Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: II. E ﬀ ects of Imprecisely Known Microscopic Scattering Cross Sections

: This work continues the presentation commenced in Part I of the second-order sensitivity analysis of nuclear data of a polyethylene-reﬂected plutonium (PERP) benchmark using the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM). This work reports the results of the computations of the ﬁrst- and second-order sensitivities of this benchmark’s computed leakage response with respect to the benchmark’s 21,600 parameters underlying the computed group-averaged isotopic scattering cross sections. The numerical results obtained for the 21,600 ﬁrst-order relative sensitivities indicate that the majority of these were small, the largest having relative values of O (10 − 2 ). Furthermore, the vast majority of the (21600) 2 second-order sensitivities with respect to the scattering cross sections were much smaller than the corresponding ﬁrst-order ones. Consequently, this work shows that the e ﬀ ects of variances in the scattering cross sections on the expected value, variance, and skewness of the response distribution were negligible in comparison to the corresponding e ﬀ ects stemming from uncertainties in the total cross sections, which were presented in Part I. On the other hand, it was found that 52 of the 21600 × 180 mixed second-order sensitivities of the leakage response with respect to the scattering and total microscopic cross sections had values that were signiﬁcantly larger than the unmixed second-order sensitivities of the leakage response with respect to the group-averaged scattering microscopic cross sections. The ﬁrst- and second-order mixed sensitivities of the PERP benchmark’s leakage response with respect to the scattering cross sections and the other benchmark parameters (ﬁssion cross sections, average number of neutrons per ﬁssion, ﬁssion spectrum, isotopic atomic number densities, and source parameters) have also been computed and will be reported in subsequent works.


Introduction
In continuation of the results presented in Part I [1], this work presents the numerical results for the first-and second-order sensitivities of the leakage response of the polyethylene-reflected plutonium (PERP) benchmark described in [2] with respect to the benchmark's group-averaged isotopic scattering cross sections. This work also presents the results for the mixed second-order sensitivities to both the scattering and total cross sections. As has been described in Part I [1], the numerical model of the PERP benchmark includes 180 (J σt = I × G) imprecisely-known parameters for the group-averaged total B g (α)ϕ g (r, Ω) = Q g (r), g = 1, . . . , G, (1) ϕ g (r d , Ω) = 0, r d ∈ S b , Ω · n < 0, g = 1, . . . , G, 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) G g =1 4π (νΣ) g f (r) ϕ g r, Ω dΩ ; (3) and where α denotes the "vector of imprecisely-known model parameters", as defined in Part I [1]. The PARTISN [4] calculations used MENDF71X 618-group cross sections [6] collapsed to G = 30 energy groups, with group boundaries, E g , as presented in Part I [1]. The MENDF71X library uses ENDF/B-VII.1 Nuclear Data [7]. As has been discussed in [1], the fundamental quantities (i.e., system responses) of interest for subcritical benchmarks (such as the PERP benchmark) are singles counting rate, doubles counting rate, the leakage multiplication, and the total leakage. The total leakage is physically more meaningful than count rates because it does not depend on the detector configuration. For this reason, many systems are characterized for practical applications by their total leakage rather than by the count rate that a particular detector would see at a particular distance. For this reason, this work considers the total leakage from the PERP benchmark to be the paradigm response of interest for sensitivity analysis; sensitivities analyses of counting rates and other responses can be performed in an analogous manner, i.e., by following the general ideas that will be presented in this work (and in subsequent related works).
Mathematically, the total neutron leakage from the PERP sphere, denoted as L(α), will depend (indirectly, through the neutron flux) on all of the imprecisely-known model parameters and is defined as follows: dΩ Ω · n ϕ g (r, Ω).
(5) Figure 1 shows the histogram plot of the leakage for each energy group for the PERP benchmark. The total leakage computed using Equation (5) for the PERP benchmark is 1.7648 × 10 6 neutrons/sec. other responses can be performed in an analogous manner, i.e., by following the general ideas that will be presented in this work (and in subsequent related works).
Mathematically, the total neutron leakage from the PERP sphere, denoted as   L α , will depend (indirectly, through the neutron flux) on all of the imprecisely-known model parameters and is defined as follows: (5) Figure 1 shows the histogram plot of the leakage for each energy group for the PERP benchmark. The total leakage computed using Equation Error! Reference source not found. for the PERP benchmark is  6 1.7648 10 neutrons/sec. Ω from energy group  ', ' 1,..., g g G into energy group   , 1,..., g g G is computed in terms of the l -th order Legendre coefficient    , , g g s l i , of the Legendre-expanded microscopic scattering cross section from energy group  g into energy group g for isotope i . Since the cross-sections for every material are treated in the PARTISN [4] calculations as being space-independent within the respective material, the variable r will henceforth no longer appear in the arguments of the various cross sections. The coefficients     The scattering transfer cross section Σ g →g s r, Ω → Ω from energy group g , g = 1, . . . , G into energy group g, g = 1, . . . , G, is computed in terms of the l-th order Legendre coefficient σ g →g s,l,i , of the Legendre-expanded microscopic scattering cross section from energy group g into energy group g for isotope i. Since the cross-sections for every material are treated in the PARTISN [4] calculations as being space-independent within the respective material, the variable r will henceforth no longer appear in the arguments of the various cross sections. The coefficients σ g →g s,l,i are tabulated parameters, and the finite-order Legendre-expansion of Σ g →g s Ω → Ω has the following expression: where ISCT = 3 denotes the order of the respective finite expansion in Legendre polynomial.
The total cross section Σ g t for energy group g, g = 1, . . . , G, and material m is computed for the PERP benchmark using the following expression: where σ g f ,i and σ g c,i denote, respectively, the tabulated group microscopic fission and neutron capture cross sections for group g, g = 1, . . . , G. Other nuclear reactions, including (n,2n) and (n,3n) reactions are not present in the PERP benchmark. The expressions in Equations (6) and (7) indicate that the zeroth order (i.e., l = 0) scattering cross sections must be considered separately from the higher order (i.e., l ≥ 1) scattering cross sections, since the l = 0 scattering cross sections contribute to the total cross sections, while the l ≥ 1 scattering cross sections do not contribute to the total cross sections.
In Equations (8)- (10), the dagger denotes "transposition," σ g t,i denotes the microscopic total cross section for isotope i and energy group g, N i,m denotes the respective isotopic number density, and J n denotes the total number of isotopic number densities in the model. Thus, the vector t comprises a total of J t = J σt + J n = 30 × 6 + 6 = 186 imprecisely-known "model parameters" as its components.
Recall from Part I [1] that the components of the vector of first-order sensitivities of the leakage response with respect to the model parameters are denoted as S (1) (α), which is defined as follows: The symmetric matrix of second-order sensitivities of the leakage response with respect to the model parameters is denoted as S (2) (α), and is defined as follows: The results as well as their impact on the uncertainties induced in the leakage response by the first-and second-order sensitivities ∂L(α)/∂σ t and, respectively, ∂ 2 L(α)/∂σ t ∂σ t , were reported in Part I [1]. This work will report the computational results for the first-order sensitivities ∂L(α)/∂σ s and the second-order sensitivities ∂ 2 L(α)/∂σ s ∂σ s and ∂ 2 L(α)/∂σ s ∂σ t , along with their effects on the uncertainties induced in the leakage response.

First-Order Sensitivities ∂L(α)/∂σ s
The equations needed for deriving the expressions of the first-order sensitivities of ∂L/∂s j , j = 1, . . . , J σs will differ from each other depending on whether the parameters s j correspond to the zeroth-order (l = 0) or to the higher order (l ≥ 1) scattering cross sections. There are two distinct cases, as follows: (1) ∂L(α) ∂s j (s=σ s,l=0 ) , j = 1, . . . , J σs,l=0 , where the quantities s j refer to the parameters underlying the zeroth-order scattering microscopic cross sections; and (2) ∂L(α) ∂s j (s=σ s,l≥1 ) , j = 1, . . . , J σ s,l≥1 , where the quantities s j refer to the parameters underlying the l th -order (l ≥ 1) scattering microscopic cross sections. , j = 1, . . . , J σs,l=0 The first-order sensitivities of the leakage response with respect to zeroth-order scattering microscopic cross sections are computed by particularizing Equations (150) and (151) in [3], where Equation (151) provides the contributions arising directly from the scattering cross sections, while Equation (150) provides contributions arising indirectly through the total cross sections. The expression obtained by particularizing Equation (151) in [3] to the PERP benchmark yields: where the multigroup adjoint fluxes ψ (1),g (r, Ω), g = 1, . . . , G are the solutions of the following first-Level Adjoint Sensitivity System (1st-LASS) presented in Equations (156) and (157) in [3]: ψ (1),g (r d , Ω) = 0, Ω · n > 0, g = 1, . . . , G, where r d is the radius of the PERP sphere, and where the adjoint operator A (1),g (α) takes on the following particular form of Equation (149) in [3]: The contributions stemming from the total cross sections are computed using Equation (150) in [3] in conjunction with the relations ∂L Adding Equations (13) and (17) yields the following complete expression: For the PERP benchmark, when the parameters s j correspond to the zeroth-order scattering microscopic cross sections, i.e., s j ≡ σ g j →g j s,l j =0,i j , the following relations hold: where the subscripts i j , l j , g j ,g j and m j refer to the isotope, order of Legendre expansion, energy groups, and material associated with the parameter s j , respectively, and where δ g j g and δ g j g denote the Kronecker-delta functionals (e.g., δ g j g = 1 if g j = g; δ g j g = 0 if g j g). Inserting Equations (19) and (20) into Equation (18), using the addition theorem for spherical harmonics in one-dimensional geometry, performing the respective angular integrations, and finally setting l j = 0 in the resulting expression yields the following expression: (1),g j 0 (r) − N i j ,m j V dV 4π dΩ ψ (1),g j (r, Ω)ϕ g j (r, Ω), where the forward and adjoint flux moments ϕ g j 0 (r) and ξ (1),g j 0 (r) are defined as follows: , j = 1, . . . , σ s,l≥1 The first-order sensitivities of the leakage response with respect to the l th -order (l ≥ 1) microscopic scattering cross sections are computed by particularizing Equation (151) in [3]: Inserting Equation (19) into Equation (24), using the addition theorem for spherical harmonics in one-dimensional geometry and performing the respective angular integrations, yields the following expression: where the forward and adjoint flux moments ϕ g j l j (r) and ξ (1),g j l j (r) are defined as follows: dΩ P l (Ω)ψ (1),g (r,Ω).
(62) While computing the sensitivities S (2) σ g →g s,l,i , σ h →h s,l ,k , it has been verified, within the first five significant digits, that the numerical values obtained using Equation (59) are the same as the corresponding numerical values obtained using Equation (60). The numerical values of the second-order relative sensitivities of the leakage response with respect to the scattering cross sections are small by comparison to the corresponding leakage sensitivities to the total cross sections presented in Part I [1], the largest of them being of the order of 10 −2 . The results for the second-order sensitivities of the leakage response with respect to the 0th-order scattering cross sections of isotope 1 ( 239 Pu) and to the second-order scattering cross sections of all of the other isotopes, i.e., S (2) Table 1. The dimensions of each of the submatrices presented in Table 1 are 900 × 900. As shown in the table, these second-order relative sensitivities are all much smaller than 1.0. Table 1. Overview of second-order relative sensitivities of the leakage response with respect to the zeroth-order (l = 0) scattering cross sections of isotope 1 ( 239 Pu) and to the zeroth-order (l = 0) scattering cross sections of all isotopes, S (2) The largest of all of the sensitivities summarized in Table 1 are included among the elements . . , 30, which comprises the second-order relative sensitivities in submatrix of the leakage response with respect to the zeroth-order scattering cross sections of isotope 1 ( 239 Pu). Moreover, the largest 10 relative sensitivities comprised in s,l=0,k=1 , g, g , h, h = 1, . . . , 30, are listed in Table 2. All of these sensitivities are with respect to the zeroth-order self-scattering cross sections, rather than the in-scattering or out-scattering cross sections. In particular, the largest second-order sensitivity is S (2) σ 12→12 s,l=0,i=1 , σ 13→13 s,l=0,k=1 = 3.579 × 10 −2 , which corresponds to the second-order sensitivity of the leakage response with respect to the self-scattering cross section parameters of σ 12→12 s,l=0,i=1 and σ 13→13 s,l=0,k=1 . Table 2. Largest ten relative sensitivities comprised in S (2) σ g →g s,l=0,i=1 , σ h →h s,l=0,k=1 ; g, g , h, h = 1, . . . , 30 (second-order sensitivities of the leakage with respect to the zeroth-order scattering cross sections of 239 Pu).

Rank
Relative Sensitivity Rank Relative Sensitivity Tables 3-5 present an overview of the second-order relative sensitivities of the leakage response with respect to the zeroth-order scattering cross sections of isotope 1 ( 239 Pu) and to the l th -order scattering cross sections of all isotopes, defined as S (2) s,l=0,i=1 σ h →h s,l ,k /L , k = 1, . . . , 6; g, g , h, h = 1, . . . , 30, for l = 1, 2, 3, respectively. The results presented in these tables indicate that the higher the order of scattering cross sections, the smaller the mixed second-order sensitivities. Table 3. Overview of the second-order mixed relative sensitivities of the leakage response with respect to the zeroth-order (l = 0) scattering cross sections of 239 Pu and to the first-order (l = 1) scattering cross sections of all other isotopes: Table 4. Overview of second-order mixed relative sensitivities of the leakage response with respect to the zeroth-order (l = 0) scattering cross sections of 239 Pu and to the second-order (l = 2) scattering cross sections of all other isotopes: Table 5. Overview of second-order mixed relative sensitivities of the leakage response with respect to the zeroth-order (l = 0) scattering cross sections of 239 Pu and to the third-order (l = 3) scattering cross sections of all other isotopes: The first-order sensitivities of the leakage response with respect to the zeroth-order self-scattering cross sections can be compared directly to the corresponding unmixed second-order sensitivities. These comparisons are presented in Tables 6-11 for all six of the isotopes contained in the PERP benchmark. The main conclusions that can be drawn from these comparisons are as follows: (i) both the first-and second-order unmixed sensitivities of the leakage response with respect to the zeroth-order self-scattering cross sections are very small; and (ii) the absolute values of the second-order unmixed relative sensitivities are much smaller, by at least an order of magnitude, than the corresponding first-order sensitivities (except for the second-order unmixed sensitivity of the leakage with respect to the self-scattering cross section of isotopes C and 1 H in their respective lowest-energy group).      The results presented in Tables 6-11 indicate that the largest values for both the first-and second-order relative sensitivities for the isotopes 239 Pu, 240 Pu, 69 Ga, and 71 Ga, are for the energy group 12. For the isotope C, the largest values for the first-and second-order relative sensitivities are for the 12th energy group and the 16th energy group, respectively. For the isotope 1 H, the largest values for the first-and second-order relative sensitivities are for the 12th energy group and the 30th energy group, respectively. It is noteworthy that all of the first-order relative sensitivities of the leakage response with respect to the zeroth-order scattering cross sections of isotopes C and 1 H are positive, signifying that an increase in the corresponding microscopic cross sections will cause an increase in the value of the response L (i.e., more neutrons will leak out of the sphere). These sensitivities indicate that an increase in low energy scattering moderates and reflects slow neutrons into the plutonium, which increases the induced fission rate in 239 Pu, thus increasing the neutron flux, which in turn increases the neutron leakage.

Mixed Second-Order Sensitivities of the PERP Total Leakage Response with respect to the Parameters Underlying the Benchmark's Scattering and Total Cross Sections
This section presents the computation and analysis of the numerical results for the second-order mixed sensitivities of the leakage response with respect to the group-averaged scattering and total microscopic cross sections of all isotopes of the PERP benchmark. As has been shown by Cacuci [3], these mixed sensitivities can be computed using two distinct expressions, involving distinct second-level adjoint systems and the corresponding adjoint functions, by considering either the computation of ∂ 2 L/∂s j ∂t m 2 , j = 1, . . . , J σs ; m 2 = 1, . . . , J σt or the computation of ∂ 2 L/∂t j ∂s m 2 , j = 1, . . . , J σt ; m 2 = 1, . . . , J σs . These two distinct paths for computing the 2nd-order sensitivities with respect to the group-averaged scattering and total microscopic cross sections will be presented in Sections 3.1 and 3.2, respectively. Of course, the end results produced by these two distinct paths must be identical, thus providing a mutual "solution verification" that the respective computations were performed correctly.

Second-Order Sensitivities
The expression of ∂ 2 L ∂s j ∂t m 2 (s=σ s,l=0 ,t=σ t ) must also include the contributions stemming from the total cross sections, since the total cross sections comprises the zeroth-order scattering cross sections. The contributions are computed by particularizing Equation (158) in [3] to the PERP benchmark and by noting that , to obtain: (1) (s=σ s,l=0 ,t=σ t ) In Equation (65), the parameters s j correspond to the zeroth-order scattering cross sections, so that s j ≡ σ g j →g j s,l j =0,i j , while the parameters t m 2 correspond to the total cross sections, so that t m 2 ≡ σ g m 2 t,i m 2 , where the subscripts i m 2 and g m 2 denote the isotope and energy group associated with t m 2 , respectively.
As shown in Table 12, the largest absolute values of the mixed second-order sensitivities mostly involve the zeroth-order self-scattering cross sections in the 12th energy group of the isotopes, and either the total cross sections for the 12th energy group for isotopes 239 Pu, 240 Pu, 69 Ga and 71 Ga, or the total cross sections for the 30th energy group for isotopes C and 1 H.
Additional information regarding the three submatrices in Table 12 that have elements with absolute values greater than 1.0 is provided below: (1) The eight elements in the submatrix S (2) σ g t,i=6 , σ g →h s,l=0,k=1 , g, g , h = 1, . . . , 30 (of second-order sensitivities of the leakage response with respect to the total cross sections of 1 H and to the zeroth-order scattering cross sections of 239 Pu) that have values greater than 1.0 are presented in Table 14. All of these relative sensitivities are with respect to the same total cross section parameter σ g=30 t, 6 and to the zeroth-order self-scattering cross sections. The relative sensitivities with respect to the 0th-order in-scattering and out-scattering cross sections are all smaller than 1.0.
As shown in Table 15, the largest absolute values of the mixed second-order sensitivities involve mostly the first-order self-scattering cross sections in the 7th, 12th, or 30th energy groups of the isotopes, along (mostly) with either the total cross sections for the 7th or 12th energy group for isotopes 239 Pu, 240 Pu, 69 Ga and 71 Ga, or (occasionally) the total cross sections for the 30th energy group for isotopes C and 1 H.
(2) The matrix S (2) s,l=1,k=6 /L , g, g , h = 1, . . . , 30, of second-order sensitivities of the leakage response with respect to the total cross sections of 1 H and the first-order scattering cross sections of 1 H, comprises 13 elements that have values greater than 1.0 which are listed in Table 16. All the 13 sensitivities presented in this table are with respect to the total cross section parameter σ g=30 t,i=6 . The largest sensitivity is S (2) σ 30 t,i=6 , σ 30→30 s,l=1,k=6 = 6.996.  Table 17 summarizes the results obtained for the second-order relative sensitivities of the leakage response with respect to the total cross sections and the second-order scattering cross sections between all isotopes, S (2) Table 17, the largest values of the mixed second-order sensitivities in each of the respective submatrix involve the second-order self-scattering cross sections in the 7th or 12th energy groups of the isotopes, and the total cross sections corresponding either to the 7th energy group for isotopes 239 Pu, 240 Pu, 69 Ga and 71 Ga, or to the 30th energy group for isotopes C and 1 H, respectively. Table 17. Summary of second-order relative sensitivities of the leakage response with respect to the total cross sections and the second-order (l = 2) scattering cross sections for all isotopes: S (2) σ g t,i , σ g →h s,l=2,k = ∂ 2 L/∂σ g t,i ∂σ g →h s,l=2,k σ g t,i σ g →h s,l=2,k /L , l = 2; i, k = 1, . . . , 6; g, g , h = 1, . . . , 30.
s,l=3,k /L , l = 3; i, k = 1, . . . , 6; g , g, h = 1, . . . , 30, comprising the second-order relative sensitivities of the leakage response with respect to the total cross sections and the third-order scattering cross sections for all isotopes. The largest absolute values of these mixed second-order sensitivities involve the third-order self-scattering cross sections in the 6th or 7th or 12th energy group, and either the total cross sections for the 7th or 12th energy group for isotopes 239 Pu, 240 Pu, 69 Ga and 71 Ga, or the total cross sections for the 30th energy group for isotopes C and 1 H, respectively. All of these relative sensitivities have values much smaller than 1.0; the largest value is S (2) σ 30 t,i=6 , σ 12→12 s,l=3,k=6 = 7.13 × 10 −2 . Table 18. Summary of the second-order relative sensitivities of the leakage response with respect to the total cross sections and the third-order (l = 3) scattering cross sections between all isotopes: s,l=3,k /L , i, k = 1, . . . , 6; g , g, h = 1, . . . , 30.

Uncertainties in the PERP Leakage Response Induced by Uncertainties in Scattering Cross Sections
Since correlations among the group cross sections are not available for the PERP benchmark, the maximum entropy principle (see, e.g., [8]) indicates that neglecting them minimizes the inadvertent introduction of spurious information into the computations of the various response moments. As has the respective sensitivities computed in Section 2 in Equations (84)-(89). The results thus obtained are presented in Table 19, considering uniform parameter standard deviations of 1%, 5%, and 10%, respectively. These results indicate that the effects of both the first-and second-order sensitivities on the expected response value, its standard deviation and skewness are negligible, which is not surprising in view of the values for the first-and second-order sensitivities already presented in Tables 6-11.  The contributions to the leakage response moments stemming from the group-averaged uncorrelated microscopic scattering cross sections are much smaller than the corresponding contributions stemming from the group-averaged uncorrelated microscopic total cross sections. This fact can be readily illustrated by considering standard deviations of 10% for all of the group-averaged uncorrelated microscopic scattering and total cross sections, and by comparing the corresponding results in Table 19 and Table 25  It is noteworthy that several mixed second-order sensitivities of the leakage response with respect to the total and scattering cross sections, as shown in Section 3, have values that are significantly larger (by several orders of magnitude) than the values of the unmixed sensitivities. Recall that the following sensitivities have absolute values larger than 1.0: (a) 8 elements of the matrix S (2) σ g t,i=6 , σ g →h s,l=0,k=1 , g, g , h = 1, . . . , 30, presented in Table 14; (b) 3 elements of the matrix S (2) σ g t,i=6 , σ g →h s,l=0,k=5 , g, g , h = 1, . . . , 30, as listed in Table 12; (c) 26 elements of the matrix S (2) σ g t,i=6 , σ g →h s,l=0,k=6 , g, g , h = 1, . . . , 30, as listed in Table 13; (d) 2 elements of the matrix S (2) σ g t,i=6 , σ g →h s,l=1,k=1 , g, g , h = 1, . . . , 30, as listed in Table 15; (e) 13 elements of the matrix S (2) σ g t,i=6 , σ g →h s,l=1,k=6 , g, g , h = 1, . . . , 30, as listed in Table 16.
The above results indicate that it would be very important to obtain correlations among the various model parameter, since these correlations could contribute, in conjunction with the mixed second-order sensitivities, to the ultimate values of the response moments. Since the mixed second-order sensitivities of the leakage response to the group-averaged total and scattering microscopic cross sections are significantly larger than the unmixed second-order sensitivities of the leakage response to the group-averaged scattering microscopic cross sections, it is likely that the correlations among the respective total and scattering cross sections could provide significantly larger contributions to the response moments than just the standard deviations of the scattering cross sections.

Conclusions
This work has presented results for the first-and second-order sensitivities of the PERP total leakage response with respect to the benchmark's group-averaged microscopic scattering and total cross sections. 1.
The first-order sensitivities of the leakage response with respect to the zeroth-order self-scattering cross sections can be compared directly to the corresponding unmixed second-order sensitivities. For all six of the isotopes contained in the PERP benchmark, both the first-and the second-order unmixed relative sensitivities of the leakage response with respect to the zeroth-order self-scattering cross sections are small, and the second-order relative sensitivities are much smaller, by at least an order of magnitude, than the corresponding first-order relative sensitivities.

5.
This work has not taken into consideration the effects of the mixed second-order sensitivities of the leakage response with respect to the scattering and total microscopic cross section parameters since no correlations among these parameters are available. However, several mixed second-order sensitivities of the leakage response to the group-averaged microscopic total and scattering cross sections are significantly larger than the unmixed second-order sensitivities of the leakage response with respect to the group-averaged microscopic scattering cross sections. Therefore, it would be very important to obtain correlations among the respective total and scattering cross sections, since these correlations could provide, through the mixed second-order sensitivities, significantly larger contributions to the response moments than just the contributions from the standard deviations of the scattering cross sections.