Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: IV. Effects of Imprecisely Known Source Parameters

By applying the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to the polyethylene-reflected plutonium (PERP) benchmark, this work presents results for the first- and second-order sensitivities of this benchmark’s leakage response with respect to the spontaneous fission source parameters. The numerical results obtained for these sensitivities indicate that the 1st-order relative sensitivity of the leakage response to the source parameters for the two fissionable isotopes in the benchmark are all positive, signifying that an increase in the source parameters will cause an increase in the total neutron leakage from the PERP sphere. The 1st- and 2nd-order relative sensitivities with respect to the source parameters for 239Pu are very small (10−4 or less). In contradistinction, the 1st-order and several 2nd-order relative sensitivities of the leakage response with respect to the source parameters of 240Pu are large. Large values (e.g., greater than 1.0) are also displayed by several mixed 2nd-order relative sensitivities of the leakage response with respect to parameters involving the source and: (i) the total cross sections; (ii) the average neutrons per fission; and (iii) the isotopic number densities. On the other hand, the values of the mixed 2nd-order relative sensitivities with respect to parameters involving the source and: (iv) the scattering cross sections; and (v) and the fission cross sections are smaller than 1.0. It is also shown that the effects of the 1st- and 2nd-order sensitivities of the PERP benchmark’s leakage response with respect to the benchmark’s source parameters on the moments (expected value, variance and skewness) of the PERP benchmark’s leakage response distribution are negligibly smaller than the corresponding effects (on the response distribution) stemming from uncertainties in the total, fission and scattering cross sections.


Introduction
In previous works [1][2][3], the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) conceived by Cacuci [4][5][6] has been applied to the subcritical polyethylene-reflected plutonium (acronym: PERP) metal OECD/NEA-benchmark [7], to compute efficiently the exact values of the 1st-order and 2nd-order sensitivities of the PERP's leakage response with respect to the following model parameters: (i) 180 group-averaged total microscopic cross sections [1]; (ii) 21,600 group-averaged scattering microscopic cross sections [2]; and (iii) 120 fission process parameters [3]. This work, designated as Part IV, presents the results of having applied the 2nd-ASAM to compute the 1st-and 2nd-order sensitivities of the PERP's leakage response with respect to the PERP benchmark's 12 source parameters. The remaining results obtained by applying the 2nd-ASAM to compute the 1st-and 2nd-order sensitivities of the PERP's leakage response with respect to the PERP's 6 isotopic number densities will be presented in Part V [8]. The overall conclusions drawn from this massive sensitivity analysis endeavor will be presented in Part VI [9].
Although the physical characteristics of the PERP metal sphere benchmark have been detailed in Part I [1], it is convenient, for easy reference, to recall the dimensional and material composition of the benchmark in Table 1. The PERP benchmark has no delayed neutron or (α, n) source; the sole source of neutrons is provided by the spontaneous fissions stemming from 239 Pu (Isotope 1) and 240 Pu (Isotope 2). This source has been computed using the code SOURCES4C [10]. For an actinide nuclide k, where k = 1, 2 for the PERP benchmark, the spontaneous source depends on the following 12 model parameters [10]: the decay constant λ k , the atom density N k , the average number of neutrons per spontaneous fission ν SF k , the spontaneous fission branching ratio F SF k , and the two parameters a k and b k used in a Watt's fission spectra to approximate the spontaneous fission neutron spectra. The nominal values of these parameters (except for N k ) are available from a library file contained in SOURCES4C [10], and the nominal values for N k are specified from the PERP benchmark. These imprecisely known source parameters also contribute to the accuracy of the neutron transport calculation. To evaluate the uncertainties induced in the leakage response by the imprecisely known source parameters, the 1st-order and 2nd-order sensitivities of the leakage response with respect to the source parameters will be computed by specializing the general expressions derived by Cacuci [6] to the PERP benchmark.
This work is organized, as follows: Section 2 presents the computational results for the 12 first-order sensitivities and 12 × 12 second-order sensitivities of the leakage response with respect to the benchmark's source parameters. Section 3 reports the numerical results for the 12 × 180 mixed 2nd-order sensitivities to the source parameters and total microscopic cross sections. Section 4 reports the numerical results for the 12 × 21, 600 matrix of mixed 2nd-order sensitivities to the source parameters and scattering microscopic cross sections. Section 5 presents the numerical results for the 12 × 60 mixed 2nd-order sensitivities to the source parameters and fission microscopic cross sections. Section 6 reports the computational results for the 12 × 60 mixed 2nd-order sensitivities to source parameters and the average number of neutrons per fission of all the fissionable isotopes in the PERP benchmark. Section 7 reports the numerical results for the 12 × 6 mixed 2nd-order sensitivities to source parameters and the isotopic number densities of all isotopes in the PERP benchmark. Section 8 presents the impact of the 1st-and 2nd-order sensitivities on the uncertainties induced in the PERP's leakage response by the imprecisely known source parameters. Section 9 offers conclusions based upon the computational results presented in this work.

Computation of 1st-and 2nd-Order Sensitivities of the PERP Leakage Response to Source Parameters
As described in Part I [1], the neutron flux is computed by solving numerically the neutron transport equation using the PARTISN [11] multigroup discrete ordinates transport code. These PARTISN [11] computations were performed using the MENDF71X 618-group cross sections [12] collapsed to G = 30 energy groups, with group boundaries, E g , as presented in [1]. The MENDF71X library uses ENDF/B-VII.1 Nuclear Data [13].
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), (6) where S b is the external surface area of the PERP ball. Figure 1 shows the histogram plot of the leakage for each energy group for the PERP benchmark. The total leakage computed using Equation (6) for the PERP benchmark is 1.7648 × 10 6 neutrons/sec. Table 2 summarizes the integrals for the source, fission source, absorption, in-scattering, self-scattering, out-scattering, and particle balance.  The sub-sections to follow will report computational results for the 1st-and 2nd-order sensitivities of the leakage response with respect to the source parameters for

First-Order Sensitivities ∂L(α α α)/∂q
In view of Equation (4), the source Q g (r) for the PERP benchmark depends on the vector of model parameters q, having components defined as follows: q q 1 , . . . , q J q Table 3. Nominal values of the source parameters for the PERP benchmark [10]. The first-order sensitivity of the PERP leakage response to the source parameters are computed from the following particular form of Equation (154) from Reference [6]: dΩ ψ (1),g (r, Ω) ∂Q g (q; r, Ω) ∂q j , j = 1, . . . , J q .
Performing the integration over the energy interval in Equation (4) yields the following expression for the PERP benchmark's spontaneous fission source: with: where: The first-order derivatives of the spontaneous fission source with respect to the parameters q j=1 ≡ λ i=1 and q j=2 ≡ λ i=2 are as follows: The first-order derivatives of the spontaneous fission source with respect to the parameters q j=3 ≡ F SF i=1 and q j=4 ≡ F SF i=2 are as follows: The first-order derivatives of the spontaneous fission source with respect to the parameters q j=5 ≡ a i=1 and q j=6 ≡ a i=2 are as follows: where: The first-order derivatives of the spontaneous fission source with respect to the parameters q j=7 ≡ b i=1 and q j=8 ≡ b i=2 are as follows: where: The first-order derivatives of the spontaneous fission source with respect to the parameters q j=9 ≡ ν SF i=1 and q j=10 ≡ ν SF i=2 are as follows: The first-order derivatives of the spontaneous fission source with respect to the parameters q j=11 ≡ N 1,1 and q j=12 ≡ N 2,1 are as follows: Inserting the expressions obtained in Equations (18)-(25) into Equation (8) yields the following expressions for the 1st-order sensitivities of the leakage response with respect to the source parameters: For j = 5, 6 : For j = 7, 8 : For j = 9, 10 : For j = 11, 12 : where: ξ The 1st-order absolute sensitivities of the PERP's leakage response with respect to the source parameters for the PERP benchmark are computed using Equations (26)-(31). It is important to note that the parameters and q j=10 ≡ ν SF i=2 appear solely in the expression of the spontaneous fission source, Q g SF , for the PERP benchmark. Therefore, the expressions provided in Equations (26)-(30) represent the total 1st-order sensitivities of the leakage response with respect to these parameters. In contradistinction, however, the isotopic densities N 1,1 and N 2,1 appear not only in the expression of the PERP's source Q g SF , but also appear as parameters in the definitions of the various macroscopic cross sections that enter as coefficients of the various terms in the definition of the forward and adjoint Boltzmann operator (i.e., on the left side of the various forward and adjoint transport equations). Therefore, the expression shown on the right-most side of Equation (31) represents the partial 1st-order sensitivity of the PERP's leakage response with respect to the isotopic densities N 1,1 and N 2,1 appearing solely in the source Q g SF . This fact has been emphasized by using the notation ∂L α; Q g SF in Equation (31). The sensitivities obtained in Equations (26)-(31) are absolute, as opposed to relative sensitivities, which makes it difficult to rank the importance of these sensitivities in affecting to the PERP's leakage response. Therefore, to facilitate the direct comparison of the importance ranking of the sensitivities obtained in Equations (26)-(31), the numerical results for these sensitivities will be presented in unit-less values of the respective relative sensitivities, which are denoted as S (1) q j and are defined as follows: Applying Equation (33) to Equations (26), (27), (30) and (31), yields the same expression for the 1st-order relative sensitivities for S (1) and S (1) (N i,1 ) for i = 1, 2, namely: which means that these sensitivities will all have the same relative values, although their absolute values differ from each other.
The numerical values of the 1st-order relative sensitivities of the PERP leakage response with respect to the source parameters are presented in Tables 4 and 5, below. All the values obtained for the 1st-order sensitivities, as shown in Tables 4 and 5, have been independently verified with the results calculated from the central-difference estimates obtained by repeated forward PARTISN computations, in which the source parameters were individually perturbed by a small amount. These verifications showed good agreements between the sensitivities computed using the 1st-LASS and the corresponding ones computed using central-difference methods. Table 4. First-order relative sensitivities S (1) q j for isotope 239 Pu.  Table 5. First-order relative sensitivities S (1) q j for isotope 240 Pu.
The results shown in Table 4 indicate that the 1st-order relative sensitivities with respect to the source parameters of isotope 239 Pu are very small, in the order of 10 −4 or less. However, as shown in Table 5, the 1st-order relative sensitivities with respect to the source parameters λ 2 , F SF 2 , ν SF 2 , and N 2,1 of isotope 240 Pu are quite large, with values close to 1.0. Also, it can be seen that the leakage response is less sensitive to spectrum effects (i.e., to parameters a and b of the normalized Watt's spectrum) than to the parameters affecting the magnitudes of the respective sources. Moreover, the 1st-order relative sensitivities with respect to the Watt's coefficients a 2 and b 2 of isotope 240 Pu are also much larger than the ones with respect to the Watt's coefficients a 1 and b 1 of isotope 239 Pu.
As indicated in Table 4, the 1st-order sensitivities of the leakage response with respect to the source parameters of isotope 239 Pu (i.e., λ 1 , F SF 1 , ν SF 1 , a 1 , b 1 and N 1,1 ) are all negligibly small by comparison to the corresponding results shown in Table 5 for 240 Pu.

Second-Order Sensitivities
The equations needed for deriving the expression of the 2nd-order sensitivities ∂ 2 L(α)/∂q∂q are obtained by particularizing Equation (208) from Reference [6] to the PERP benchmark, which yields: Computing the unmixed 2nd-order derivatives of the spontaneous fission source with respect to λ i , F SF i , ν SF i and N i,1 shows that they vanish, i.e., ∂Q g (q; r, Ω) ∂λ i ∂λ i = ∂Q g (q; r, Ω) The mixed 2nd-order derivatives with respect to the source parameters that do not belong to the same isotope also vanish, i.e., ∂Q g (q; r, Ω) where i j and i m 2 denote the isotope associated with the source parameters q j and q m 2 , respectively. The expressions of the five non-zero 2nd-order derivatives of the spontaneous fission source with respect to q j=1,2 ≡ λ i and other source parameters are provided in Equations (38) through (42), below: The expressions of the four non-zero 2nd-order derivatives of the spontaneous fission source with respect to q j=5,6 ≡ a i and other source parameters are provided in Equations (47), (49), (51) and (52), below: where: where: The three 2nd-order derivatives of the spontaneous fission source with respect to q j=7,8 ≡ b i and other source parameters are provided in Equations (53), (55) and (56), below: where: The 2nd-order derivatives of the spontaneous fission source with respect to q j=9,10 ≡ ν SF i and q j=11,12 ≡ N i,1 are as follows: Inserting the 2nd-order derivatives obtained in Equations (38)-(57) into Equation (35), yields the following expressions for the 2nd-order sensitivities of the leakage response with respect to the source parameters: For j = 1, m 2 = 3 or j = 2, m 2 = 4: (58) For j = 1, m 2 = 5 or j = 2, m 2 = 6: For j = 1, m 2 = 7 or j = 2, m 2 = 8: For j = 1, m 2 = 9 or j = 2, m 2 = 10: For j = 1, m 2 = 11 or j = 2, m 2 = 12: For j = 3, m 2 = 5 or j = 4, m 2 = 6: For j = 3, m 2 = 7 or j = 4, m 2 = 8: For j = 3, m 2 = 9 or j = 4, m 2 = 10: For j = 3, m 2 = 11 or j = 4, m 2 = 12: For j = 5, m 2 = 5 or j = 6, m 2 = 6: For j = 5, m 2 = 7 or j = 6, m 2 = 8: For j = 5, m 2 = 9 or j = 6, m 2 = 10: For j = 5, m 2 = 11 or j = 6, m 2 = 12: For j = 7, m 2 = 7 or j = 8, m 2 = 8: For j = 7, m 2 = 9 or j = 8, m 2 = 10: For j = 7, m 2 = 11 or j = 8, m 2 = 12: For j = 9, m 2 = 11 or j = 10, m 2 = 12: The 2nd-order absolute sensitivities of the leakage response with respect to the source parameters for the PERP benchmark are computed using Equations (58)-(74). The corresponding relative sensitivities are defined as follows: It is noteworthy that the 2nd-order relative sensitivities for S (2) , and S (2) ν SF i , N i,1 for i = 1, 2 all have the same expression, namely: Furthermore, the right side of Equation (76) is the same as rightmost side of Equation (34). Hence, the respective mixed 2nd-order relative sensitivities have the same values as the 1st-order sensitivities of the leakage response with respect to the source parameters λ i , F SF i , ν SF i , and N i,1 , for i = 1, 2, namely, Similarly, the following relations hold for the 2nd-order sensitivities with respect to the Watt's spectrum coefficients a i and b i : (78) The computations which were performed to obtain the numerical values of the corresponding 2nd-order sensitivities of the PERP leakage response with respect to the source parameters for 239 Pu have yielded results that are several orders of magnitude smaller than the corresponding 1st-order sensitivities shown in Table 4. Therefore, the 2nd-order sensitivities of the leakage response with respect to the source parameters for 239 Pu will not be presented in this work, since they are inconsequential for applications to uncertainty quantification and/or predictive modeling. Hence, the remainder of this work will present only the 2nd-order sensitivities of the leakage response with respect to the source parameters (i.e., λ 2 , F SF 2 , ν SF 2 , a 2 , b 2 and N 2,1 ) for isotope 240 Pu. The numerical results obtained for the 2nd-order relative sensitivities to the source parameters of isotope 240 Pu are presented in Table 6. Since the matrix S (2) q j , q m 2 , j, m 2 = 1, . . . , J q , is symmetrical with respect to its main diagonal, only the results for the elements of the upper triangular segment of this matrix are shown in Table 6. Table 6. 2nd-order relative sensitivities S (2) q j , q m 2 , j, m 2 = 1, . . . , J q of the leakage response with respect to the source parameters of isotope 240 Pu.
The results shown in Table 6 indicate that the 2nd-order relative sensitivities of the leakage response with respect to the source parameters are all positive. The unmixed 2nd-order sensitivities, i.e., the elements on the main diagonal in Table 6, are mostly zero, except for S (2) (a 2 , a 2 ) and S (2) (b 2 , b 2 ). The largest 2nd-order sensitivities are S (2) 9998 which are the same as the 1st-order sensitivities S (1) same values as the 1st-order relative sensitivity of S (1) (a 2 ), which was presented in Table 5. Similarly, the 2nd-order sensitivities S (2) same values as the 1st-order relative sensitivity of S (1) (b 2 ). The unmixed 2nd-order relative sensitivity S (2) (a 2 , a 2 ) with respect to the Watt's coefficient a 2 , is about 50% larger than the corresponding 1st-order sensitivity. However, the value of the 2nd-order relative sensitivity S (2) (b 2 , b 2 ) with respect to the Watt's coefficient b 2 is about 1/7 of the value of the corresponding 1st-order sensitivity S (1) (b 2 ).

Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Benchmark's Source Parameters and Total Cross Sections
This Section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities ∂ 2 L(α)/∂q∂σ t , of the PERP's leakage response with respect to the source parameters and group-averaged total microscopic cross sections of all isotopes of the PERP benchmark. As has been shown by Cacuci [6], these mixed sensitivities can be computed using either one of two distinct expressions, involving distinct 2nd-level adjoint systems and corresponding adjoint functions, by considering either the computation of ∂ 2 L(α)/∂q∂σ t or the computation of ∂ 2 L(α)/∂σ t ∂q. These two distinct paths will be presented in Sections 3.1 and 3.2, respectively. The corresponding end results produced by these two distinct paths must be identical to one another, thus providing a mutual "solution verification", ensuring that the respective computations were performed correctly.

Computing the Second-Order Sensitivities
The equation needed for deriving the expression of the 2nd-order sensitivities ∂ 2 L(α)/∂q∂σ t is obtained by particularizing Equation (204) from Reference [6] to the PERP benchmark, which takes on the following form: where the 2nd-level adjoint functions h (2),g 1, j , j = 1, . . . , J n ; g = 1, . . . , G, are the solutions of the following 2nd-Level Adjoint Sensitivity System (2nd-LASS) presented in Equations (200) and (202) of [6]: The derivatives appearing on the right-side of Equation (81) have been defined previously in Equations (18)-(25) for each of the respective source parameters.
In Equation (80), the parameters t m 2 correspond to the total cross sections, i.e., t m 2 ≡ σ g m 2 t,i m 2 , where the subscripts i m 2 , g m 2 and m m 2 denote the isotope, energy group and material associated with the parameter t m 2 , respectively. The following relation holds: where δ g m 2 g denotes the Kronecker-delta functional (δ g m 2 g = 1 if g m 2 = g; δ g m 2 g = 0 if g m 2 g). Inserting the result obtained in Equation (83) into Equation (80) yields: The equation needed for deriving the expression for ∂ 2 L(α)/∂σ t ∂q is obtained by particularizing Equation (162) from Reference [6] to the PERP benchmark, which yields: where the adjoint functions ψ are the solutions of the 2nd-Level Adjoint Sensitivity System (2nd-LASS) presented in Equations (34) and (40) of Part I [1], which are reproduced below for convenient reference: The parameters t j and q m 2 in Equation (85) correspond to the total cross sections and source parameters, respectively. Inserting the results obtained in Equations (18)-(25) into Equation (85), and performing the respective angular integrations, yields the following simplified expressions for Equation (85): (88) For j = 1, . . . , J σt ; m 2 = 5, 6 : For j = 1, . . . , J σt ; m 2 = 9, 10 : For j = 1, . . . , J σt ; m 2 = 11, 12 : where: ξ and where the subscripts i j = 1, . . . , 6 and i m 2 = 1, 2 denote the isotopes associated with the parameters t j and q m 2 , respectively.
. The corresponding matrix for the 2nd-order relative sensitivities is defined as follows: Applying Equation (95) to Equations (88), (89), (92) and (93) yields the following relations: Therefore, the mixed 2nd-order relative sensitivities S (2) t,k and S (2) N i,1 , σ g t,k of the PERP's leakage response with respect to the total cross section parameter σ g t,k and the source parameters λ i , F SF i , ν SF i , N i,1 , have the same value, which can also be confirmed by using Equation (84) together with Equation (95).
To facilitate the presentation and interpretation of the numerical results, the matrix S (2) q j , t m 2 has been partitioned into J q × I = 12 × 6 submatrices, each of dimensions 1 × G = 1 × 30. The summary of the main features of these submatrices involving the source parameters of isotope 240 Pu is presented in Table 7 in the following form: when a submatrix comprises elements with relative sensitivities having absolute values greater than 1.0, the total number of such elements is shown in the shaded cells of the table. Otherwise, if the relative sensitivities of all the elements of a submatrix have values that lie in the interval (−1.0, 1.0), only the element having the largest absolute value in the submatrix is listed in Table 7, together with the phase-space coordinates of that element. The submatrices in Table 7, which comprise components with absolute values greater than 1.0, will be discussed in detail in subsequent sub-sections of this Section. Table 7. Summary presentation of the matrix S (2) q j , σ g t,k , j = 2, 4, 6, 8, 10, 12 ; k = 1, . . . , 6; g = 1, . . . , 30, for 2nd-order relative sensitivities of the leakage response with respect to the source parameters of isotope 240 Pu and total cross sections for all isotopes.

Second-Order Relative Sensitivities
of the leakage response with respect to source parameter λ 2 , F SF 2 , ν SF 2 , N 2,1 , respectively, of 240 Pu and the 12th energy group of the total cross section for 239 Pu. Table 8.
Second-Order Relative Sensitivities S (2) 6 and S (2) N 2,1 , σ g t, 6 , g = 1, . . . , 30 Table 9 shows the results obtained for the 2nd-order mixed relative sensitivity of the leakage response with respect to the source parameters (λ 2 , F SF 2 , ν SF 2 , N 2,1 ) of isotope 2 ( 240 Pu) and the total cross sections of isotope 6 ( 1 H). These submatrices are denoted as S (2) and S (2) N 2,1 , σ g t, 6 , respectively. As has been shown in Equation (96), the corresponding elements in these four submatrices have the same values. In each submatrix, 6 elements (shown in bold) have relative 2nd-order sensitivities with absolute values greater than 1.0; these large mixed 2nd-order relative sensitivities involve the total cross sections of isotope 1 H for energy groups g = 16, . . . , 20 and g = 30, respectively. The most negative value in the respective submatrix is attained by the elements = −9.364, involving the 30th energy group of the total cross section of isotope 1 H. Table 9.

Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Parameters Underlying the Benchmark's Source Parameters and Scattering Cross Sections
This Section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities ∂ 2 L(α)/∂q∂σ s of the leakage response with respect to the source parameters and group-averaged scattering microscopic cross sections of all isotopes contained in the PERP benchmark. The 2nd-order mixed sensitivities ∂ 2 L(α)/∂q∂σ s can also be computed using the alternative expressions for ∂ 2 L(α)/∂σ s ∂q. These two distinct paths will be presented in Sections 4.1 and 4.2, respectively. As will be discussed in detail in Section 4.3, the pathway for computing ∂ 2 L(α)/∂q∂σ s turns out to be about 590 times more efficient than the pathway for computing ∂ 2 L(α)/∂σ s ∂q.

Computing the Second-Order Sensitivities
The equations needed for deriving the expressions of the 2nd-order sensitivities ∂ 2 L/∂q j ∂s m 2 , j = 1, . . . , J q ; 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, because the 0th-order scattering cross sections contribute to the total cross sections while the higher-order scattering cross sections do not. Therefore, the 0th-order order scattering cross sections must be considered separately from the higher order scattering cross sections. As described in [1][2][3] and Appendix A, the total number of 0th-order scattering cross sections comprised in σ s is denoted as J σs,l=0 , where J σs,l=0 = G × G × I, while the total number of higher order 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 , where ISCT is the total number of Legendre moments in the finite expansion of the scattering cross sections. There are two distinct cases, as follows: , j = 1, . . . , J q ; m 2 = 1, . . . , J σs,l=0 , where the quantities q j refer to the source parameters while the quantities s m 2 refer to the parameters underlying the 0th-order (l = 0) scattering microscopic cross sections; and , j = 1, . . . , J q ; m 2 = 1, . . . , σ s,l≥1 ,, where the quantities q j refer to the source parameters while the quantities s m 2 refer to the parameters underlying the l th -order (l ≥ 1) scattering microscopic cross sections.
As shown in Table 10, the absolute values of all elements in the matrix S (2) q j , σ g →g s,l=0,k are smaller than 1.0. The overall largest value in the matrix S (2) q j , σ g →g s,l=0,k is attained by the sensitivities S (2) λ 2 , σ 12→12 s,l=0,5 = S (2) F SF 2 , σ 12→12 s,l=0,5 = S (2) ν SF 2 , σ 12→12 s,l=0,5 = S (2) N 2,1 , σ 12→12 s,l=0,5 = 0.681, all of which involve the 0th-order self-scattering cross section for the 12th energy group of isotope 5 (C). The values of the mixed 2nd-order relative sensitivities S (2) λ 2 , σ g →g  Table 10, the values of all of the largest elements of each of the respective sub-matrices are positive; most of these elements involve the 0th-order self-scattering cross sections for the 12th energy group of isotopes 239 Pu, 240 Pu, 69 Ga, 71 Ga and C, while the others involve the 0th-order out-scattering cross section σ 16→17 s,l=0,k=6 for isotope 1 H. Table 10. Summary presentation of the matrix S (2) q j , σ g →g s,l=0,k , for 2nd-order relative sensitivities of the leakage response with respect to the source parameters of isotope 240 Pu and the 0th-order (l = 0) scattering cross sections for all isotopes in the PERP benchmark.
The mixed 2nd-order relative sensitivities with respect to the source parameters a 2 , b 2 of isotope 240 Pu and the 0th-order scattering cross sections for all isotopes, namely, S (2) a 2 , σ g →g s,l=0,k , S (2) b 2 , σ g →g s,l=0,k , for k = 1, . . . , 6, are small, having absolute values of the order of 10 −2 or less. As shown in Table 10, the values of the largest elements of the respective sub-matrix are all positive, except for S (2) a 2 , σ 16→17 s,l=0,k=6 and S (2) b 2 , σ 16→17 s,l=0,k=6 , which have small negative values; these elements involve (most of the time) either the 0th-order self-scattering cross sections for the 7th energy group of isotopes 239 Pu, 240 Pu, 69 Ga, 71 Ga and C, or (occasionally) the 0th-order out-scattering cross section σ 16→17 s,l=0,k=6 for isotope 1 H.
The elements of S (2) Table 11 also indicates that all of the nonzero values of the elements of the matrices S (2) s,l=1,k=6 , which comprise the mixed 2nd-order relative sensitivities with respect to the source parameters λ 2 , F SF 2 , ν SF 2 , N 2,1 of isotope 240 Pu and the 1st-order scattering cross sections of isotope 1 H, are negative. The results presented in the Table 11 also reveal that the largest elements of the respective sub-matrix are all negative, involving either the 1st-order self-scattering cross sections for the 7th energy group of isotopes 239 Pu, 240 Pu, 69 Ga and 71 Ga (i.e., σ 7→7 s,l=1,k , k = 1, . . . , 4) or the 12th energy group of isotope C (i.e., σ 12→12 s,l=0,k=5 ), or the 1st-order out-scattering cross section σ 12→13 s,l=0,k=6 of isotope 1 H. As also shown in Table 11, the elements of S (2) a 2 , σ g →g s,l=1,k , S (2) b 2 , σ g →g s,l=1,k for k = 1, . . . , 6 [i.e., the mixed 2nd-order relative sensitivities of the leakage response with respect to the source parameters a 2 , b 2 of isotope 240 Pu and the 1st-order scattering cross sections for all isotopes] are all small and can have either positive or negative values. The value of the largest elements of these submatrices are generally negative, except for S (2) a 2 , σ 16→16 s,l=1,k=6 and S (2) b 2 , σ 16→16 s,l=1,k=6 , which have positive values. The majority of these elements involve the 1st-order self-scattering cross sections for the 7th energy group of isotopes 239 Pu, 240 Pu, 69 Ga, 71 Ga and C (namely, σ 7→7 s,l=0,k , k = 1, . . . , 5) while a minority involve the 16th energy group of isotope 1 H (namely, σ 16→16 s,l=1,k=6 ).

Results for the Relative Sensitivities
The matrix S (2) q j , σ g →g s,l=2,k , j = 2, 4, 6, 8, 10, 12; k = 1, . . . , 6; g , g = 1, . . . , 30, comprises the 2nd-order mixed relative sensitivities of the leakage response with respect to the source parameters of isotope 240 Pu and the 2nd-order scattering cross sections for all isotopes in the PERP benchmark. As expected, based on the work previously performed in Part II [2], these 2nd-order mixed relative sensitivities with respect to the higher-order scattering cross sections are very small, of the order of 10 −2 or less. The overall largest element in this matrix is S (2) Due to the small values of its elements, the detailed features of S (2) q j , σ g →g s,l=2,k are not presented in work.

Results for the Relative Sensitivities
The matrix S (2) q j , σ g →g s,l=3,k , j = 2, 4, 6, 8, 10, 12; k = 1, . . . , 6; g , g = 1, . . . , 30, comprises the 2nd-order mixed relative sensitivities of the leakage response with respect to the source parameters of isotope 240 Pu and the 3rd-order scattering cross sections for all isotopes in the PERP benchmark. The elements S (2) s,l=3,k=6 = −5.35 × 10 −3 have the largest absolute values; the remaining elements are even smaller and will therefore not be discussed further.

Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Benchmark's Source Parameters and Fission Cross Sections
This Section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities ∂ 2 L(α)/∂q∂σ f , of the leakage response with respect to the source parameters and group-averaged fission microscopic cross sections of all isotopes of the PERP benchmark. These 2nd-order mixed sensitivities can also be obtained by alternatively computing the matrix ∂ 2 L(α)/∂σ f ∂q. As illustrated in Sections 5.1 and 5.2, respectively, these two distinct paths use distinct 2nd-level adjoint functions and therefore provide an intrinsic verification of the accuracy of the respective computations.
and where N f = 2 denotes the total number of fissionable isotopes in the PERP benchmark. The matrix of 2nd-order relative sensitivities corresponding to ∂ 2 L/∂q j ∂ f m 2 , j = 1, . . . , J q ; m 2 = 1, . . . , J σ f , will be denoted as S (2) q j , σ g f ,k and is defined as follows: Applying Equation (150) to Equations (143), (144), (147) and (148) yields the following result: (2),g 2, j;0 (r) + U (2),g 2, j;0 (r) Q g SF,i , f or i = 1, 2; k = 1, 2; g = 1, . . . , 30. (151) As indicated by Equation (151), the mixed 2nd-order relative sensitivities with respect to the microscopic fission cross section σ g f ,k and the source parameters f ,k and S (2) N i,1 , σ g f ,k , have the same value. This result has also been confirmed by using Equation (137) together with Equation (150). Table 12 summarizes the results for the J q × J σ f (= 6 × 60) matrix S (2) q j , σ g f ,k , j = 2, 4, 6, 8, 10, 12 ; k = 1, 2; g = 1, . . . , 30, which comprises the 2nd-order relative sensitivities of the leakage response with respect to the source parameters of isotope 240 Pu and the fission cross sections for all isotopes in the PERP benchmark. To facilitate the presentation of the numerical results, the matrix S (2) q j , σ g f ,k has been partitioned into J q × N f (= 6 × 2) submatrices, each of dimensions 1 × G = 1 × 30. It has been found that the absolute values of all elements of S (2) q j , σ g f ,k are all smaller than 1.0. Of the sensitivities summarized in Table 12, the single largest relative value is  Table 12. Summary of the matrix S (2) q j , σ g f ,k , j = 2, 4, 6, 8, 10, 12 ; k = 1, 2; g = 1, . . . , 30, for 2nd-order relative sensitivities of the leakage response with respect to the source parameters of isotope 240 Pu and fission cross sections for all fissionable isotopes.

Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Benchmark's Source Parameters and the Average Number of Neutrons per Fission
This Section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities ∂ 2 L(α)/∂q∂ν of the leakage response with respect to the source parameters and the average number of neutrons per fission of all isotopes in the PERP benchmark. These 2nd-order mixed sensitivities can also be computed by using the alternative expression ∂ 2 L(α)/∂ν∂q, which requires adjoint functions that are distinct from those required for computing ∂ 2 L(α)/∂q∂ν. These two distinct paths are illustrated in Sections 6.1 and 6.2, respectively, as follows.

Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Benchmark's Source Parameters and Isotopic Number Densities
The 2nd-order sensitivities of the leakage response with respect to the source parameters , and , have been computed using Equations (62), (66), (70), (73) and (74), respectively, and the respective numerical results have been presented in Table 6.
As denoted by the presence of Q g SF in the argument of the leakage response L α; Q g SF , only the contributions stemming from the spontaneous fission source were considered in the computation of these mixed 2nd-order sensitivities.
In order to account for the partial contributions stemming from the macroscopic total, scattering and fission cross sections, as well as the source, this Section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities ∂ 2 L(α)/∂q∂N of the leakage response with respect to the source parameters and isotopic number densities of all (including the non-fissionable) isotopes of the PERP benchmark. Note that the 2nd-order mixed sensitivities ∂ 2 L(α)/∂q∂N can also be computed using the alternative expressions for ∂ 2 L(α)/∂N∂q. These two distinct paths are illustrated in Sections 7.1 and 7.2, respectively.

Quantification of Uncertainties in the PERP Leakage Response due to Uncertainties in Source Parameters
Correlations between the source parameters or correlations between these source parameters and other cross section parameters are not available for the PERP benchmark. As discussed in [1][2][3], when such correlations are unavailable, the maximum entropy principle (see, e.g., Ref. [14]) 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. Considering the PERP leakage response 1st-and 2nd-order sensitivities with respect to the PERP benchmark's source parameters, the formulas for computing the expected value, variance and skewness of the leakage response distribution are as follows: 1) The expected value, [E(L)] q , of the leakage response L(α) has the following expression: where the superscript "U" indicates contributions solely from the uncorrelated source parameters, and where the term [E(L)] (2,U) q , which provides the 2nd-order contributions, is given by the following expression: In Equation (193), the quantity s q i denotes the standard deviation associated with the imprecisely known model parameter q i , i = 1, . . . , J q .
2) Taking into account contributions solely from the uncorrelated and normally-distributed source parameters (which will be indicated by using the superscript "(U,N)" in the following equations), the 3) Considering contributions solely from the uncorrelated normally-distributed source parameters, the third-order moment, [µ 3 (L)] (U,N) q , of the leakage response for the PERP benchmark takes on the following form: As Equation (197) The effects of the first-and second-order sensitivities on the response's expected value, variance and skewness are quantified by considering typical values for the standard deviations for the uncorrelated source parameters, using these values together with the respective sensitivities computed in Section 2 in Equations (192) through (198). The results thus obtained are presented in Table 16, 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 expected in view of the small values for the first-and second-order sensitivities presented in Tables 4 and 5.  The relative effects of uncertainties in the source parameters can be compared to the corresponding effects stemming from the total and scattering cross sections, respectively, by considering standard deviations of 10% for all of these parameters and by comparing the corresponding results shown in Table 16 with the corresponding results presented in Table 25 from Part I [1] and Table 19 from Part II [2]. This comparison reveals that the following relations hold: The above relations indicate that the contributions to the expected value, second-order variance and skewness stemming from the uncorrelated source parameters are much smaller than the corresponding contributions stemming from the group-averaged uncorrelated microscopic scattering and total cross sections. However, the contributions to the first-order variance stemming from uncorrelated source parameters are larger than those stemming from the uncorrelated microscopic scattering cross sections but are much smaller than those stemming from the uncorrelated microscopic total cross sections.
It would be very important to establish if correlations among the model parameters mentioned in items (a)-(c), above, since such correlations could contribute, in conjunction with the respective mixed second-order sensitivities, to the values of the response moments. Since the mixed second-order sensitivities of the leakage response to the source parameters and group-averaged total microscopic cross sections are significantly larger than the unmixed second-order sensitivities of the leakage response to the source parameters, it is likely that the correlations among the respective source parameters and the total cross sections could provide significantly larger contributions to the response moments than just the standard deviations of the source parameters.

Discussions and Conclusions Related to the Sensitivities and Uncertainties to the Source Parameters
This work has presented results for the first-order sensitivities, ∂L(α)/∂q, and the second-order sensitivities ∂ 2 L(α)/∂q∂q of the PERP total leakage response with respect to the source parameters. In addition, this work has also presented the results for the mixed second-order sensitivities ∂ 2 L(α)/∂q∂σ t , ∂ 2 L(α)/∂q∂σ s , ∂ 2 L(α)/∂q∂σ f , ∂ 2 L(α)/∂q∂ν and ∂ 2 L(α)/∂q∂N. 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 source parameters for the fissionable isotopes are all positive, signifying that an increase in the source parameters will cause an increase in the total neutron leakage from the PERP sphere. (2) The 1st-order relative sensitivities for S (1) (λ i ), S (1) F SF i , S (1) ν SF i and S (1) (N i,1 ) for i = 1, 2 have the same value, although their absolute sensitivities differ from each other. The 1st-order relative sensitivities with respect to the source parameters of isotope 239 Pu are very small, of the order of 10 −4 or less. However, the 1st-order relative sensitivities with respect to the source parameters λ 2 , F SF 2 , ν SF 2 , and N 2,1 of isotope 240 Pu are quite large, with values close to 1.0. (3) The following relations hold for the 1st-and 2nd-order sensitivities to the source parameters: 1 ; S (1) (a i ) = S (2) (λ i , a i ) = S (2) F SF i , a i = S (2) a i , ν SF i = S (2) (a i , N i,1 ); and S (1) (b i ) = S (2) (λ i , b i ) = S (2) F SF i , b i = S (2) b i , ν SF i = S (2) (b i , N i,1 ) for i = 1, 2. (4) The 2nd-order sensitivities ∂ 2 L(α)/∂q∂q are all positive. The 2nd-order relative sensitivities of the leakage response with respect to the source parameters of isotope 239 Pu are very small, of the order of 10 −4 or less. However, several mixed 2nd-order relative sensitivities of the leakage response with respect to the source parameters of isotope 240 Pu are quite large, having values close to 1.0. The unmixed 2nd-order sensitivities in the matrix S (2) q j , q m 2 , j, m 2 = 1, . . . , J q are mostly zero, except for S (2) (a i , a i ) and S (2) (b i , b i ), i = 1, 2. Moreover, the unmixed 2nd-order relative sensitivity with respect to the Watt's coefficient a 2 , namely, S (2) (a 2 , a 2 ), is about 50% larger than the corresponding 1st-order one; whereas the value of the 2nd-order relative sensitivity with respect to the Watt's coefficient b 2 , namely, S (2) (b 2 , b 2 ), is about 1/7 of the value of the corresponding 1st-order sensitivity S (1) (b 2 ). (5) For the 2nd-order mixed sensitivities ∂ 2 L(α)/∂q∂σ t , among the J q × J σt (= 2160) elements of the matrix S (2) q j , σ g t,k , j = q j ; k = 1, . . . , 6; g = 1, . . . , 30, 32 elements have relative sensitivities greater than 1.0. These large sensitivities involve the total cross sections of isotopes 239 Pu or 1 H. However, when the source parameters a i or b i , or the total cross sections of isotopes 240 Pu, 69 Ga, 71 Ga and C are involved, the absolute values of the mixed 2nd-order relative sensitivities are all smaller than 1.0. The largest absolute values in the matrix S (2) q j , σ g t,k are S (2)  = −9.364. Also, all the elements in the submatrices S (2) λ 2 , σ g t,k , S (2) F SF 2 , σ g t,k , S (2) ν SF 2 , σ g t,k and S (2) N 2,1 , σ g t,k have negative values; whereas the elements in submatrices S (2) a 2 , σ g t,k and S (2) b 2 , σ g t,k can have positive or negative values, depending on the energy group as well as the isotope of the microscopic total cross sections. (6) For the 2nd-order mixed sensitivities ∂ 2 L(α)/∂q∂σ s , the corresponding relative sensitivities are all smaller than 1.0. The overall largest value in the matrix S (2) q j , σ g →g s,l=0,k is S (2) λ 2 , σ 12→12 s,l=0,5 = S (2) F SF 2 , σ 12→12 s,l=0,5 = S (2) ν SF 2 , σ 12→12 s,l=0,5 = S (2) N 2,1 , σ 12→12 s,l=0,5 = 0.681. All of these (largest) sensitivities are related to the 0th-order self-scattering cross section for the 12th energy group of isotope 5 (C). For the 2nd-order mixed relative sensitivities with respect to the source parameters and the 0th-order (i.e., l = 0) scattering microscopic cross sections, the values of the relative sensitivities can be positive or negative, but there are more positive values than negative ones. For the 2nd-order mixed relative sensitivities with respect to the source parameters and the 1st-order (i.e., l = 1) scattering microscopic cross sections, the overall largest (absolute) value is S (2) λ 2 , σ 12→13 s,l=1,k=6 = S (2) F SF 2 , σ 12→13 s,l=1,k=6 = S (2) ν SF 2 , σ 12→13 s,l=1,k=6 = S (2) N 2,1 , σ 12→13 s,l=1,k=6 = −0.104; these sensitivities involve the 1st-order out-scattering cross section σ 12→13 s,l=0,k=6 of isotope 1 H. In addition, for the scattering order l = 1, the values of the relative sensitivities can also be positive or negative, but there are more negative values than positive ones. The values for the 2nd-order mixed relative sensitivities of the leakage response with respect to the source parameters of isotope 240 Pu and the higher-order (i.e., l = 2, 3) scattering cross sections for all isotopes in the PERP benchmark are all very small, in the order of 10 −2 or less. (7) For the 2nd-order mixed sensitivities ∂ 2 L(α)/∂q∂σ f , it has been found that the values of the corresponding relative sensitivities are all smaller than 1.0. The single largest relative value is S (2) λ 2 , σ g=12 f ,1 in the energy region g = 7, . . . , 14, with the largest portion contained in group 12, and the next largest contained in group 7. (9) For the 2nd-order mixed sensitivities ∂ 2 L(α)/∂q∂N, it has been found that among the 60 elements in the relative sensitivity matrix S (2) q j , N k,m , there are 9 elements having values greater than 1.0; these are: S (2) (λ 2 , N 1,1 ) = S (2) F SF 2 , N 1,1 = S (2) ν SF 2 , N 1,1 = 5.967; S (2) (λ 2 , N 2,1 ) = S (2) F SF 2 , N 2,1 = S (2) (λ 2 , N 2,1 ) = 1.219; and S (2) (λ 2 , N 6,2 ) = S (2) F SF 2 , N 6,2 =S (2) ν SF 2 , N 6,2 = 1.001. The elements S (2) λ 2 , N k,m , S (2) F SF 2 , N k,m , and S (2) ν SF 2 , N k,m , k = 1, . . . 6; m = 1, 2, have identical values. Also, all of the mixed 2nd-order sensitivities of the leakage response with respect to the source parameters and the isotopic number densities are positive, except for S (2) (a 2 , N 6,2 ) and S (2) (b 2 , N 6,2 ), which have negative values. (9) By considering typical values for the standard deviations for the uncorrelated source parameters, it has been found that the effects of both the first-and second-order sensitivities on the expected response value, its standard deviation and skewness are negligible. However, many mixed 2nd-order sensitivities in matrices ∂ 2 L(α)/∂q∂σ t , ∂ 2 L(α)/∂q∂ν and ∂ 2 L(α)/∂q∂N are significantly larger than the unmixed 2nd-order sensitivities of the leakage response with respect to the source parameters. Therefore, it would be very important to obtain correlations among the various model parameters, since the correlations among the source parameters and other model parameters (e.g., total cross sections, average number of neutrons per fission, and isotopic number densities) could provide significantly larger contributions to the response moments than the standard deviations of the source parameters.
Author Contributions: D.G.C. conceived and directed the research reported herein, developed the general theory of the second-order comprehensive adjoint sensitivity analysis methodology to compute 1st-and 2nd-order sensitivities of flux functionals in a multiplying system with source, and the uncertainty equations for response moments. R.F. derived the expressions of the various derivatives with respect to the model parameters to the PERP benchmark and performed all the numerical calculations. All authors have read and agreed to the published version of the manuscript.