplified expression for computational purposes: The numerical values of the 1st-order relative sensitivities, S (1) ν g i ∂L/∂ν g i ν g i /L , i = 1, 2; g = 1, . . . , 30, of the leakage response with respect to the average number of neutrons per fission for the two fissionable isotopes contained in the PERP benchmark will be presented in Section 5.3, below, in tables that will also include comparisons with the numerical values of the corresponding 2nd-order unmixed relative sensitivities S (2) ν g i , ν

Numerical Results for ∂ 2 L(α α α)/∂ν ν ν∂ν ν ν
The 2nd-order absolute sensitivities of the leakage response with respect to the parameters underlying the average number of neutrons per fission, i.e., ∂ 2 L/∂ν g i ∂ν g k , i, k = 1, . . . , N f ; g, g = 1, . . . , G, for the N f = 2 fissionable isotopes and G = 30 energy groups of the PERP benchmark are computed using Equation (123). The (Hessian) matrix g k and are defined as follows: The numerical results obtained for the matrix S (2) ν g i , ν g k , i, k = 1, 2; g, g = 1, . . . , 30 have been partitioned into N f × N f = 4 submatrices, each of dimensions G × G(= 30 × 30), and the summary of the main features of each submatrix is presented in Table 12. Max. value = 1.54 × 10 −1 at g = 12, g = 12 Max. value = 1.54 × 10 −1 at g = 12, g = 12 Max. value = 8.01 × 10 −3 at g = 12, g = 12 The 2nd-order mixed sensitivities S (2) ν g i , ν g k , i, k = 1, 2; g, g = 1, . . . , 30 are all positive. Most of the J ν × J ν (= 60 × 60) elements are very small, but 52 elements have very large relative sensitivities, with values greater than 1.0, as summarized in Table 12. All of these 52 large sensitivities belong to the sub-matrix S (2) ν g 1 , ν g 1 , and relate to the parameters corresponding to the average number of neutrons per fission in isotope 239 Pu. The overall maximum relative sensitivity is S (2) ν 12 1 , ν 12 1 = 2.963. Additional details about the sub-matrix S (2) ν g 1 , ν g 1 , g, g = 1, . . . , 30 is provided in the following Section. Also noted in Table 12 is that all of the mixed 2nd-order relative sensitivities involving ν  Table 13 presents the results obtained for the 1st-and 2nd-order unmixed relative sensitivities with respect to the average number of neutrons per fission ν for isotope 1 ( 239 Pu). These results indicate that for energy groups g = 7, . . . , 14, the values of the 2nd-order sensitivities are significantly larger than the corresponding values of the 1st-order sensitivities for the same energy group; for other energy groups, the 2nd-order relative sensitivities are smaller than the corresponding values of the 1st-order sensitivities. All of the 1st-and 2nd-order relative sensitivities are positive, and the largest values for the 1st-order and 2nd-order relative sensitivities are both related to the 12th energy group. Table 14 presents the 1st-order and 2nd-order unmixed relative sensitivities for isotope 2 ( 240 Pu). The results in this table indicate that the values for both the 1st-and 2nd-order relative sensitivities are all very small, and the values of the 2nd-order unmixed relative sensitivities are at least one order of magnitude smaller than the corresponding values of the 1st-order ones for all energy groups. The largest 1st-order relative sensitivity is S (1) ν 12 i=2 = 6.316 × 10 −2 , and the largest 2nd-order unmixed relative sensitivity is S (2) ν 12 i=2 , ν 12 k=2 = 8.011 × 10 −3 , both occur for the 12th energy group of the average number of neutrons per fission for isotope 240 Pu.  Table 15 presents the 2nd-order mixed relative sensitivity results obtained for S (2)

Second-Order Relative Sensitivities
. . . , 30, for the leakage response with respect to the parameters underlying the average number of neutrons per fission of isotope 239 Pu. The majority of the larger 2nd-order relative sensitivities are concentrated in the energy region confined by the energy groups g = 7, . . . , 14 and g = 7, . . . , 14. Shown in bold in Table 15 are the numerical values of 52 elements that have values greater than 1.0. The largest value among these sensitivities is attained by the relative 2nd-order unmixed sensitivity S (2) ν  In addition to the sensitivities presented in Table 15, the following 2nd-order relative sensitivities in the matrix S (2) ν g 1 , ν g 1 , g, g = 1, . . . , 30 have values greater than 1.0: . Also, as shown in Table 15, the values of the mixed sensitivities in row g = 12 are the largest among all g = 1, . . . , 30 rows. Likewise, the values of the mixed sensitivities in column g = 12 are the largest among all groups g = 1, . . . , 30.

Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Parameters Underlying the Average Number of Neutrons Per Fission and Total Cross Sections
This section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities ∂ 2 L(α)/∂ν∂σ t of the leakage response with respect to the average number of neutrons per fission and total microscopic cross sections of all isotopes of the PERP benchmark. Similarly, these mixed sensitivities can be computed using either the computation of ∂ 2 L(α)/∂ν∂σ t or the computation of ∂ 2 L(α)/∂σ t ∂ν. These two distinct paths will be presented in Sections 6.1 and 6.2, respectively.
The matrix of 2nd-order relative sensitivities corresponding to , is defined as follows: To facilitate the presentation and interpretation of the numerical results, the J ν × J σt (= 60 × 180) matrix S (2) ν g i , σ g t,k has been partitioned into N f × I = 2 × 6 submatrices, each of dimensions G × G = 30 × 30. The main features of each of these submatrices is presented in Table 16. 72 elements with absolute values > 1.0 Min. value = −2.39 × 10 −1 at g = 12, g = 12 Min. value = −1.08 × 10 −2 at g = 12, g = 12 Min. value = −7.31 × 10 −3 at g = 12, g = 12 5 7 elements with absolute values > 1.0 99 elements with absolute values > 1.0 Min. value = −1.97 × 10 −1 at g = 12, g = 12 Min. value = −1.25 × 10 −2 at g = 12, g = 12 Min. value = −5.60 × 10 −4 at g = 12, g = 12 Min. value = −3.80 × 10 −4 at g = 12, g = 12 Min. value = −8.41 × 10 −2 at g = 12, g = 30 6 1 element with absolute value > 1.0 Most of the values of the J ν × J σt (= 10, 800) elements in the matrix S (2) ν g i , σ g t,k , i = 1, 2; k = 1, . . . , 6; g, g = 1, . . . , 30 are very small, and the majority (10,780 out of 10,800) of these elements have negative values. The results in Table 16 indicate that, when the 2nd-order mixed relative sensitivities involve ν g 2 , g = 1, . . . , 30 or the total cross sections of isotopes 240 Pu, 69 Ga and 71 Ga, their absolute values are all smaller than 1.0, except for one element in the submatrix S (2) ν g 2 , σ g t, 6 . The element with the most negative value in each of the submatrices is always related to ν g i , i = 1, 2 for the 12th energy group and σ g t,k , k = 1, . . . , 6 for either the 12th or the 30th energy group. There are 179 elements with large relative sensitivities, having absolute values greater than 1.0, as indicated in Table 16. Those large sensitivities reside in the submatrices S (2) and S (2) ν g 2 , σ g t, 6 , respectively, and 178 out of the 179 large sensitivities involve the average number of neutrons per fission of isotope 239 Pu, namely, ν g 1 , and the total cross sections of isotopes 239 Pu, C and 1 H. Of the sensitivities summarized in Table 16, the single largest relative value is S (2) ν 12 1 , σ 30 t,6 = −19.29. The submatrix S (2) /L comprises the 2nd-order mixed relative sensitivity results obtained for, g, g = 1, . . . , 30, for the leakage response with respect to the average number of neutrons per fission of 239 Pu and to the total microscopic cross sections of 239 Pu. All elements in this submatrix have negative 2nd-order relative sensitivities. The largest 2nd-order mixed relative sensitivities are concentrated in the energy region confined by the energy groups g = 7, . . . , 14 and g = 7, . . . , 16. The numerical values of these large elements are presented in Table 17, which indicates (in bold) the 72 elements that have values greater than 1.0. The largest absolute value in this submatrix is attained by the relative 2nd-order mixed sensitivity S (2) t,k=1 = −3.785, involving the parameters representing the average number of neutrons per fission and total cross section of isotope 239 Pu in the 12th energy group. In addition to the sensitivities presented in Table 17, the following 2nd-order relative sensitivities in the matrix S (2) ν g i=1 , σ g t,k=1 , g, g = 1, . . . , 30 have absolute values greater than 1.0: The absolute values of the mixed sensitivities in row g = 12 are the largest among all g = 1, . . . , 30 rows, including rows not presented in Table 17. Similarly, the values of the mixed sensitivities in group g = 12 are the most negative among all groups g = 1, . . . , 30, except for the sensitivity value located in groups g = 13 and g = 12, which is less negative than the value located in groups g = 13 and g = 13.
6.3.2. Second-Order Relative Sensitivities S (2) ν g 1 , σ g t,5 , g, g = 1, . . . , 30 As presented in Table 18, the submatrix S (2) ν g 1 , σ g t,5 , g, g = 1, . . . , 30, comprising the 2nd-order relative sensitivities of the leakage response with respect to the average number of neutrons per fission of isotope 1 ( 239 Pu) and the total cross sections of isotope 5 (C), includes 7 elements that have values greater than 1.0. All of these 7 large elements involve the total cross section σ g =30 t,5 for group g = 30 of isotope 5 (C).   Tables 19  and 20. Of these 99 elements, 71 elements are located in the energy phase-space confined by the energy groups g = 7, . . . , 14 and g = 14, . . . , 29, while the other 28 elements are located in energy groups g = 30 or g = 30; some of these sensitivities have very large negative values. The largest negative value is displayed by the 2nd-order relative sensitivity of the leakage response with respect to the 12th energy group of the parameter underlying the average number of neutrons per fission for 239 Pu and the 30th energy group of the total cross section for 1 H, namely, S (2) ν 12 1 , σ 30 t,6 = −19.29. The submatrix S (2) ν g 2 , σ g t, 6 , g, g = 1, . . . , 30, comprising the 2nd-order sensitivities of the leakage response with respect to the average number of neutrons per fission of isotope 2 ( 240 Pu) and the total cross sections of isotope 6 ( 1 H), contains a single large element that has an absolute value greater than 1.0, namely, S (2) ν 12 2 , σ 30 t,6 = −1.003.

Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Parameters Underlying the Average Number of Neutrons Per Fission and Scattering Cross Sections
This Section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities, ∂ 2 L(α)/∂ν∂σ s , of the leakage response with respect to the average number of neutrons per fission and scattering microscopic cross sections of all isotopes of the PERP benchmark.

Computation of the Second-Order Sensitivities
The equations needed for deriving the expression of the 2nd-order mixed sensitivities are obtained by particularizing Equations (177) and (178) presented in [5] to the PERP benchmark, where Equation (178) provides the contributions arising directly from the parameters underlying the average number of neutrons per fission and scattering cross sections, while Equation (177) provides contributions arising indirectly through the total cross sections, since the 0th-order scattering cross sections are part of the total cross sections. The expression obtained by particularizing Equation (178) in [5] to the PERP benchmark yields: Inserting the results obtained in Equations (68) and (69) into Equation (132), using the addition theorem for spherical harmonics in one-dimensional geometry, performing the respective angular integrations, and setting l m 2 = 0 in the resulting expression yields the following simplified form for Equation (132): where the 0th-order moments ϕ , are as follows: Inserting the result obtained in Equation (73) into Equation (134) yields the following simplified expression: Collecting the partial contributions obtained in Equations (133) and (135), yields the following result:

Second-Order Sensitivities
When computing the 2nd-order sensitivities ∂ 2 L/∂ f j ∂s m 2 ( f =ν,s=σ s, l≥1 ) , the parameters f j ≡ ν correspond to the l th -order (l ≥ 1) scattering cross sections, neither of which contribute to the total cross sections. Thus, the expression of ∂ 2 L/∂ f j ∂s m 2 ( f =ν,s=σ s, l≥1 ) is obtained by particularizing Equation (178) in [5] to the PERP benchmark, which yields, Inserting the results obtained in Equations (68) and (69) into Equation (137), using the addition theorem for spherical harmonics in one-dimensional geometry and performing the respective angular integrations yields the following simplified form for Equation (137): where the moments ϕ Due to the symmetry of the mixed 2nd-order sensitivities, the results computed using Equations (136) and (138) for ∂ 2 L(α)/∂ν∂σ s can be verified by computing the expressions of the sensitivities ∂ 2 L(α)/∂σ s ∂ν, which also requires separate consideration of the zeroth-order scattering cross sections. The two cases involved are as follows: (1) , j = 1, . . . , J σs,l=0 ; m 2 = J σ f + 1, . . . , J σ f + J ν , where the quantities s j refer to the parameters underlying the 0th-order scattering cross sections, while the quantities f m 2 refer to the parameters underlying the average number of neutrons per fission; (2) , j = 1, . . . , J s,l≥1 ; m 2 = J σ f + 1, . . . , J σ f + J ν , where the quantities s j refer to the parameters underlying the l th -order (l ≥ 1) scattering cross sections, and the quantities f m 2 refer to the parameters underlying the average number of neutrons per fission.

Second-Order Sensitivities
The equations needed for deriving the expression of the 2nd-order mixed sensitivities are obtained by particularizing Equations (160) and (169) in [5] to the PERP benchmark. Particularizing Equation (169) in [5] to the PERP benchmark yields the following expression: where the 2nd-level adjoint functions, θ In Equation (139), the parameters indexed by s j correspond to the 0th-order scattering cross sections, so that s j ≡ σ g j →g j s,l j =0,i j , while the parameters indexed by f m 2 correspond to the average number of neutrons per fission, so that f m 2 ≡ ν g m 2 i m 2 . Inserting the results obtained in Equations (121) and (122) into Equation (139), yields the following simplified expression for Equation (139): where the 0th-order moments Θ (2),g m 2 1, j;0 (r) and Θ (2),g 2, j;0 (r) have been previously defined in Equations (94) and (95), respectively.
Inserting the results obtained in Equations (121) and (122) into Equation (141), yields the following simplified expression for Equation (141): Collecting the partial contributions obtained in Equations (140) and (142), yields the following result:
1.14 × 10 −3 at g = 12, g = 12 → h = 12        Table 23, all of the largest absolute values of the mixed 2nd-order sensitivities involve ν g=12 i , i = 1, 2 for the 12th energy group or the 7th energy group of isotopes 239 Pu or 240 Pu, and (most of the time) involve either the 2nd-order self-scattering cross sections σ 7→7 s,l=2,k , k = 1, . . . , 5 for the 7th energy group of isotopes 239 Pu, 240 Pu, 69 Ga, 71 Ga and C or (occasionally) the 2nd-order self-scattering cross sections σ 12→12 s,l=2,i=6 for the 12th energy group of isotope 1 H. As shown in Table 23, all of the largest elements in the respective sub-matrix are positive, and the vast majority of them are very small. The overall largest element in the matrix S (2) s,l= 2,k=6 = 9.03 × 10 −2 .   Table 24, the mixed 2nd-order sensitivities having the largest absolute values involve ν g=12 i , i = 1, 2 for the 12th energy group or (occasionally) the 7th energy group of isotopes 239 Pu or 240 Pu, and the 3rd-order self-scattering cross sections σ 7→7 s,l=3,k , k = 1, . . . , 5 for the 7th energy group of isotopes 239 Pu, 240 Pu, 69 Ga, 71 Ga and C, or (occasionally) the 3rd-order self-scattering cross sections σ 12→12 s,l=3,i=6 for the 12th energy group of isotope 1 H. The overall largest (in absolute value) element of the matrix S (2)

Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Parameters Underlying the Average Number of Neutrons per Fission and Fission Cross Sections
This Section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities ∂ 2 L(α)/∂ν∂σ f of the leakage response with respect to the average number of neutrons per fission and fission microscopic cross sections of all isotopes of the PERP benchmark. Likewise, the numerical values of the 2nd-order mixed sensitivities ∂ 2 L(α)/∂ν∂σ f can also be computed by using the alternative expression for ∂ 2 L(α)/∂σ f ∂ν. The formulas for computing the 2nd-order mixed sensitivities ∂ 2 L(α)/∂ν∂σ f are presented in Section 8.1, while the formulas for computing, alternatively, the expressions for ∂ 2 L(α)/∂σ f ∂ν are presented in Section 8.2.
The numerical results obtained for the matrix S (2) ν g i , σ g f ,k , i, k = 1, 2; g, g = 1, . . . , 30 have been partitioned into N f × N f = 4 submatrices, each of dimensions G × G(= 30 × 30). The summary of the main features of each submatrix is presented in Table 25.
28 elements with absolute values > 1.0 Max. value = 1.04 × 10 −1 at g = 12, g = 12 i = 2 ( 240 Pu) Max. value = 1.05 × 10 −1 at g = 12, g = 12 Max. value = 6.86 × 10 −2 at g = 12, g = 12 The 2nd-order mixed sensitivities ∂ 2 L(α)/∂ν∂σ f are mostly positive. Among the J ν × J σ f (= 60 × 60) elements in the matrix S (2) ν g i , σ g f ,k , i, k = 1, 2; g, g = 1, . . . , 30, 3557 out of 3600 elements have positive values, and most of them are very small; however, 28 out of these 3600 elements have large relative sensitivities, with values greater than 1.0, as noted inTable 25. All of these larger sensitivities reside in the sub-matrix S (2) , σ g f ,k=1 , and relate to the fission parameters for isotope 239 Pu. The overall maximum relative sensitivity is S (2) ν 12 1 , σ 12 f ,1 = 3.225. Additional details about the sub-matrix S (2) , σ g f ,k=1 are provided in the following section. The results in Table 25 also indicate that all of the mixed 2nd-order relative sensitivities involving the fission parameters (either ν The numerical results for the elements of the submatrix S (2) f ,k=1 /L , g, g = 1, . . . , 30, of 2nd-order mixed relative sensitivities of the leakage response with respect to the average number of neutrons per fission and fission cross sections of isotope 239 Pu, indicate that the majority (899 out of 900) of the elements of this submatrix have positive 2nd-order relative sensitivities; only 1 element is negative. Table 26 presents the 28 elements (in bold) of S (2) , σ g f ,k=1 , g, g = 1, . . . , 30 which have values greater than 1.0. The largest value among these sensitivities is attained by the relative 2nd-order mixed sensitivity S (2) ν 12 1 , σ 12 f ,1 = 3.225.

Quantification of Uncertainties in the PERP Leakage Response Due to Uncertainties in Fission Cross Sections
Correlations between the group-averaged microscopic fission cross sections or correlations between these cross sections and other cross sections are not available for the PERP benchmark. When such correlations are unavailable, the maximum entropy principle (see, e.g., [9]) indicates that neglecting them minimizes the inadvertent introduction of spurious information into the computations of the various moments of the response's distribution in parameter space. The formulas for computing the expected value, variance and skewness of the response distribution by taking into account the 2nd-order response sensitivities together with the standard deviations of the group-averaged fission microscopic cross sections parameter correlations are as follows: 1. The expected value, [E(L)] f , of the leakage response L(α) has the following expression: expected response value, standard deviation and skewness are small, which is not surprising in view of the small values for the 1st-and 2nd-order sensitivities already presented in Tables 2 and 3.  6 , g, g = 1, . . . , 30, presented in Table 5. However, the effects of these sensitivities on the uncertainties in the response distribution can be taken into account only if the corresponding correlations among the various model parameters were available.

Uncertainties in the PERP Leakage Response Stemming from Uncertainties in the Average Number of Neutrons per Fission
The correlations between the average number of neutrons per fission are unknown, so these parameters will be assumed to be uncorrelated, since this assumption is the least biased, according to the maximum entropy principle [9] in avoiding the introduction of spurious information in the uncertainty quantification computations. Similar to those formulas presented in Section 9, upto 2nd-order response sensitivities, the expected value, [E(L)] ν , of the leakage response L(α) has the following expression: where the subscript "ν" and superscript "U" indicate contributions solely from the group-averaged uncorrelated parameters underlying the average number of neutrons per fission, and where the term [E(L)] (2,U) ν , which provides the 2nd-order contributions, is given by the following expression: In Equation (168), the quantity s g ν,i denotes the standard deviation associated with the imprecisely known model parameter ν Again, taking into account contributions solely from the group-averaged uncorrelated parameters underlying the average number of neutrons per fission, the third-order moment, [µ 3 (L)] (U,N) ν , of the leakage response for the PERP benchmark takes on the following form:  Table 28 shows the results computed using Equations (167)-(173) together with the 1st-and 2nd-order respective sensitivity values presented in Section 5.3, for uniform parameter standard deviations of 1%, 5%, and 10% of ν g i , i = 1, 2; g = 1, . . . , 30, respectively.  The relative effects on the leakage response of uncertainties in the average number of neutrons per fission can be compared to the corresponding effects stemming from the fission and total cross sections. A final comparison, with corresponding conclusions, will be made after all of the 2nd-order sensitivities of the PERP leakage response to the PERP benchmark's underlying nuclear data are obtained. Thus, comparing the results shown in Table 28 for standard deviations of 10% with the corresponding results presented in Table 27 for the fission cross sections and Table 25  The above comparisons indicate that the contributions to the leakage response moments stemming from the group-averaged uncorrelated parameters underlying the average number of neutrons per fission are much smaller than the corresponding contributions stemming from the group-averaged uncorrelated microscopic total cross sections but are bigger than the corresponding contributions stemming from the group-averaged uncorrelated microscopic fission cross sections. Again, it is important to note that the results presented in Table 28 consider only the standard deviations of the uncorrelated parameters underlying the average number of neutrons per fission, since correlations between these parameters are unavailable. On the other hand, the results presented in Sections 5-7 indicated that the largest values are displayed by several mixed 2nd-order sensitivities of the leakage response with respect to ν and σ t , and with respect to ν and σ f , which are much larger than the values of the unmixed sensitivities. Recall that the following sensitivities have absolute values larger than 1.0: (a) 52 elements of the matrix S (2) ν g 1 , ν g 1 , g, g , h = 1, . . . , 30, as summarized in Table 15; only 6 of these are included in the computations leading to the results shown in Table 28; (b) 72 elements of the matrix S (2) ν g 1 , σ g t,1 , g, g , h = 1, . . . , 30, presented in Table 17; (c) 7 elements of the matrix S (2) ν g 1 , σ g t,5 , g, g = 1, . . . , 30 as listed in Table 18; (d) 99 elements of the matrix S (2) ν g 1 , σ g t, 6 , g, g = 1, . . . , 30 presented in Tables 19 and 20; (e) 1 element of the matrix S (2) ν g 2 , σ g t, 6 , g, g = 1, . . . , 30 presented in Section 6.3.4; and (f) 28 elements of the matrix S (2) ν g i=1 , σ g f ,k=1 , g, g = 1, . . . , 30 presented in Table 26. However, the effect of these large sensitivities on the uncertainties in the response distribution cannot be considered presently because the corresponding correlations among the various model parameters are not available.

Conclusions
This work has presented results for the first-order sensitivities, ∂L(α)/∂σ f , and the second-order sensitivities ∂ 2 L(α)/∂σ f ∂σ f of the PERP total leakage response with respect to the group-averaged microscopic fission cross sections, and the mixed second-order sensitivities ∂ 2 L(α)/∂σ f ∂σ t and ∂ 2 L(α)/∂σ f ∂σ s of the leakage response with respect to the group-averaged microscopic fission/total cross sections and corresponding fission and scattering cross sections. In addition, this work has also presented results for ∂L(α)/∂ν and ∂ 2 L(α)/∂ν∂ν, i.e., the first-and, respectively, second-order sensitivities of the PERP total leakage response with respect to the parameters underlying the benchmark's average number of neutrons per fission, as well as results for the mixed second-order sensitivities for ∂ 2 L(α)/∂ν∂σ t , ∂ 2 L(α)/∂ν∂σ s , and ∂ 2 L(α)/∂ν∂σ f .
For the sensitivities with respect to the fission cross sections, the following conclusions can be drawn from the results reported in this work: 1.
The 1st-order relative sensitivities of the PERP leakage response with respect to the group-averaged microscopic fission cross sections for the two fissionable PERP isotopes are positive, as shown in Tables 2 and 3, signifying that an increase in σ g f ,i , i = 1, 2; g = 1, . . . , 30 will cause an increase in the PERP leakage response L (i.e., more neutrons will leak out of the sphere). The 2nd-order unmixed relative sensitivities of the PERP leakage response with respect to the group-averaged microscopic fission cross sections are positive for the energy groups g = 7, . . . , 15, but are negative for the other energy groups; 2.
Comparing the results for the 1st-order relative sensitivities to those obtained for the 2nd-order unmixed relative sensitivities for isotope 1 ( 239 Pu) indicates that the values of the 2nd-order sensitivities are close to, and generally smaller than, the corresponding values of the 1st-order sensitivities for the same energy group, except for the 12th energy group, where the 2nd-order relative sensitivity is larger. For isotope 2 ( 240 Pu), the values for both the 1st-and 2nd-order relative sensitivities are very small, and the values of the 2nd-order unmixed relative sensitivities are at least an order of magnitude smaller than the corresponding values of the 1st-order ones.
The largest values of the 1st-order and 2nd-order relative sensitivities are always related to the 12th energy group for both isotopes 239 Pu and 240 Pu; 3.
The 1st-order relative sensitivities with respect to the fission cross sections are up to 50% smaller than the corresponding values with respect to the total cross sections, and are approximately one order of magnitude larger than the corresponding 1st-order relative sensitivities with respect to the 0th-order scattering cross sections for isotope 239 Pu. Likewise, the absolute values of the 2nd-order unmixed relative sensitivities with respect to the fission cross sections are 50-90% smaller than the corresponding values with respect to total cross sections but are approximately one to two orders of magnitudes larger than the 2nd-order sensitivities corresponding to the 0th-order scattering cross sections for 239 Pu; 4.

6.
The J σ f × J σs (= 60 × 21, 600) dimensional matrix S (2) σ g f ,i , σ g →h s,l,k comprises more elements having positive (rather than negative) values when involving even-orders (l = 0, 2) scattering cross sections, and vice-versa when involving odd-orders (l = 1, 3) scattering cross sections. Overall, however, the total number of positive elements in this matrix is comparable to that of negative elements in the sensitivity matrix. As shown in Tables 8-11, in each submatrix of S (2) σ g f ,i , σ g →h s,l,k , l = 0, . . . , 3; i = 1, 2; k = 1, . . . , 6; g, g , h = 1, . . . , 30, the largest absolute values of the 2nd-order relative sensitivities corresponding to even-order scattering parameters are all positive, while those corresponding to odd-orders scattering parameters are all negative;

7.
The absolute values of all the J σ f × J σs (= 60 × 21, 600) elements of the matrix S (2) σ g f ,i , σ g →h s,l,k are less than 1.0, and the vast majority of them are very small; also, the higher the order of scattering cross sections, the smaller the absolute values of these sensitivities. Also, it is observed that the largest absolute value of the 2nd-order relative sensitivities in each submatrix of S (2) σ g f ,i , σ g →h s,l,k , l = 0, . . . , 3; i = 1, 2; k = 1, . . . , 6; g, g , h = 1, . . . , 30, generally involve the fission cross sections for the 12th energy group of isotopes 239 Pu or 240 Pu, and the self-scattering cross sections in the 12th or 7th energy group for all isotopes. The largest sensitivity comprised in S (2) s,l= 0,1 = 3.03 × 10 −1 , i.e., the 2nd-order mixed sensitivity of the PERP leakage response with respect to the 12th energy group of the fission and 0th-order self-scattering cross sections of isotope 239 Pu; 8.
The alternative paths for computing the mixed 2nd-order sensitivities, which are due to the symmetry of these sensitivities, provide multiple reciprocal "solution verifications" possibilities, ensuring that the respective computations were performed correctly. However, one of the alternative paths is much more efficient computationally than the other. For example, computing ∂ 2 L(α)/∂σ f ∂σ t is around 3 times more efficient than computing alternatively the symmetric sensitivities ∂ 2 L(α)/∂σ t ∂σ f . Also, computing ∂ 2 L(α)/∂σ f ∂σ s is about 60 times more efficient than computing alternatively the sensitivities ∂ 2 L(α)/∂σ s ∂σ f ;

9.
Many mixed 2nd-order sensitivities of the leakage response to the group-averaged fission and total microscopic cross sections are significantly larger than the unmixed 2nd-order sensitivities of the leakage response with respect to the group-averaged fission microscopic cross sections. Therefore, it would be very important to obtain correlations among the various model parameter, since the correlations among the respective fission and total cross sections could provide significantly larger contributions to the response moments than the standard deviations of the fission cross sections.
For the sensitivities with respect to the parameters underlying the average number of neutrons per fission, the following conclusions can be drawn from the results reported in this work: 10. The 1st-order relative sensitivities of ∂L(α)/∂ν for the two fissionable PERP isotopes are positive, as shown in Tables 13 and 14, signifying that an increase in ν g i , i = 1, 2; g = 1, . . . , 30 will cause an increase in the PERP leakage response L. The 2nd-order unmixed relative sensitivities of the leakage response with respect to the average number of neutrons per fission are also positive; 11. Comparing the results for the 1st-order relative sensitivities of ∂L(α)/∂ν to those 2nd-order unmixed relative sensitivities for isotope 1 ( 239 Pu) indicate that, for energy groups g = 7, . . . , 14, the values of the 2nd-order unmixed sensitivities are significantly larger than the corresponding values of the 1st-order sensitivities for the same energy group, and they are smaller for other energy groups. For isotope 2 ( 240 Pu), the values for both the 1st-and 2nd-order relative sensitivities are all very small, and the values of the 2nd-order unmixed relative sensitivities are at least an order of magnitude smaller than the corresponding values of the 1st-order ones. The largest values of the 1st-order and 2nd-order unmixed relative sensitivities are always related to the 12th energy group of the parameters underlying the average number of neutrons per fission for both isotopes 239 Pu and 240 Pu; 12. The 1st-order relative sensitivities of ∂L(α)/∂ν are comparable to the corresponding values with respect to the total cross sections for energy groups g = 7, . . . , 12, but for energy groups g = 13, . . . , 22, they are considerably smaller. On the other hand, the 1st-order relative sensitivities of ∂L(α)/∂ν are 30% to 50% larger than the corresponding values to ∂L(α)/∂σ f for 239 Pu. Likewise, the values of the 2nd-order unmixed relative sensitivities with respect to the average number of neutrons per fission are significantly smaller than the corresponding values with respect to total cross sections, but larger than the corresponding values with respect to fission cross sections; 13. The 2nd-order mixed sensitivities ∂ 2 L(α)/∂ν∂ν are all positive. Among the J ν × J ν = (60 × 60) elements in the matrix S (2) ν g i , ν g k , i, k = 1, 2; g, g = 1, . . . , 30, 52 elements have relative sensitivities greater than 1.0. All of these 52 large sensitivities belong to the submatrix S (2) ν g 1 , ν g 1 , and involve the parameters underlying the average number of neutrons per fission of isotope 239 Pu. The largest of these sensitivities is S (2) ν 12 1 , ν 12 1 = 2.963. The values of the mixed 2nd-order relative sensitivities involving the parameters underlying the average number of neutrons per fission of isotope 240 Pu are all smaller than 1.0; 14. The 2nd-order mixed sensitivities ∂ 2 L(α)/∂ν∂σ t are mostly negative. Among the J ν × J σt (= 10, 800) elements of the matrix S (2) ν g i , σ g t,k , i = 1, 2; k = 1, . . . , 6; g, g = 1, . . . , 30, there are 179 elements belonging to the submatrices S (2) ν g 1 , σ g t,1 , S (2) ν g 1 , σ g t, 5 and S (2) ν g 1 , σ g t, 6 which have absolute values greater than 1.0; 178 of these large sensitivities involve the parameters underlying the average number of neutrons per fission of isotope 239 Pu, and the total cross sections of isotopes 239 Pu, C and 1H. The largest (negative) relative sensitivity is S (2) ν 12 1 , σ 30 t,6 = −19.29. In addition, the mixed 2nd-order relative sensitivities involving isotopes 240 Pu, 69Ga and 71Ga generally have absolute values smaller than 1.0; 15. The J ν × J σs (= 60 × 21, 600) dimensional matrix S (2) ν g i , σ g →h s,l,k comprises more elements having positive (rather than negative) values for even-orders ( l = 0, 2) scattering cross sections and vice-versa when involving odd-orders (l = 1, 3) scattering cross sections. Overall, however, this matrix contains about as many positive elements as negative ones. As shown in Tables 21-24,