Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: V. Computation of Mixed 2nd-Order Sensitivities Involving Isotopic Number Densities

Abstract

This work applies the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to compute the mixed 2nd-order sensitivities of a polyethylene-reflected plutonium (PERP) benchmark’s leakage response with respect to the benchmark’s imprecisely known isotopic number densities and the other benchmark imprecisely known parameters, including: (i) the 6 × 180 mixed 2nd-order sensitivities involving the total microscopic cross sections; (ii) the 6 × 21,600 mixed 2nd-order sensitivities involving the scattering microscopic cross sections; (iii) the 6 × 60 mixed 2nd-order sensitivities involving the fission microscopic cross sections; and (iv) the 6 × 60 mixed 2nd-order sensitivities involving the average number of neutrons produced per fission. It is shown that many of these mixed 2nd-order sensitivities involving the isotopic number densities have very large values. Most of the large sensitivities involve the isotopic number density of ²³⁹Pu, and the microscopic total, scattering or fission cross sections for the 12th or 30th energy groups of ²³⁹Pu or ¹H, respectively. The 2nd-order mixed sensitivity of the PERP leakage response with respect to the isotopic number density of ²³⁹Pu and the microscopic total cross section for the 30th energy group of ¹H is the largest of the above mentioned sensitivities, attaining the value −94.91.

1. Introduction

In Parts I−IV [1,2,3,4], which are precursors of this work, the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) conceived by Cacuci [5,6,7] has been successfully applied to the subcritical polyethylene-reflected plutonium (acronym: PERP) metal fundamental physics benchmark [8], to compute the exact values of the 1st-order and 2nd-order sensitivities of the PERP’s benchmark leakage response with respect to the 180 group-averaged total microscopic cross sections [1], 21,600 group-averaged scattering microscopic cross sections [2], 120 fission process parameters [3], and 10 source parameters [4]. This work presents the results obtained for the mixed 2nd-order sensitivities of the PERP benchmark’s leakage response with respect to the benchmark’s six isotopic number densities.

Table 1. Dimensions and material composition of the PERP benchmark.

Materials	Isotopes	Weight Fraction	Density (g/cm3)	Zones
Material 1 (plutonium metal)	Isotope 1 (239Pu)	9.3804 × 10−1	19.6	Material 1 is assigned to zone 1, which has a radius of 3.794 cm.
	Isotope 2 (240Pu)	5.9411 × 10−2
	Isotope 3 (69Ga)	1.5152 × 10−3
	Isotope 4 (71Ga)	1.0346 × 10−3
Material 2 (polyethylene)	Isotope 5 (C)	8.5630 × 10−1	0.95	Material 2 is assigned to zone 2, which has an inner radius of 3.794 cm and an outer radius of 7.604 cm.
	Isotope 6 (1H)	1.4370 × 10−1

summarizes the dimensions and material composition of the PERP benchmark; additional details are presented in Part I [1]. As shown in Table 1, the six isotopic number densities correspond to each of the isotopes contained in the PERP benchmark, respectively. The isotopic number density is one of important parameters that contribute to the accuracy of the neutron transport calculation, as it appears in the definitions/constructions of the total, scattering and fission macroscopic cross sections, as well as the source term of the neutron transport equation.

The mixed 2nd-order sensitivities of the leakage response with respect to the isotopic number densities are computed by specializing the general expressions derived by Cacuci [5,6,7] to the PERP benchmark. This work is organized as follows: Section 2 presents the numerical results for the $6\times 180$ mixed 2nd-order sensitivities with respect to the isotopic number densities and total microscopic cross sections. Section 3 summarizes the numerical results for the $6\times 21600$ matrix of mixed 2nd-order sensitivities to the isotopic number densities and scattering microscopic cross sections. Section 4 presents the numerical results for the $6\times 60$ mixed 2nd-order sensitivities to the isotopic number densities and fission microscopic cross sections. Section 5 reports the numerical results for the $6\times 60$ mixed 2nd-order sensitivities to the isotopic number densities and the average number of neutrons per fission. The conclusions drawn from the numerical results presented in this work are summarized in Section 6.

2. Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Parameters Underlying the Benchmark’s Isotopic Number Densities and Total Cross Sections

As described in Part I [1], the neutron flux is computed by solving numerically the neutron transport equation using the PARTISN [9] multigroup discrete ordinates transport code. For the PERP benchmark under consideration, PARTISN [9] solves the following multi-group approximation of the neutron transport equation with a spontaneous fission source provided by the code SOURCES4C [10]:

$$B^g\left(\alpha \right){\phi }^g\left(r,\mathsf{\Omega }\right)=Q^g\left(r\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(1)

$${\phi }^g\left(r_d,\mathsf{\Omega }\right)=0,\mathsf{\Omega }\cdot n<0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(2)

where $r_d$ denotes the external radius of the PERP benchmark, and where

$$\begin{array}{l}{B}^g\left(\alpha \right){\phi }^g\left(r,\mathsf{\Omega }\right)\triangleq \mathsf{\Omega }·\nabla {\phi }^g\left(r,\mathsf{\Omega }\right)+{\Sigma }_t^g\left(r\right)\hspace{0.17em}{\phi }^g\left(r,\mathsf{\Omega }\right)\\ -{\displaystyle \sum _{g^{\prime }=1}^G{\displaystyle \underset{4\pi }{\int }{\Sigma }_s^{g^{\prime }\to g}\left(r,{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega }\right)\hspace{0.17em}{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)d{\mathsf{\Omega }}^{\prime }}\hspace{0.17em}}\hspace{0.17em}-{\chi }^g\left(r\right){\displaystyle \sum _{g^{\prime }=1}^G{\displaystyle \underset{4\pi }{\int }{\left(\nu \Sigma \right)}_f^{g^{\prime }}\left(r\right)\hspace{0.17em}{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)d{\mathsf{\Omega }}^{\prime }}\hspace{0.17em}},\end{array}$$

(3)

$$Q^g\left(r\right)\triangleq {\displaystyle \sum _{k=1}^{N_f}{\lambda }_k{N}_{k,1}{F}_k^{SF}{\nu }_k^{SF}\left(\frac{2}{\sqrt{\pi a_k{}^3{b}_k}}{e}^{-\frac{a_k{b}_k}{4}}\right){\displaystyle {\int }_{E^{g+1}}^{E^g}dE\hspace{0.33em}{e}^{-E/a_k}\sinh \sqrt{b_{k}E}}}.$$

(4)

In Equation (1), the vector $\alpha$ denotes the “vector of imprecisely known model parameters”, which has been defined in Part I [1] as $\alpha \triangleq {\left[{\sigma }_t;{\sigma }_s;{\sigma }_f;\nu ;p;q;N\right]}^{†}$. The vector-valued components ${\sigma }_t$, ${\sigma }_s$, ${\sigma }_f$, $\nu$, $p$, $q$ and $N$ 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. For convenient reference, the components of the vector of model parameters $\alpha$ are reproduced in Appendix A.

The PARTISN [9] calculations used the MENDF71X [11] 618-group cross sections 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 [12].

The total neutron leakage from the PERP sphere, denoted as $L\left(\alpha \right)$, will depend (indirectly, through the neutron flux) on the imprecisely known model parameters and is defined as follows:

$$L\left(\alpha \right)\triangleq {\displaystyle \underset{S_b}{\int }dS}{\displaystyle \sum _{g=1}^G {\displaystyle \underset{\mathsf{\Omega }\cdot n>0}{\int }d\mathsf{\Omega }}}\hspace{0.33em}\mathsf{\Omega }\cdot n\hspace{0.17em}{\phi }^g\left(r,\mathsf{\Omega }\right).$$

(5)

As has been shown by Cacuci [5], the 2nd-order mixed sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_t$ can be computed using two distinct expressions, involving distinct 2nd-level adjoint systems and corresponding 2nd-level adjoint functions, by considering either the computation of ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_t$ or the computation of ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_t\partial N$. These two distinct paths will be presented in Section 2.1 and Section 2.2, respectively. The end results produced by these two distinct paths must be identical to one another, thus providing a mutual “solution verification” which ensures that the respective computations were performed correctly. Moreover, the computation of ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_t$ can be significantly more efficient than the computation of ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_t\partial N$, as will be further illustrated by the numerical results presented in Section 2.3.

2.1. Computing the Second-Order Sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_t$

The PERP benchmark comprises six distinct isotopes and two distinct materials; the respective isotopes are not repeated in the two materials. Therefore, only the following isotopic number densities exist for this benchmark: $N_{1,1},N_{2,1},N_{3,1},N_{4,1},N_{5,2},N_{6,2}$, so that the vector $N$ is defined as follows:

$$N\triangleq {\left[n_1,\dots ,n_{J_n}\right]}^{{}^{†}}\triangleq {\left[N_{1,1},N_{2,1},N_{3,1},N_{4,1},N_{5,2},N_{6,2}\right]}^{†},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{J}_n=6.$$

(6)

The vector ${\sigma }_t$ is defined in Part I [1] and in Equation (A6) in Appendix A.

The equations needed for deriving the expression of the 2nd-order sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_t$ are obtained by particularizing Equations (158), (167), (177) and (204) in [5] to the PERP benchmark and adding their respective contributions. The expression obtained by particularizing Equation (158) in [5] takes on the following form:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial t_{m_2}}\right)}^{\left(1\right)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{\psi }^{(1),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\frac{{\partial }^2{\Sigma }_t{}^g\left(t\right)}{\partial n_j\partial t_{m_2}}}\\ -{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[{\psi }_{1,i}^{(2),g}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,i}^{(2),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial t_{m_2}}},\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.33em}{m}_2=1,\dots ,J_{\sigma t}.\end{array}$$

(7)

The multigroup adjoint fluxes ${\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right),\hspace{0.33em}g=1,\dots ,G$, appearing in Equation (7) are the solutions of the following 1st-Level Adjoint Sensitivity System (1st-LASS) presented in Equations (156) and (157) of [5]:

$$A^{\left(1\right),g}\left(\alpha \right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)=\mathsf{\Omega }\cdot n\delta \left(r-r_d\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(8)

$${\psi }^{\left(1\right),g}\left(r_d,\mathsf{\Omega }\right)=0,\mathsf{\Omega }\cdot n>0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}g=1,\dots ,G,$$

(9)

where the adjoint operator $A^{\left(1\right),g}\left(\alpha \right)$ takes on the following particular form of Equation (149) in [5]:

$$\begin{array}{l}{A}^{\left(1\right),g}\left(\alpha \right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)\\ \triangleq -\mathsf{\Omega }·\nabla {\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+{\Sigma }_t^g\left(t\right)\hspace{0.17em}{\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)\hspace{0.17em}-{\displaystyle \sum _{g^{\prime }=1}^G{\displaystyle \underset{4\pi }{\int }d{\mathsf{\Omega }}^{\prime }}{\Sigma }_s^{g\to g^{\prime }}\left(s;\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime }\right)\hspace{0.17em}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\hspace{0.17em}}\\ \hspace{0.17em}-\nu {\Sigma }_f^g\left(f\right){\displaystyle \sum _{g^{\prime }=1}^G{\displaystyle \underset{4\pi }{\int }d{\mathsf{\Omega }}^{\prime }}{\chi }^{g^{\prime }}\hspace{0.17em}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G}.\end{array}$$

(10)

The 2nd-level adjoint functions ${\psi }_{1,i}^{(2),g}$ and ${\psi }_{2,j}^{(2),g},\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G$, appearing in Equation (7) are the solutions of the following 2nd-Level Adjoint Sensitivity System (2nd-LASS) obtained from Equations (164)–(166) of [5]:

$$B^g\left({\alpha }^0\right){\psi }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right)=-{\sigma }_{t,i_j}^g{\phi }^g\left(r,\mathsf{\Omega }\right),\hspace{0.33em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.33em}g=1,\dots ,G,$$

(11)

$${\psi }_{1,i}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n<0;\hspace{0.33em}\hspace{0.33em}i=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G,$$

(12)

$$A^{(1),g}\left({\alpha }^0\right){\psi }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right)=-{\sigma }_{t,i_j}^g{\psi }^{(1),g}\left(r,\mathsf{\Omega }\right),\hspace{0.33em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(13)

$${\psi }_{2,i}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n>0;\hspace{0.33em}\hspace{0.33em}i=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G.$$

(14)

The parameters $n_j$ and $t_{m_2}$ in Equation (7) correspond to the isotopic number densities and microscopic total cross sections, respectively, and will therefore be denoted as $n_j\equiv N_{i_j,m_j}$ and $t_{m_2}\equiv {\sigma }_{t,i_{m_2}}^{g_{m_2}}$, where the subscripts $i_j$ and $m_j$ denote the isotope and material associated with the parameter $n_j$, while 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 derivatives will be used in Equation (7) and subsequently in this work:

$$\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial t_{m_2}}=\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial {\sigma }_{t,i_{m_2}}^{g_{m_2}}}=\frac{\partial \left({\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}{\sigma }_{t,i}^g}}\right)}{\partial {\sigma }_{t,i_{m_2}}^{g_{m_2}}}={\delta }_{g_{m_2}g}{N}_{i_{m_2},m_{m_2}},$$

(15)

$$\begin{array}{c}\frac{{\partial }^2{\Sigma }_t{}^g\left(t\right)}{{\partial }_{n_j}\partial t_{m_2}}=\frac{{\partial }^2{{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial N_{i_j,m_j}\partial {\sigma }_{t,i_{m_2}}^{g_{m_2}}}=\frac{\partial \left[\partial {{\Sigma }_t}^g\left(\mathbf{t}\right)/\partial N_{i_j,m_j}\right]}{\partial {\sigma }_{t,i_{m_2}}^{g_{m_2}}}=\frac{\partial \left[\partial \left({\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{N}_{i,m}{\sigma }_{t,i}^g\right)/\partial N_{i_j,m_j}\right]}{\partial {\sigma }_{t,i_{m_2}}^{g_{m_2}}}\hfill \\ \\ =\frac{\partial \left[{\sigma }_{t,i_j}^g\right]}{\partial {\sigma }_{t,i_{m_2}}^{g_{m_2}}}={\delta }_{i_j{i}_{m_2}}{\delta }_{g_{m_2}g},\hfill \end{array}$$

(16)

where ${\delta }_{g_{m_2}g}$ denotes the Kronecker-delta functionals (e.g., ${\delta }_{g_{m_2}g}=1\hspace{0.17em}$ if $g_{m_2}=g$; ${\delta }_{g_{m_2}g}=0$ if $g_{m_2}\ne g$). Inserting the results obtained in Equations (15) and (16) into Equation (7), yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial t_{m_2}}\right)}^{\left(1\right)}=-{\delta }_{i_j{i}_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{\psi }^{(1),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\\ -N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[{\psi }_{1,i}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g_{m_2}}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,i}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\right],\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.33em}{m}_2=1,\dots ,J_{\sigma t}.\end{array}$$

(17)

The contributions stemming from Equation (167) in [5] have the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial t_{m_2}}\right)}^{\left(2\right)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[{\theta }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial t_{m_2}}},\\ for\hspace{0.33em}j=1,\dots ,J_n\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma t}\hspace{0.17em},\end{array}$$

(18)

where the 2nd-level adjoint functions ${\theta }_{1,j}^{(2),g}$, and ${\theta }_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G$, are the solutions of the following 2nd-Level Adjoint Sensitivity System resulted from Equations (164)–(166) of [5]:

$$B^g\left({\alpha }^0\right){\theta }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right)={\displaystyle \sum _{g’=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\sigma }_{s,l,i_j}^{g^{\prime }\to g}{P}_l\left(\mathsf{\Omega }\right){\varphi }_l^{g^{\prime }}\left(r\right)},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(19)

$${\theta }_{1,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n<0;j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(20)

$$A^{(1),g}\left({\alpha }^0\right){\theta }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right)={\displaystyle \sum _{g’=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right){\sigma }_{s,l,i_j}^{g\to g^{\prime }}}{P}_l\left(\mathsf{\Omega }\right){\xi }_l^{(1),g^{\prime }}\left(r\right)},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G,$$

(21)

$${\theta }_{2,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n>0;\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G.$$

(22)

In Equations (19) and (21), ${\sigma }_{s,l,i_j}^{g^{\prime }\to g}$ denotes the $l^{th}$-order Legendre-expanded microscopic scattering cross section from energy group $g^{\prime }$ into energy group $g$ for isotope $i_j$, as defined in Equation (A7) in Appendix A. The $l^{th}$-moments ${\phi }_l^g\left(r\right)$ and ${\xi }_l^{(1),g}\left(r\right)$ are defined as follows:

$${\phi }_l^g\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{P}_l\left(\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right),$$

(23)

$${\xi }_l^{(1),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{P}_l\left(\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right).$$

(24)

Inserting Equation (15) into Equation (18) yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial t_{m_2}}\right)}^{\left(2\right)}=-N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[{\theta }_{1,j}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g_{m_2}}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\right],\\ for\hspace{0.33em}j=1,\dots ,J_n\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma t}\hspace{0.17em}.\end{array}$$

(25)

The contributions stemming from Equation (177) in [5] have the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial t_{m_2}}\right)}^{\left(3\right)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[u_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial t_{m_2}}},\\ for\hspace{0.33em}j=1,\dots ,J_n\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma t}\hspace{0.17em},\end{array}$$

(26)

where the 2nd-level adjoint functions $u_{1,j}^{(2),g}$, and $u_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G$, are the solutions of the following 2nd-Level Adjoint Sensitivity System obtained from Equations (183)–(185) of [5]:

$$B^g\left({\alpha }^0\right)u_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right)={\chi }^g{\displaystyle \sum _{g’=1}^G{\nu }_{i_j}^{g^{\prime }}{\sigma }_{f,i_j}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(27)

$$u_{1,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n<0;j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(28)

$$A^{(1),g}\left({\alpha }^0\right)u_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right)={\nu }_{i_j}^g{\sigma }_{f,i_j}^g{\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G,$$

(29)

$$u_{2,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n>0;\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G.$$

(30)

In Equations (27) and (29), the 0th-moments are defined as follows:

$${\phi }_0^g\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{\phi }^g\left(r,\mathsf{\Omega }\right),$$

(31)

$${\xi }_0^{(1),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{\psi }^{(1),g}\left(r,\mathsf{\Omega }\right).$$

(32)

Inserting Equation (15) into Equation (26) yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial t_{m_2}}\right)}^{\left(3\right)}=-N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[u_{1,j}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g_{m_2}}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\right],\\ for\hspace{0.33em}j=1,\dots ,J_n\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma t}\hspace{0.17em}.\end{array}$$

(33)

Finally, the contribution stemming from Equation (204) in [5] has the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial t_{m_2}}\right)}^{\left(4\right)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{g}_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right)\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial t_{m_2}}},\\ for\hspace{0.33em}j=1,\dots ,J_n\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma t}\hspace{0.17em},\end{array}$$

(34)

where the 2nd-level adjoint functions $g_{1,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G$, are the solutions of the following 2nd-Level Adjoint Sensitivity System resulted from Equations (200) and (202) of [5]:

$$B^g\left({\alpha }^0\right)g_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right)=\frac{Q_{SF,i_j}^g}{n_j},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(35)

$$g_{1,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n<0;j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G.$$

(36)

The $\left(\alpha ,n\right)$ source is zero for the PERP benchmark. Only the spontaneous fission source is present in the PERP benchmark, which implies that

$$Q^g=Q_{SF}^g={\displaystyle \sum _{m=1}^M{\displaystyle \sum _{k=1}^{N_f}{Q}_{SF,k}^g}},$$

(37)

where the spontaneous source rate density in group $g$ for isotope $k$ is defined as follows [10]:

$$Q_{SF,k}^g={\lambda }_k{N}_{k,m}{\chi }_{SF,k}^g={\lambda }_k{N}_{k,m}{F}_k^{SF}{\nu }_k^{SF}\left(\frac{2}{\sqrt{\pi a_k{}^3{b}_k}}{e}^{-\frac{a_k{b}_k}{4}}\right){\displaystyle {\int }_{E^{g+1}}^{E^g}dE\hspace{0.33em}{e}^{-E/a_k}\sinh \sqrt{b_{k}E}}.$$

(38)

In Equation (38), the quantity ${\lambda }_k$ denotes the decay constant for isotope $k$, while ${\chi }_{SF,k}^g$ includes the spontaneous fission branch ratio and the spontaneous fission neutron spectra, which are approximated by a Watt’s fission spectra using two evaluated parameters ($a_k$ and $b_k$).

Inserting Equation (15) into Equation (34) yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial t_{m_2}}\right)}^{\left(4\right)}=-N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[g_{1,j}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g_{m_2}}\left(r,\mathsf{\Omega }\right)\right],\\ for\hspace{0.33em}j=1,\dots ,J_n\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma t}\hspace{0.17em}.\end{array}$$

(39)

Combining the partial contributions obtained in Equations (17), (25), (33) and (39) yields the following expression:

$$\begin{array}{c}\frac{{\partial }^{2}L}{\partial n_j\partial t_{m_2}}=\sum _{i=1}^4{\left(\frac{{\partial }^{2}L}{\partial n_j\partial t_{m_2}}\right)}^{\left(i\right)}=-{\delta }_{i_j{i}_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\psi }_{1,i}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g_{m_2}}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,i}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\right]\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\theta }_{1,j}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g_{m_2}}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\right]\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[u_{1,j}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g_{m_2}}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\right]\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[g_{1,j}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g_{m_2}}\left(r,\mathsf{\Omega }\right)\right],for j=1,\dots ,J_n;m_2=1,\dots ,J_{\sigma t}.\end{array}$$

(40)

2.2. Alternative Path: Computing the Second-Order Sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_t\partial N$

As mentioned earlier, the mixed 2nd-order sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_t$ can be alternatively computed by using the symmetric expression ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_t\partial N$. The equations needed for deriving the expression for ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_t\partial N$ are obtained by particularizing Equations (158), (159), (160) and (162) in [5] to the PERP benchmark. The combined expression obtained by particularizing these equations takes on the following form:

$$\begin{array}{c}\frac{{\partial }^{2}L}{\partial t_j\partial n_{m_2}}=-\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\frac{{\partial }^2{{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial t_j\partial n_{m_2}}\\ -\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\psi }_{1,i}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,i}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial Q^g\left(\mathbf{q};r,\mathsf{\Omega }\right)}{\partial n_{m_2}},for j=1,\dots ,J_{\sigma t};m_2=1,\dots ,J_n,\end{array}$$

(41)

where the adjoint functions ${\psi }_{1,j}^{(2),g}$ and ${\psi }_{2,j}^{(2),g},j=1,\dots ,J_{\sigma t};\hspace{0.33em}g=1,\dots ,G$ are the solutions of the 2nd-LASS presented in Equations (32), (34), (39) and (40) of Part I [1], which are reproduced below for convenient reference:

$$B^g\left({\alpha }^0\right){\psi }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right)=-{\delta }_{g_{j}g}{N}_{i_j,m_j}{\phi }^g\left(r,\mathsf{\Omega }\right),\hspace{0.33em}\hspace{0.17em}j=1,\dots ,J_{\sigma t};\hspace{0.17em}\hspace{0.17em}\hspace{0.33em}g=1,\dots ,G,$$

(42)

$${\psi }_{1,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n<0;\hspace{0.33em}j=1,\dots ,J_{\sigma t};\hspace{0.33em}g=1,\dots ,G,$$

(43)

$$A^{(1),g}\left({\alpha }^0\right){\psi }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right)=-{\delta }_{g_{j}g}{N}_{i_j,m_j}{\psi }^{(1),g}\left(r,\mathsf{\Omega }\right),\hspace{0.33em}\hspace{0.17em}j=1,\dots ,J_{\sigma t};\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(44)

$${\psi }_{2,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n>0;\hspace{0.33em}j=1,\dots ,J_{\sigma t};\hspace{0.33em}g=1,\dots ,G.$$

(45)

The parameters $t_j$ and $n_{m_2}$ in Equation (41) correspond to the total cross sections and isotopic number densities, and are therefore denoted as $t_j\equiv {\sigma }_{t,i_j}^{g_j}$ and $n_{m_2}\equiv N_{i_{m_2},m_{m_2}}$, respectively. The following results will be used in subsequent derivations:

$$\frac{{\partial }^2{\Sigma }_t{}^g\left(t\right)}{\partial t_j\partial n_{m_2}}=\frac{{\partial }^2{\Sigma }_t{}^g\left(t\right)}{\partial {\sigma }_{t,i_j}^{g_j}\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial \left[\partial \left({\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}{\sigma }_{t,i}^g}}\right)/\partial {\sigma }_{t,i_j}^{g_j}\right]}{\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial \left[{\delta }_{g_{j}g}{N}_{i_j,m_j}\right]}{\partial N_{i_{m_2},m_{m_2}}}={\delta }_{i_j{i}_{m_2}}{\delta }_{g_{j}g},$$

(46)

$$\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial n_{m_2}}=\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial \left({\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}{\sigma }_{t,i}^g}}\right)}{\partial N_{i_{m_2},m_{m_2}}}={\sigma }_{t,i_{m_2}}^g,$$

(47)

$$\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(s;\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial n_{m_2}}=\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(s;\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial N_{i_{m_2},m_{m_2}}}={\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right){\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{P}_l\left({\mathsf{\Omega }}^{\prime }\cdot \mathsf{\Omega }\right)},$$

(48)

$$\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(s;{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial n_{m_2}}=\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(s;{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial N_{i_{m_2},m_{m_2}}}={\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right){\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{P}_l\left({\mathsf{\Omega }}^{\prime }\cdot \mathsf{\Omega }\right)},$$

(49)

$$\frac{\partial {\left(\nu {\Sigma }_f\right)}^g\left(f\right)}{\partial n_{m_2}}=\frac{\partial {\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\left(\nu {\sigma }_f\right)}_i^g}{\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial {\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\nu }_i^g{\sigma }_{f,i}^g}{\partial N_{i_{m_2},m_{m_2}}}={\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g,$$

(50)

$$\frac{\partial {\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(f\right)}{\partial n_{m_2}}=\frac{\partial {\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\left(\nu {\sigma }_f\right)}_i^{g^{\prime }}}{\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial {\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\nu }_i^{g^{\prime }}{\sigma }_{f,i}^{g^{\prime }}}{\partial N_{i_{m_2},m_{m_2}}}={\nu }_{i_{m_2}}^{g^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }},$$

(51)

$$\frac{\partial Q^g\left(q;r,\mathsf{\Omega }\right)}{\partial n_{m_2}}=\frac{\partial Q_{SF}^g}{\partial n_{m_2}}=\frac{\partial {\displaystyle \sum _{m=1}^M{\displaystyle \sum _{k=1}^{N_f}{\lambda }_k{N}_{k,m}{\chi }_{SF,k}^g}}}{\partial N_{i_{m_2},m_{m_2}}}={\lambda }_{m_2}{\chi }_{SF,i_{m_2}}^g=\frac{Q_{SF,i_{m_2}}^g}{N_{i_{m_2},m_{m_2}}}=\frac{Q_{SF,i_{m_2}}^g}{n_{m_2}}.$$

(52)

Inserting the results obtained in Equations (46)–(52) into Equation (41), and performing the respective angular integrations yields the following expression for Equation (41):

$$\begin{array}{c}\frac{{\partial }^{2}L}{\partial t_j\partial n_{m_2}}=-{\delta }_{i_j{i}_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g_j}\left(r,\mathsf{\Omega }\right){\phi }^{g_j}\left(r,\mathsf{\Omega }\right)\\ -\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\psi }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]{\sigma }_{t,i_{m_2}}^g\\ +\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{\xi }_{1,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{\left(1\right),g^{\prime }}\left(r\right)+\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{\xi }_{2,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\chi }^g{\xi }_{2,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\nu }_{i_{m_2}}^{g^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)+\sum _{g=1}^G{\int }_{V}dV{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{\xi }_{1,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)\\ +\frac{1}{n_{m_2}}\sum _{g=1}^G{\int }_{V}dV{\xi }_{2,j;0}^{\left(2\right),g}\left(r\right)Q_{SF,i_{m_2}}^g,for j=1,\dots ,J_{\sigma t};m_2=1,\dots ,J_n,\end{array}$$

(53)

where

$${\xi }_{1,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{P}_l\left(\mathsf{\Omega }\right){\psi }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right),$$

(54)

$${\xi }_{2,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{P}_l\left(\mathsf{\Omega }\right){\psi }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right),$$

(55)

$${\xi }_{1,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{\psi }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right),$$

(56)

$${\xi }_{2,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{\psi }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right).$$

(57)

2.3. Numerical Results for ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_t$

The second-order absolute sensitivities, ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_t$, of the leakage response with respect to the isotopic number densities and the total cross sections for all isotopes of the PERP benchmark have been computed using Equation (40), and have been independently verified by computing ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_t\partial N$ using Equation (53). Regarding computational requirements: computing ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_t$ requires 16 forward and adjoint PARTISN transport computations for obtaining the various 2nd-level adjoint functions ${\psi }_{1,j}^{(2),g},{\psi }_{2,j}^{(2),g},{\theta }_{1,j}^{(2),g},{\theta }_{2,j}^{(2),g},u_{1,j}^{(2),g},u_{2,j}^{(2),g}$ and $g_{1,j}^{(2),g},j=1,\dots ,J_n;g=1,\dots ,G$ needed in Equation (40). In contradistinction, computing ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_t\partial N$ using Equation (53) would require $J_{\sigma t}=G\times I=30\times 6=180$ forward and adjoint PARTISN computations for obtaining the adjoint functions ${\psi }_{1,j}^{(2),g}$ and ${\psi }_{2,j}^{(2),g},j=1,\dots ,J_{\sigma t};\hspace{0.33em}g=1,\dots ,G$ which are needed in Equation (53). It is thus evident that computing ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_t$ using Equation (40) is about 10 times more efficient than computing ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_t\partial N$ using Equation (53).

The matrix ${\partial }^{2}L/\partial n_j\partial t_{m_2},\hspace{0.33em}j=1,\dots ,J{}_n;\hspace{0.33em}{m}_2=1,\dots ,J{}_{\sigma t}$ has dimensions $J_n\times J_{\sigma t}\hspace{0.33em}(=6\times 180)$; corresponding to this matrix is the matrix denoted as $S^{(2)}\left(N_{i,m},{\sigma }_{t,k}^g\right)$ of 2nd-order relative sensitivities, which is defined as follows:

$$S^{(2)}\left(N_{i,m},{\sigma }_{t,k}^g\right)\triangleq \frac{{\partial }^{2}L}{\partial N_{i,m}\partial {\sigma }_{t,k}^g}\left(\frac{N_{i,m}{\sigma }_{t,k}^g}{L}\right),\hspace{0.17em}\hspace{0.33em}i,\hspace{0.17em}k=1,\dots ,6;\hspace{0.33em}m=1,2;\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30.$$

(58)

To facilitate the presentation and interpretation of the numerical results, the $J_n\times J_{\sigma t}\hspace{0.33em}(=6\times 180)$ matrix $S^{(2)}\left(N_{i,m},{\sigma }_{t,k}^g\right)$ has been partitioned into $J_n\times I\hspace{0.33em}=6\times 6$ submatrices, each of dimensions $1\times G=1\times 30$. The summary of the main features of each of these submatrices is presented in

Table 2. Summary presentation of the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{t,k}^g\right)\hspace{0.17em},\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}m=1,2;\hspace{0.33em}g=1,\dots ,30$

	$$\begin{gathered} \mathit{k}=1 \\ {(}^{239}\mathbf{Pu}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=2 \\ {(}^{240}\mathbf{Pu}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=3 \\ {(}^{69}\mathbf{Ga}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=4 \\ {(}^{71}\mathbf{Ga}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=5 \\ \left(\mathbf{C}\right) \end{gathered}$$	$$\begin{gathered} \mathit{k}=6 \\ {(}^1\mathbf{H}) \end{gathered}$$
$i=1$ (239Pu)	$S^{(2)}\left(N_{1,1},{\sigma }_{t,1}^g\right)$ 18 elements with absolute values >1.0	$S^{(2)}\left(N_{1,1},{\sigma }_{t,2}^g\right)$ 1 element with absolute value >1.0	$S^{(2)}\left(N_{1,1},{\sigma }_{t,3}^g\right)$ Min. value = −4.51 × 10−2 at g = 12	$S^{(2)}\left(N_{1,1},{\sigma }_{t,4}^g\right)$ Min. value = −3.06 × 10−2 at g = 12	$S^{(2)}\left(N_{1,1},{\sigma }_{t,5}^g\right)$ 12 elements with absolute value >1.0	$S^{(2)}\left(N_{1,1},{\sigma }_{t,6}^g\right)$ 22 elements with absolute values >1.0
$i=2$ (240Pu)	$S^{(2)}\left(N_{2,1},{\sigma }_{t,1}^g\right)$ 10 elements with absolute values >1.0	$S^{(2)}\left(N_{2,1},{\sigma }_{t,2}^g\right)$ Min. value = −2.05 × 10−1 at g = 12	$S^{(2)}\left(N_{2,1},{\sigma }_{t,3}^g\right)$ Min. value = −5.45 × 10−3 at g = 12	$S^{(2)}\left(N_{2,1},{\sigma }_{t,4}^g\right)$ Min. value = −3.70 × 10−3 at g = 12	$S^{(2)}\left(N_{2,1},{\sigma }_{t,5}^g\right)$ 1 element with absolute value >1.0	$S^{(2)}\left(N_{2,1},{\sigma }_{t,6}^g\right)$ 9 elements with absolute values >1.0
$i=3$ (69Ga)	$S^{(2)}\left(N_{3,1},{\sigma }_{t,1}^g\right)$ Min. value = −7.14 × 10−3 at g = 12	$S^{(2)}\left(N_{3,1},{\sigma }_{t,2}^g\right)$ Min. value = −4.52 × 10−4 at g = 12	$S^{(2)}\left(N_{3,1},{\sigma }_{t,3}^g\right)$ Min. value = −3.78 × 10−3 at g = 12	$S^{(2)}\left(N_{3,1},{\sigma }_{t,4}^g\right)$ Min. value = −1.40 × 10−5 at g = 13	$S^{(2)}\left(N_{3,1},{\sigma }_{t,5}^g\right)$ Min. value = −3.06 × 10−3 at g = 30	$S^{(2)}\left(N_{3,1},{\sigma }_{t,6}^g\right)$ Min. value = −3.66 × 10−2 at g = 30
$i=4$ (71Ga)	$S^{(2)}\left(N_{4,1},{\sigma }_{t,1}^g\right)$ Min. value = −4.51 × 10−3 at g = 12	$S^{(2)}\left(N_{4,1},{\sigma }_{t,2}^g\right)$ Min. value = −2.85 × 10−4 at g = 12	$S^{(2)}\left(N_{4,1},{\sigma }_{t,3}^g\right)$ Min. value = −1.32 × 10−5 at g = 13	$S^{(2)}\left(N_{4,1},{\sigma }_{t,4}^g\right)$ Min. value = −2.56 × 10−3 at g = 12	$S^{(2)}\left(N_{4,1},{\sigma }_{t,5}^g\right)$ Min. value = −1.95 × 10−3 at g = 30	$S^{(2)}\left(N_{4,1},{\sigma }_{t,6}^g\right)$ Min. value = −2.33 × 10−2 at g = 30
$i=5$ (C)	$S^{(2)}\left(N_{5,2},{\sigma }_{t,1}^g\right)$ 9 elements with absolute values >1.0	$S^{(2)}\left(N_{5,2},{\sigma }_{t,2}^g\right)$ Min. value = −1.14 × 10−1 at g = 12	$S^{(2)}\left(N_{5,2},{\sigma }_{t,3}^g\right)$ Min. value = −5.13 × 10−3 at g = 12	$S^{(2)}\left(N_{5,2},{\sigma }_{t,4}^g\right)$ Min. value = −3.51 × 10−3 at g = 22	$S^{(2)}\left(N_{5,2},{\sigma }_{t,5}^g\right)$ 1 element with absolute value >1.0	$S^{(2)}\left(N_{5,2},{\sigma }_{t,6}^g\right)$ 11 elements with absolute values >1.0
$i=6$ (1H)	$S^{(2)}\left(N_{6,2},{\sigma }_{t,1}^g\right)$ 11 elements with absolute values >1.0	$S^{(2)}\left(N_{6,2},{\sigma }_{t,2}^g\right)$ Min. value = −1.83 × 10−1 at g = 12	$S^{(2)}\left(N_{6,2},{\sigma }_{t,3}^g\right)$ Min. value = −8.21 × 10−3 at g = 12	$S^{(2)}\left(N_{6,2},{\sigma }_{t,4}^g\right)$ Min. value = −6.23 × 10−3 at g = 22	$S^{(2)}\left(N_{6,2},{\sigma }_{t,5}^g\right)$ 1 element with absolute value >1.0	$S^{(2)}\left(N_{6,2},{\sigma }_{t,6}^g\right)$ 19 elements with absolute values >1.0

in the following form: when a submatrix comprises elements with relative sensitivities with absolute values that are greater than 1.0, the total number of such elements are counted and 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 $\left(-1.0,1.0\right)$, only the element having the largest absolute value in the submatrix is listed in Table 2, together with the phase-space coordinates of that element. The sub-matrices in Table 2, which comprise components with absolute values greater than 1.0, will be discussed in detail in subsequent sub-sections of this section.

Almost all [i.e., 1072 out of $J_n\times J_{\sigma t}\hspace{0.33em}(=1080)$] of the elements of the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{t,k}^g\right)\hspace{0.17em},\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}m=1,2;\hspace{0.33em}g=1,\dots ,30$ have negative values. The remaining 8 elements have very small (of the order of 10⁻⁴ or less) positive values; they are all related to the isotopic number densities of isotopes ²⁴⁰Pu or ⁷¹Ga. The results shown in Table 2 indicate that, 125 elements (of the total of 1080 elements) have very large relative sensitivities, greater than 1.0. The majority (123 out of 125) of those large sensitivities involve the isotopic number densities of ²³⁹Pu or ¹H (namely, $N_{1,1}$ and $N_{6,2}$) and/or the microscopic total cross sections of isotopes ²³⁹Pu or ¹H (namely, ${\sigma }_{t,1}^g$ and ${\sigma }_{t,6}^g$). Of the sensitivities summarized in Table 2, the single largest relative value is $S^{(2)}\left(N_{1,1},{\sigma }_{t,6}^{30}\right)=-94.91$. The results in Table 2 also indicate that when the 2nd-order mixed relative sensitivities $S^{(2)}\left(N_{i,m},{\sigma }_{t,k}^g\right)$ involve the isotopic number densities of isotopes ⁶⁹Ga and ⁷¹Ga or the microscopic total cross sections of isotopes ⁶⁹Ga and ⁷¹Ga, their absolute values are all smaller than 1.0. Moreover, as shown in Table 2, the element with the most negative value in each of the submatrices involves the microscopic total cross sections for the 12th or the 30th energy group of the isotopes.

2.3.1. Second-Order Relative Sensitivities $S^{(2)}\left(N_{1,1},{\sigma }_{t,1}^g\right),g=1,\dots ,30$

Table 3. Second-Order Relative Sensitivities $S^{(2)}\left(N_{1,1},{\sigma }_{t,1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$.

g	Relative Sensitivities	g	Relative Sensitivities
1	−0.005	16	−10.430
2	−0.009	17	−4.783
3	−0.026	18	−2.885
4	−0.122	19	−2.242
5	−0.621	20	−1.883
6	−1.795	21	−1.631
7	−10.307	22	−1.168
8	−9.440	23	−0.934
9	−10.951	24	−0.597
10	−10.978	25	−0.687
11	−10.064	26	−0.732
12	−17.172	27	−0.219
13	−15.138	28	−0.044
14	−12.627	29	−0.392
15	−9.217	30	−5.241

shows the results obtained for the 2nd-order mixed relative sensitivity of the leakage response with respect to the isotopic number density and the microscopic total cross sections of isotope 1 (²³⁹Pu), denoted as $S^{(2)}\left(N_{1,1},{\sigma }_{t,1}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=1,m=1}\partial {\sigma }_{t,k=1}^g\right)\left(N_{1,1}{\sigma }_{t,1}^g/L\right),\hspace{0.33em}g=1,\dots ,30$. The 18 elements that have values greater than 1.0, shown bold in this table, involve the total cross sections of isotope ²³⁹Pu for the energy groups $g=6,\dots ,22$ and $g=30$. The largest negative value in this submatrix is attained by the relative 2nd-order mixed sensitivity $S^{(2)}\left(N_{1,1},{\sigma }_{t,1}^{g=12}\right)=-17.172$, which involves the isotopic number density and the 12th energy group of the total cross sections of isotope ²³⁹Pu.

2.3.2. Second-Order Relative Sensitivities $S^{(2)}\left(N_{1,1},{\sigma }_{t,2}^g\right),g=1,\dots ,30$

The matrix $S^{(2)}\left(N_{1,1},{\sigma }_{t,2}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=1,m=1}\partial {\sigma }_{t,k=2}^g\right)\left(N_{1,1}{\sigma }_{t,2}^g/L\right),\hspace{0.33em}g=1,\dots ,30$, comprising the 2nd-order sensitivities of the leakage response with respect to the isotopic number density of isotope 1 (²³⁹Pu) and the microscopic total cross sections of isotope 2 (²⁴⁰Pu), contains a single large element that has an absolute value greater than 1.0, which is $S^{(2)}\left(N_{1,1},{\sigma }_{t,2}^{g=12}\right)=-1.005$.

2.3.3. Second-Order Relative Sensitivities $S^{(2)}\left(N_{1,1},{\sigma }_{t,5}^g\right),g=1,\dots ,30$

The submatrix $S^{(2)}\left(N_{1,1},{\sigma }_{t,5}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=1,m=1}\partial {\sigma }_{t,k=5}^g\right)\left(N_{1,1}{\sigma }_{t,5}^g/L\right),\hspace{0.33em}g=1,\dots ,30$, comprising the 2^nd-order relative sensitivities of the leakage response with respect to the isotopic number density of isotope 1 (²³⁹Pu) and the microscopic total cross sections of isotope 5 (C), is presented in

Table 4. Second-Order Relative Sensitivities $S^{(2)}\left(N_{1,1},{\sigma }_{t,5}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$.

g	Relative Sensitivities	g	Relative Sensitivities
1	−4.720 × 10−6	16	−2.034
2	−2.276 × 10−5	17	−1.657
3	−1.017 × 10−4	18	−1.441
4	−6.664 × 10−4	19	−1.315
5	−7.945 × 10−3	20	−1.219
6	−0.019	21	−1.136
7	−0.218	22	−1.040
8	−0.328	23	−0.962
9	−0.332	24	−0.870
10	−0.387	25	−0.824
11	−0.439	26	−0.758
12	−1.118	27	−0.674
13	−1.363	28	−0.623
14	−1.402	29	−0.605
15	−1.245	30	−7.952

. This submatrix includes 12 elements that have absolute values greater than 1.0, noted in bold, involving the total cross sections of isotope C for the energy groups $g=12,\dots ,22$ and $g=30$. The largest negative value is displayed by the 2nd-order relative sensitivity of the leakage response with respect to the isotopic number density for ²³⁹Pu and the 30th energy group of the total cross section for C, namely, $S^{(2)}\left(N_{1,1},{\sigma }_{t,5}^{g=30}\right)=-7.952$.

2.3.4. Second-Order Relative Sensitivities $S^{(2)}\left(N_{1,1},{\sigma }_{t,6}^g\right),g=1,\dots ,30$

Table 5. Second-Order Relative Sensitivities $S^{(2)}\left(N_{1,1},{\sigma }_{t,6}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$.

g	Relative Sensitivities	g	Relative Sensitivities
1	−4.001 × 10−6	16	−11.421
2	−2.325 × 10−5	17	−11.674
3	−1.087 × 10−4	18	−11.427
4	−8.747 × 10−4	19	−10.979
5	−7.585 × 10−3	20	−10.380
6	−0.040	21	−9.744
7	−0.447	22	−8.947
8	−0.646	23	−8.287
9	−1.001	24	−7.499
10	−1.250	25	−7.115
11	−1.478	26	−6.559
12	−3.696	27	−5.860
13	−4.693	28	−5.479
14	−5.379	29	−5.490
15	−5.577	30	−94.909

lists the values of the components of the submatrix $S^{(2)}\left(N_{1,1},{\sigma }_{t,6}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=1,m=1}\partial {\sigma }_{t,k=6}^g\right)\left(N_{1,1}{\sigma }_{t,6}^g/L\right),\hspace{0.33em}g=1,\dots ,30$, comprising the 2nd-order mixed relative sensitivities of the leakage response with respect to the isotopic number density of isotope 1 (²³⁹Pu) and the microscopic total cross sections of isotope 6 (¹H). This submatrix includes 22 elements that have absolute values greater than 1.0, shown in bold; the large sensitivities involve the total cross sections of isotope ¹H for energy groups $g=9,\dots ,30$. The largest negative value is for the 2nd-order relative sensitivity of the leakage response with respect to isotopic number density of isotope ²³⁹Pu and the 30th energy group of the total cross section of isotope ¹H, i.e., $S^{(2)}\left(N_{1,1},{\sigma }_{t,6}^{g=30}\right)=-94.909$.

2.3.5. Second-Order Relative Sensitivities $S^{(2)}\left(N_{2,1},{\sigma }_{t,1}^g\right),g=1,\dots ,30$

Table 6. Second-Order Relative Sensitivities $S^{(2)}\left(N_{2,1},{\sigma }_{t,1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$.

g	Relative Sensitivities	g	Relative Sensitivities
1	−4.881 × 10−4	16	−1.136
2	−9.701 × 10−4	17	−0.525
3	−2.796 × 10−3	18	−0.320
4	−0.013	19	−0.254
5	−0.068	20	−0.216
6	−0.198	21	−0.190
7	−1.148	22	−0.137
8	−1.052	23	−0.106
9	−1.221	24	−0.067
10	−1.223	25	−0.084
11	−1.121	26	−0.090
12	−1.914	27	−0.005
13	−1.676	28	0.0002
14	−1.386	29	−0.049
15	−1.007	30	−0.639

shows the results obtained for the matrix $S^{(2)}\left(N_{2,1},{\sigma }_{t,1}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=2,m=1}\partial {\sigma }_{t,k=1}^g\right)\left(N_{2,1}{\sigma }_{t,k=1}^g/L\right),\hspace{0.33em}g=1,\dots ,30$, comprising the 2nd-order mixed relative sensitivity of the leakage response with respect to the isotopic number density of ²⁴⁰Pu and the total cross sections of ²³⁹Pu. As highlighted in bold in this table, 10 elements of this submatrix have relative sensitivities with absolute values greater than 1.0. These large mixed relative sensitivities involve the total cross sections of isotope ²³⁹Pu for energy groups $g=7,\dots ,16$, respectively. The largest negative value in this submatrix is $S^{(2)}\left(N_{2,1},{\sigma }_{t,1}^{g=12}\right)=-1.914$, also occurring in the 12th energy group of the total cross sections.

2.3.6. Second-Order Relative Sensitivities $S^{(2)}\left(N_{2,1},{\sigma }_{t,5}^g\right),g=1,\dots ,30$

The matrix $S^{(2)}\left(N_{2,1},{\sigma }_{t,5}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=2,m=1}\partial {\sigma }_{t,k=5}^g\right)\left(N_{2,1}{\sigma }_{t,5}^g/L\right),\hspace{0.33em}g=1,\dots ,30$, comprising the 2nd-order sensitivities of the leakage response with respect to the isotopic number density of isotope 2 (²⁴⁰Pu) and the microscopic total cross sections of isotope 5 (C), has only one large element with absolute value greater than 1.0, namely, $S^{(2)}\left(N_{2,1},{\sigma }_{t,5}^{g=30}\right)=-1.067$, which occurs for the 30th energy group of the total cross sections of isotope C.

2.3.7. Second-Order Relative Sensitivities $S^{(2)}\left(N_{2,1},{\sigma }_{t,6}^g\right),g=1,\dots ,30$

Table 7. Second-Order Relative Sensitivities $S^{(2)}\left(N_{2,1},{\sigma }_{t,6}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$.

g	Relative Sensitivities	g	Relative Sensitivities
1	−4.446 × 10−6	16	−1.553
2	−1.222 × 10−5	17	−1.568
3	−4.406 × 10−5	18	−1.526
4	−2.817 × 10−4	19	−1.463
5	−1.977 × 10−3	20	−1.381
6	−8.680 × 10−3	21	−1.295
7	−0.083	22	−1.189
8	−0.107	23	−1.099
9	−0.162	24	−0.996
10	−0.201	25	−0.944
11	−0.234	26	−0.868
12	−0.563	27	−0.763
13	−0.685	28	−0.730
14	−0.762	29	−0.740
15	−0.773	30	−12.741

summaries the 2nd-order relative sensitivities in submatrix $S^{(2)}\left(N_{2,1},{\sigma }_{t,6}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=2,m=1}\partial {\sigma }_{t,k=6}^g\right)\left(N_{2,1}{\sigma }_{t,6}^g/L\right),\hspace{0.33em}g=1,\dots ,30$, of the leakage response with respect to the isotopic number density of isotope 2 (²⁴⁰Pu) and the microscopic total cross sections of isotope 6 (¹H). As shown in bold characters in this table, 9 elements in this submatrix have absolute values greater than 1.0. These 9 elements are related to the energy groups $g=16,\dots ,23$ and $g=30$ of the total cross sections of isotope ¹H, respectively. The largest 2nd-order mixed relative sensitivity attained in this submatrix involves the isotopic number density for ²⁴⁰Pu and the 30th energy group of the total cross section for ¹H, i.e., $S^{(2)}\left(N_{2,1},{\sigma }_{t,6}^{g=30}\right)=-12.741$.

2.3.8. Second-Order Relative Sensitivities $S^{(2)}\left(N_{5,2},{\sigma }_{t,1}^g\right),g=1,\dots ,30$

Table 8. Second-Order Relative Sensitivities $S^{(2)}\left(N_{5,2},{\sigma }_{t,1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$.

g	Relative Sensitivities	g	Relative Sensitivities
1	−4.471 × 10−4	16	−1.177
2	−8.863 × 10−4	17	−0.593
3	−2.548 × 10−3	18	−0.385
4	−0.012	19	−0.306
5	−0.061	20	−0.257
6	−0.178	21	−0.223
7	−1.044	22	−0.161
8	−0.982	23	−0.129
9	−1.146	24	−0.082
10	−1.143	25	−0.096
11	−1.047	26	−0.100
12	−1.803	27	−0.027
13	−1.613	28	−0.005
14	−1.360	29	−0.055
15	−1.004	30	−0.761

presents the results for the submatrix $S^{(2)}\left(N_{5,2},{\sigma }_{t,1}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=5,m=2}\partial {\sigma }_{t,k=1}^g\right)\left(N_{5,2}{\sigma }_{t,1}^g/L\right),\hspace{0.33em}g=1,\dots ,30$, which comprises the 2nd-order relative sensitivities of the leakage response with respect to the isotopic number density of isotope 5 (C) and the microscopic total cross sections of isotope 1 (²³⁹Pu). This submatrix contains 9 elements that have absolute values greater than 1.0, as shown in bold in this table. These 9 elements are related to the total cross sections of isotope C for the energy groups $g=7$ and $g=9,\dots ,16$, respectively. The most negative value is $S^{(2)}\left(N_{5,2},{\sigma }_{t,1}^{g=12}\right)=-1.803$ for the 2nd-order relative sensitivity with respect to isotopic number density of isotope C and the 12th energy group of the total cross section of isotope ²³⁹Pu.

2.3.9. Second-Order Relative Sensitivities $S^{(2)}\left(N_{5,2},{\sigma }_{t,5}^g\right),g=1,\dots ,30$

The submatrix $S^{(2)}\left(N_{5,2},{\sigma }_{t,5}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=5,m=2}\partial {\sigma }_{t,k=5}^g\right)\left(N_{5,2}{\sigma }_{t,5}^g/L\right),\hspace{0.33em}g=1,\dots ,30$, comprising the 2nd-order mixed relative sensitivities of the leakage response with respect to the isotopic number density of isotope 5 (C) and the total cross sections of isotope 5 (C), has a single large element that has an absolute value greater than 1.0, namely $S^{(2)}\left(N_{5,2},{\sigma }_{t,5}^{g=30}\right)=-2.016$.

2.3.10. Second-Order Relative Sensitivities $S^{(2)}\left(N_{5,2},{\sigma }_{t,6}^g\right),g=1,\dots ,30$

Table 9. Second-Order Relative Sensitivities $S^{(2)}\left(N_{5,2},{\sigma }_{t,6}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$.

g	Relative Sensitivities	g	Relative Sensitivities
1	−3.183 × 10−6	16	−1.665
2	−8.920 × 10−6	17	−1.676
3	−3.198 × 10−5	18	−1.626
4	−2.007 × 10−4	19	−1.557
5	−1.438 × 10−3	20	−1.471
6	−6.575 × 10−3	21	−1.383
7	−0.070	22	−1.273
8	−0.102	23	−1.183
9	−0.161	24	−1.076
10	−0.200	25	−1.024
11	−0.232	26	−0.947
12	−0.567	27	−0.850
13	−0.715	28	−0.801
14	−0.812	29	−0.804
15	−0.827	30	−14.695

lists the values for the elements of the submatrix $S^{(2)}\left(N_{5,2},{\sigma }_{t,6}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=5,m=2}\partial {\sigma }_{t,k=6}^g\right)\left(N_{5,2}{\sigma }_{t,6}^g/L\right),\hspace{0.33em}g=1,\dots ,30$, comprising the 2nd-order mixed relative sensitivities of the leakage response with respect to the isotopic number density of isotope 5 (C) and the microscopic total cross sections of isotope 6 (¹H). This submatrix includes 11 elements, highlighted in bold, which have absolute values greater than 1.0. These 11 elements involve the total cross sections of isotope ¹H for energy groups $g=16,\dots ,25$, and $g=30$, respectively. The largest negative value in this submatrix is $S^{(2)}\left(N_{5,2},{\sigma }_{t,6}^{g=30}\right)=-14.695$.

2.3.11. Second-Order Relative Sensitivities $S^{(2)}\left(N_{6,2},{\sigma }_{t,1}^g\right),g=1,\dots ,30$

Table 10. Second-Order Relative Sensitivities $S^{(2)}\left(N_{6,2},{\sigma }_{t,1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$.

g	Relative Sensitivities	g	Relative Sensitivities
1	−7.340 × 10−4	16	−1.787
2	−1.456 × 10−3	17	−0.892
3	−4.186 × 10−3	18	−0.597
4	−0.020	19	−0.496
5	−0.101	20	−0.431
6	−0.293	21	−0.386
7	−1.697	22	−0.286
8	−1.554	23	−0.235
9	−1.808	24	−0.153
10	−1.821	25	−0.183
11	−1.680	26	−0.196
12	−2.884	27	−0.055
13	−2.542	28	−0.011
14	−2.114	29	−0.114
15	−1.546	30	−1.933

presents the results obtained for the submatrix $S^{(2)}\left(N_{6,2},{\sigma }_{t,1}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=6,m=2}\partial {\sigma }_{t,k=1}^g\right)\left(N_{6,2}{\sigma }_{t,k=1}^g/L\right),\hspace{0.33em}g=1,\dots ,30$, comprising the 2nd-order mixed relative sensitivities of the leakage response with respect to the isotopic number density of isotope 6 (¹H) and to the microscopic total cross sections of isotope 1 (²³⁹Pu). As highlighted in bold in this table, 11 elements have relative sensitivities with absolute values greater than 1.0. These large 2nd-order mixed relative sensitivities pertain to the total cross sections of isotope ²³⁹Pu for energy groups $g=7,\dots ,16$ and $g=30$, respectively. The largest negative value in this submatrix is attained by the relative 2nd-order mixed sensitivity $S^{(2)}\left(N_{6,2},{\sigma }_{t,1}^{g=12}\right)=-2.884$.

2.3.12. Second-Order Relative Sensitivities $S^{(2)}\left(N_{6,2},{\sigma }_{t,5}^g\right),g=1,\dots ,30$

The matrix $S^{(2)}\left(N_{6,2},{\sigma }_{t,5}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=6,m=2}\partial {\sigma }_{t,k=5}^g\right)\left(N_{6,2}{\sigma }_{t,5}^g/L\right),\hspace{0.33em}g=1,\dots ,30$, comprising the 2^nd-order relative sensitivities of the leakage response with respect to the isotopic number density of isotope 6 (¹H) and the microscopic total cross sections of isotope 5 (C), contains a single large element that has an absolute value greater than 1.0, which is $S^{(2)}\left(N_{6,2},{\sigma }_{t,5}^{g=30}\right)=-3.186$.

2.3.13. Second-Order Relative Sensitivities $S^{(2)}\left(N_{6,2},{\sigma }_{t,6}^g\right),g=1,\dots ,30$

Table 11. Second-Order Relative Sensitivities $S^{(2)}\left(N_{6,2},{\sigma }_{t,6}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$.

g	Relative Sensitivities	g	Relative Sensitivities
1	−9.518 × 10−6	16	−3.558
2	−2.606 × 10−5	17	−3.679
3	−9.353 × 10−5	18	−3.656
4	−5.963 × 10−4	19	−3.574
5	−4.196 × 10−4	20	−3.440
6	−0.018	21	−3.296
7	−0.169	22	−3.102
8	−0.220	23	−2.937
9	−0.332	24	−2.730
10	−0.415	25	−2.630
11	−0.489	26	−2.476
12	−1.190	27	−2.272
13	−1.465	28	−2.170
14	−1.659	29	−2.176
15	−1.724	30	−47.398

lists the values obtained for the 2nd-order relative sensitivities of the leakage response with respect to the isotopic number density of isotope 6 (¹H) and the microscopic total cross sections of ¹H, which are components of $S^{(2)}\left(N_{6,2},{\sigma }_{t,6}^g\right)\triangleq \left({\partial }^{2}L/\partial N_{i=6,m=2}\partial {\sigma }_{t,k=6}^g\right)\left(N_{6,2}{\sigma }_{t,6}^g/L\right),\hspace{0.33em}g=1,\dots ,30$. In this submatrix, 19 elements have relative sensitivities with absolute values greater than 1.0, all involving the total cross sections of isotope ¹H for energy groups $g=12,\dots ,30$. The largest negative value is attained by the 2nd-order mixed relative sensitivity $S^{(2)}\left(N_{6,2},{\sigma }_{t,6}^{g=30}\right)=-47.398$, occurring in the 30th energy group of the total cross section of isotope ¹H.

3. Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Parameters Underlying the Benchmark’s Isotopic Number Densities and Scattering Cross Sections

This Section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_s$ of the leakage response with respect to the isotopic number densities and group-averaged scattering microscopic cross sections for all isotopes of the PERP benchmark. These 2nd-order mixed sensitivities can also be computed by using the symmetric expression ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_s\partial N$. These two distinct paths for computing the 2nd-order sensitivities with respect to the isotopic number densities and group-averaged scattering microscopic cross sections will be presented in Section 3.1 and, respectively, Section 3.2. As shown in Section 3.3, below, the pathway for computing ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_s$ turns out to be about 450 times more efficient than the pathway for computing ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_s\partial N$.

3.1. Computing the Second-Order Sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_s$

The equations needed for deriving the expressions of the 2nd-order sensitivities ${\partial }^{2}L/\partial n_j\partial s_{m_2}\hspace{0.33em},j=1,\dots ,J_n;m_2=1,\dots ,J_{\sigma s}$ will differ from each other depending on whether the parameter $s_{m_2}$ corresponds to the 0th-order $\left(l=0\right)$ scattering cross sections or to the higher-order $\left(l\ge 1\right)$ scattering cross sections. This is because, as shown in Equation (A3) of Appendix A, the zeroth-order scattering cross sections contribute to the total cross sections while the higher-order scattering cross sections do not. Therefore, the zeroth-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 zeroth-order $\left(l=0\right)$ scattering cross section comprised in ${\sigma }_s$ is denoted as $J_{\sigma s,l=0}$, where $J_{\sigma s,l=0}=G\times G\times I$. The total number of higher order (i.e., $l\ge 1$) scattering cross sections comprised in ${\sigma }_s$ is denoted as $J_{\sigma s,l\ge 1}$, where $J_{\sigma s,l\ge 1}=G\times G\times I\times ISCT$, with $J_{\sigma s,l=0}+J_{\sigma s,l\ge 1}=J_{\sigma s}$. There are two distinct cases, as follows:

Case 1: ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)},\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma s,l=0},$ where the quantities $n_j$ refer to the isotopic number densities, while the quantities $s_{m_2}$ refer to the parameters underlying the 0th-order scattering microscopic cross sections; and

Case 2: ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l\ge 1}\right)},\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,{\sigma }_{s,l\ge 1},$, where the quantities $n_j$ refer to the isotopic number densities, and the quantities $s_{m_2}$ refer to the parameters underlying the $l^{th}$-order ($l\ge 1$) scattering microscopic cross sections.

3.1.1. Second-Order Sensitivities ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)},j=1,\dots ,J_n;m_2=1,\dots ,J_{\sigma s,l=0}$

The equations needed for deriving the expression of the 2nd-order mixed sensitivities ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}$ are obtained by particularizing Equations (158), (159), (167), (168), (177), (178), (204) and (205) in [5] to the PERP benchmark. Specifically, using Equation (158) in [5] to the PERP benchmark yields the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{(1)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{\psi }^{(1),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\frac{{\partial }^2{\Sigma }_t{}^g\left(t\right)}{\partial n_j\partial s_{m_2}}}\\ \hspace{0.17em}-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[{\psi }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial s_{m_2}}},\\ for\hspace{0.33em}j=1,\dots ,J_n\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma s,l=0}\hspace{0.17em},\end{array}$$

(59)

where the adjoint functions ${\psi }_{1,j}^{(2),g}$ and ${\psi }_{2,j}^{(2),g},j=1,\dots ,J_{\sigma f};\hspace{0.33em}g=1,\dots ,G$ are the solutions of the 2nd-LASS presented previously in Equations (11)–(14). In Equation (59), the parameters $n_j$ and $s_{m_2}$ correspond to the isotopic number densities and microscopic total cross sections, respectively, and are therefore denoted as $n_j\equiv N_{i_j,m_j}$ and $s_{m_2}\equiv {\sigma }_{s,l_{m_2}=0,i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}$, where the subscripts $i_{m_2},\hspace{0.33em}{l}_{m_2},\hspace{0.33em}{g^{\prime }}_{m_2}$ 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

$$\begin{array}{c}\frac{{\partial }^2{{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial n_j\partial s_{m_2}}=\frac{{\partial }^2{{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial N_{i_j,m_j}\partial {\sigma }_{s,l_{m_2}=0,i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}=\frac{\partial \left\{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{N}_{i,m}\left({\sigma }_{f,i}^g+{\sigma }_{c,i}^g+{\displaystyle \sum _{g^{\prime }=1}^G}{\sigma }_{s,l=0,i}^{g\to g^{\prime }}\right)\right]/\partial N_{i_j,m_j}\right\}}{\partial {\sigma }_{s,l_{m_2}=0,i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}\hfill \\ =\frac{\partial \left\{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{\displaystyle \sum _{g^{\prime }=1}^G}{N}_{i,m}{\sigma }_{s,l=0,i}^{g\to g^{\prime }}\right]/\partial N_{i_j,m_j}\right\}}{\partial {\sigma }_{s,l_{m_2}=0,i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}=\frac{\partial \left\{{\displaystyle \sum _{g^{\prime }=1}^G}{\sigma }_{s,l=0,i_j}^{g\to g^{\prime }}\right\}}{\partial {\sigma }_{s,l_{m_2}=0,i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}={\delta }_{i_j{i}_{m_2}}{\delta }_{{g^{\prime }}_{m_2}g},\hfill \end{array}$$

(60)

$$\begin{array}{c}\frac{\partial {\Sigma }_t^g\left(\mathbf{t}\right)}{\partial s_{m_2}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{N}_{i,m}{\sigma }_{t,i}^g\left(\mathbf{t}\right)\right]}{\partial {\sigma }_{s,l_{m_2}=0,i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}=\frac{\partial \left\{{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{N}_{i,m}\left[{\sigma }_{f,i}^g\left(\mathbf{f}\right)+{\sigma }_{c,i}^g\left(\mathbf{c}\right)+{\displaystyle \sum _{g^{\prime }=1}^G}{\sigma }_{s,l=0,i}^{g\to g^{\prime }}\left(\mathbf{s}\right)\right]\right\}}{\partial {\sigma }_{s,l_{m_2}=0,i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}\hfill \\ =\frac{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{\displaystyle \sum _{g^{\prime }=1}^G}{N}_{i,m}{\sigma }_{s,l=0,i}^{g\to g^{\prime }}\left(\mathbf{s}\right)\right]}{\partial {\sigma }_{s,l_{m_2}=0,i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}={\delta }_{{g^{\prime }}_{m_2}g}{N}_{i_{m_2},m_{m_2}},\hfill \end{array}$$

(61)

and inserting the results obtained in Equations (60) and (61) into Equation (59) yields the following expression for Equation (59):

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{\left(1\right)}=-{\delta }_{i_j{i}_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\psi }_{1,j}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,j}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)\right],\\ for j=1,\dots ,J_n,m_2=1,\dots ,J_{\sigma s,l=0}.\end{array}$$

(62)

Additional contributions stem from Equation (159) in [5], which takes on the following particular form:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{\left(2\right)}=\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial s_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial s_{m_2}},\\ for j=1,\dots ,J_n;m_2=1,\dots ,J_{\sigma s,l=0}.\end{array}$$

(63)

Noting that

$$\begin{array}{c}\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial s_{m_2}}=\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial {\sigma }_{s,l_{m_2},i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{N}_{i,m}{\sigma }_{s,i}^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })\right]}{\partial {\sigma }_{s,l_{m_2},i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}\hfill \\ =\frac{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{\displaystyle \sum _{l=0}^{ISCT}}{N}_{i,m}\left(2l+1\right){\sigma }_{s,l,i}^{g\to g^{\prime }}{P}_l\left({\mathsf{\Omega }}^{\prime }·\mathsf{\Omega }\right)\right]}{\partial {\sigma }_{s,l_{m_2},i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}={\delta }_{{g^{\prime }}_{m_2}g}{\delta }_{g_{m_2}{g}^{\prime }}{N}_{i_{m_2},m_{m_2}}\left(2l_{m_2}+1\right)P_{l_{m_2}}\left({\mathsf{\Omega }}^{\prime }·\mathsf{\Omega }\right),\hfill \end{array}$$

(64)

$$\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(s;{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial s_{m_2}}=\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(s;{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial {\sigma }_{s,l_{m_2},i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}={\delta }_{g_{m_2}g}{\delta }_{{g^{\prime }}_{m_2}{g}^{\prime }}{N}_{i_{m_2},m_{m_2}}\left(2l_{m_2}+1\right)P_{l_{m_2}}\left({\mathsf{\Omega }}^{\prime }\cdot \mathsf{\Omega }\right),$$

(65)

inserting the results obtained in Equations (64) and (65) into Equation (63), using the addition theorem for spherical harmonics in one-dimensional geometry, performing the respective angular integrations, and setting $l_{m_2}=0$, yields the following expression for Equation (63):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{(2)}=N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}\left[{\xi }_0^{(1),g_{m_2}}\left(r\right){\xi }_{1,j;0}^{(2),{g^{\prime }}_{m_2}}\left(r\right)+{\phi }_0^{{g^{\prime }}_{m_2}}\left(r\right){\xi }_{2,j;0}^{(2),g_{m_2}}\left(r\right)\right],\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.33em}{m}_2=1,\dots ,J_{\sigma s,l=0}.\end{array}$$

(66)

Using Equation (167) in [5] in conjunction with the relations $\frac{{\partial }^{2}L}{\partial s_j\partial t_{m_2}}\frac{\partial s_j}{\partial n_j}\frac{\partial t_{m_2}}{\partial s_{m_2}}=\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}$ and $\frac{\partial {\Sigma }_t{}^g}{\partial t_{m_2}}\frac{\partial t_{m_2}}{\partial s_{m_2}}=\frac{\partial {\Sigma }_t{}^g}{\partial s_{m_2}}$ yields the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{(3)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[{\theta }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial s_{m_2}}},\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.33em}{m}_2=1,\dots ,J_{\sigma s,l=0},\end{array}$$

(67)

where the 2nd-level adjoint functions ${\theta }_{1,j}^{(2),g}$, and ${\theta }_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G$, are the solutions of the 2nd-Level Adjoint Sensitivity System presented previously in Equations (19)–(22). Inserting the results obtained in Equation (61) into Equation (67), yields the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{(3)}=-N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[{\theta }_{1,j}^{(2),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{(1),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{(2),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)\right],\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.33em}{m}_2=1,\dots ,J_{\sigma s,l=0}.\end{array}$$

(68)

The contributions stemming from Equation (168) in [5] to the PERP benchmark are computed as follows:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{\left(4\right)}=\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{{\partial }^2{\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial n_j\partial s_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial s_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial s_{m_2}},\\ for j=1,\dots ,J_n;m_2=1,\dots ,J_{\sigma s,l=0}.\end{array}$$

(69)

Noting that

$$\begin{array}{c}\frac{{\partial }^2{\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial n_j\partial s_{m_2}}=\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial N_{i_j,m_j}\partial {\sigma }_{s,l_{m_2},i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}=\frac{\partial \left\{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{N}_{i,m}{\sigma }_{s,i}^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })\right]/\partial N_{i_j,m_j}\right\}}{\partial {\sigma }_{s,l_{m_2},i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}\hfill \\ =\frac{\partial \left\{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{\displaystyle \sum _{l=0}^{ISCT}}{N}_{i,m}\left(2l+1\right){\sigma }_{s,l,i}^{g^{\prime }\to g}{P}_l\left({\mathsf{\Omega }}^{\prime }·\mathsf{\Omega }\right)\right]/\partial N_{i_j,m_j}\right\}}{\partial {\sigma }_{s,l_{m_2},i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}=\frac{\partial \left\{{\displaystyle \sum _{l=0}^{ISCT}}\left(2l+1\right){\sigma }_{s,l,i_j}^{g^{\prime }\to g}{P}_l\left({\mathsf{\Omega }}^{\prime }·\mathsf{\Omega }\right)\right\}}{\partial {\sigma }_{s,l_{m_2},i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}}\\ ={\delta }_{i_j{i}_{m_2}}{\delta }_{{g^{\prime }}_{m_2}{g}^{\prime }}{\delta }_{g_{m_2}g}\left(2l_{m_2}+1\right)P_{l_{m_2}}\left({\mathsf{\Omega }}^{\prime }·\mathsf{\Omega }\right),\end{array}$$

(70)

inserting Equations (64), (65) and (70) into Equation (69), using the addition theorem for spherical harmonics in one-dimensional geometry, performing the respective angular integrations, and setting $l_{m_2}=0$ into the resulting expression yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{(4)}={\delta }_{i_j{i}_{m_2}}{\xi }_0^{(1),g_{m_2}}\left(r\right){\phi }_0^{{g^{\prime }}_{m_2}}\left(r\right)\\ +N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}\left[{\xi }_0^{(1),g_{m_2}}\left(r\right){\mathrm{\Theta }}_{1,j;0}^{(2),{g^{\prime }}_{m_2}}\left(r\right)+{\phi }_0^{{g^{\prime }}_{m_2}}\left(r\right){\mathrm{\Theta }}_{2,j;0}^{(2),g_{m_2}}\left(r\right)\right],for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.33em}{m}_2=1,\dots ,J_{\sigma s,l=0},\end{array}$$

(71)

where

$${\mathrm{\Theta }}_{1,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{\theta }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right),$$

(72)

$${\mathrm{\Theta }}_{2,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{\theta }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right).$$

(73)

Contributions from the fission cross sections are computed by particularizing Equation (177) in [5], in conjunction with the relations $\frac{{\partial }^{2}L}{\partial f_j\partial t_{m_2}}\frac{\partial f_j}{\partial n_j}\frac{\partial t_{m_2}}{\partial s_{m_2}}=\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}$ and $\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial t_{m_2}}\frac{\partial t_{m_2}}{\partial s_{m_2}}=\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial s_{m_2}}$, to obtain:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{(5)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[u_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial s_{m_2}}},\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma s,l=0},\end{array}$$

(74)

where the 2nd-level adjoint functions $u_{1,j}^{(2),g}$, and $u_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G$, are the solutions of the 2nd-Level Adjoint Sensitivity System presented previously in Equations (27)–(30). Replacing the result obtained in Equation (61) into Equation (74) yields the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{(5)}=-N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[u_{1,j}^{(2),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{(1),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{(2),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)\right],\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma s,l=0}.\end{array}$$

(75)

Contributions stemming from particularizing Equation (178) in [5] to the PERP benchmark take on the following form:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{\left(6\right)}={\displaystyle \sum _{g=1}^G}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\displaystyle \sum _{g^{\prime }=1}^G}{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial s_{m_2}}\\ +{\displaystyle \sum _{g=1}^G}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\displaystyle \sum _{g^{\prime }=1}^G}{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(\mathbf{r},{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial s_{m_2}},\\ for j=1,\dots ,J{}_n;m_2=1,\dots ,J_{\sigma s,l=0}.\end{array}$$

(76)

Inserting the results obtained in Equations (64) and (65) into Equation (76), 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 expression for Equation (76):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{(6)}=N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}\left[{\xi }_0^{(1),g_{m_2}}\left(r\right)U_{1,j;0}^{(2),{g^{\prime }}_{m_2}}\left(r\right)+{\phi }_0^{{g^{\prime }}_{m_2}}\left(r\right)U_{2,j;0}^{(2),g_{m_2}}\left(r\right)\right],\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J{}_n;\hspace{0.33em}{m}_2=1,\dots ,J_{\sigma s,l=0},\end{array}$$

(77)

where

$$U_{1,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{u}_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right),$$

(78)

$$U_{2,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{u}_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right).$$

(79)

Contributions stemming from the source are computed by particularizing Equations (204) and (205) in [5] to the PERP benchmark. Particularizing Equation (204) in [5], in conjunction with the relations $\frac{{\partial }^{2}L}{\partial q_j\partial t_{m_2}}\frac{\partial q_j}{\partial n_j}\frac{\partial t_{m_2}}{\partial s_{m_2}}=\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}$ and $\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial t_{m_2}}\frac{\partial t_{m_2}}{\partial s_{m_2}}=\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial s_{m_2}}$, yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{(7)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{g}_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right)\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial s_{m_2}}},\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma s,l=0},\end{array}$$

(80)

where the 2nd-level adjoint functions $g_{1,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G$, are the solutions of the 2nd-Level Adjoint Sensitivity System presented previously in Equations (35) and (36). Inserting the result obtained in Equation (61) into Equation (80) yields the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{(7)}=-N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{g}_{1,j}^{(2),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{(1),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right),\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma s,l=0}.\end{array}$$

(81)

Particularizing Equation (205) in [5] to the PERP benchmark yields the following contributions:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{(8)}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{g}_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }}{\psi }^{(1),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)}}\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(s;\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial s_{m_2}},\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J{}_n;\hspace{0.33em}{m}_2=1,\dots ,J_{\sigma s,l=0}.\end{array}$$

(82)

Inserting the results obtained in Equations (64) into Equation (82), 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 (82):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{(8)}=N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}{\xi }_0^{(1),g_{m_2}}\left(r\right)G_{1,j;0}^{(2),{g^{\prime }}_{m_2}}\left(r\right),\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J{}_n;\hspace{0.33em}{m}_2=1,\dots ,J_{\sigma s,l=0},\end{array}$$

(83)

where

$$G_{1,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{g}_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right).$$

(84)

Collecting the partial contributions obtained in Equations (62), (66), (68), (71), (75), (77), (81) and (83), yields the following result:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}=\sum _{i=1}^8{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l=0}\right)}^{\left(i\right)}=-{\delta }_{i_j{i}_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\psi }_{1,j}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,j}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)\right]\\ +N_{i_{m_2},m_{m_2}}{\int }_{V}dV\left[{\xi }_0^{\left(1\right),g_{m_2}}\left(r\right){\xi }_{1,j;0}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r\right)+{\phi }_0^{{g^{\prime }}_{m_2}}\left(r\right){\xi }_{2,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\right]\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\theta }_{1,j}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)\right]\\ +{\delta }_{i_j{i}_{m_2}}{\xi }_0^{\left(1\right),g_{m_2}}\left(r\right){\phi }_0^{{g^{\prime }}_{m_2}}\left(r\right)+N_{i_{m_2},m_{m_2}}{\int }_{V}dV\left[{\xi }_0^{\left(1\right),g_{m_2}}\left(r\right){\mathrm{\Theta }}_{1,j;0}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r\right)+{\phi }_0^{{g^{\prime }}_{m_2}}\left(r\right){\mathrm{\Theta }}_{2,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\right]\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[u_{1,j}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)\right]\\ +N_{i_{m_2},m_{m_2}}{\int }_{V}dV\left[{\xi }_0^{\left(1\right),g_{m_2}}\left(r\right)U_{1,j;0}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r\right)+{\phi }_0^{{g^{\prime }}_{m_2}}\left(r\right)U_{2,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\right]\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{g}_{1,j}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),{g^{\prime }}_{m_2}}\left(r,\mathsf{\Omega }\right)\\ +N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\xi }_0^{\left(1\right),g_{m_2}}\left(r\right)G_{1,j;0}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r\right),forj=1,\dots ,J_n;m_2=1,\dots ,J_{\sigma s,l=0}.\end{array}$$

(85)

3.1.2. Second-Order Sensitivities ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l\ge 1}\right)},j=1,\dots ,J_n;m_2=1,\dots ,{\sigma }_{s,l\ge 1}$

For the 2nd-order sensitivities ${\left({\partial }^{2}L/\partial n_j\partial s_{m_2}\right)}_{\left(n=N,s={\sigma }_{s,l\ge 1}\right)},\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,{\sigma }_{s,l\ge 1}$, the parameters $n_j\equiv N_{i_j,m_j}$ correspond to the isotopic number densities, and the parameters $s_{m_2}\equiv {\sigma }_{s,l_{m_2},i_{m_2}}^{{g^{\prime }}_{m_2}\to g_{m_2}}$ correspond to the $l^{th}$-order $\left(l\ge 1\right)$ scattering cross sections. For this case, the expression for ${\left({\partial }^{2}L/\partial n_j\partial s_{m_2}\right)}_{\left(n=N,s={\sigma }_{s,l\ge 1}\right)}$ is obtained by particularizing Equations (159), (168), (178) and (205) in [5] to the PERP benchmark, which yields,

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l\ge 1}\right)}=\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial s_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial s_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{{\partial }^2{\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial n_j\partial s_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial s_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial s_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial s_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(\mathbf{r},{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial s_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{g}_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial s_{m_2}},\hfill \\ forj=1,\dots ,J_n;m_2=1,\dots ,J_{\sigma s,l\ge 1},\hfill \end{array}$$

(86)

where the 2nd-level adjoint functions ${\psi }_{1,i}^{(2),g},\hspace{0.33em}{\psi }_{2,i}^{(2),g},\hspace{0.33em}{\theta }_{1,i}^{(2),g},\hspace{0.33em}{\theta }_{2,i}^{(2),g},\hspace{0.33em}{u}_{1,i}^{(2),g},\hspace{0.33em}{u}_{2,i}^{(2),g}$ and $g_{2,j}^{(2),g}$ for $j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G$ are the same as those presented in Section 3.1.1, above. Inserting the results obtained in Equations (64), (65) and (70) into Equation (86), using the addition theorem for spherical harmonics in one-dimensional geometry and performing the respective angular integrations, yields the following expression:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial s_{m_2}}\right)}_{\left(n=N,s={\sigma }_{s,l\ge 1}\right)}=N_{i_{m_2},m_{m_2}}\left(2l_{m_2}+1\right){\int }_{V}dV\left[{\xi }_{l_{m2}}^{\left(1\right),g_{m_2}}\left(r\right){\xi }_{1,j;l_{m2}}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r\right)+{\phi }_{l_{m2}}^{{g^{\prime }}_{m_2}}\left(r\right){\xi }_{2,j;l_{m2}}^{\left(2\right),g_{m_2}}\left(r\right)\right]\\ +{\delta }_{i_j{i}_{m_2}}\left(2l_{m_2}+1\right){\xi }_{l_{m_2}}^{\left(1\right),g_{m_2}}\left(r\right){\phi }_{l_{m2}}^{{g^{\prime }}_{m_2}}\left(r\right)\\ +N_{i_{m_2},m_{m_2}}\left(2l_{m_2}+1\right){\int }_{V}dV\left[{\xi }_{l_{m2}}^{\left(1\right),g_{m_2}}\left(r\right){\mathrm{\Theta }}_{1,j;l_{m2}}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r\right)+{\phi }_{l_{m2}}^{{g^{\prime }}_{m_2}}\left(r\right){\mathrm{\Theta }}_{2,j;l_{m2}}^{\left(2\right),g_{m_2}}\left(r\right)\right]\\ +N_{i_{m_2},m_{m_2}}\left(2l_{m_2}+1\right){\int }_{V}dV\left[{\xi }_{l_{m2}}^{\left(1\right),g_{m_2}}\left(r\right)U_{1,j;l_{m2}}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r\right)+{\phi }_{l_{m2}}^{{g^{\prime }}_{m_2}}\left(r\right)U_{2,j;l_{m2}}^{\left(2\right),g_{m_2}}\left(r\right)\right]\\ +N_{i_{m_2},m_{m_2}}\left(2l_{m_2}+1\right){\int }_{V}dV{\xi }_{l_{m2}}^{\left(1\right),g_{m_2}}\left(r\right)G_{1,j;l_{m2}}^{\left(2\right),{g^{\prime }}_{m_2}}\left(r\right),\\ forj=1,\dots ,J_n;m_2=1,\dots ,J_{\sigma s,l\ge 1},\end{array}$$

(87)

where

$${\mathrm{\Theta }}_{1,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{P}_l\left(\mathsf{\Omega }\right){\theta }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right),$$

(88)

$${\mathrm{\Theta }}_{2,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{P}_l\left(\mathsf{\Omega }\right){\theta }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right),$$

(89)

$$U_{1,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{P}_l\left(\mathsf{\Omega }\right)u_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right),$$

(90)

$$U_{2,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{P}_l\left(\mathsf{\Omega }\right)u_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right),$$

(91)

$$G_{1,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{P}_l\left(\mathsf{\Omega }\right)g_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right).$$

(92)

3.2. Alternative Path: Computing the Second-Order Sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_s\partial N$

The results computed using the expressions for ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_s$ obtained in Eqs. (85) and (87) can be verified by obtaining the expressions for the symmetric expression ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_s\partial N$, the computation of which also requires separate consideration of the zeroth-order scattering cross sections. The two cases involved are as follows:

Case 1: ${\left(\frac{{\partial }^{2}L}{\partial s_j\partial n_{m_2}}\right)}_{\left(s={\sigma }_{s,l=0},n=N\right)},\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_{\sigma s,l=0};m_2=1,\dots ,J_n$, where the quantity $s_j$ refers to the parameters underlying the 0th-order scattering cross sections while $n_{m_2}$ refers to the isotopic number densities;

Case 2: ${\left(\frac{{\partial }^{2}L}{\partial s_j\partial n_{m_2}}\right)}_{\left(s={\sigma }_{s,l\ge 1},n=N\right)},\hspace{0.17em}\hspace{0.17em}j=1,\dots ,{\sigma }_{s,l\ge 1};m_2=1,\dots ,J_n$, where $s_j$ refers to the parameters underlying the $l^{th}$-order ($l\ge 1$) scattering cross sections and where $n_{m_2}$ refers to the isotopic number densities.

3.2.1. Second-Order Sensitivities ${\left({\partial }^{2}L/\partial s_j\partial n_{m_2}\right)}_{\left(s={\sigma }_{s,l=0},n=N\right)},j=1,\dots ,J_{\sigma s,l=0};m_2=1,\dots ,J_n$

The equations needed for deriving the expression of the 2nd-order mixed sensitivities ${\left({\partial }^{2}L/\partial s_j\partial n_{m_2}\right)}_{\left(s={\sigma }_{s,l=0},n=N\right)},\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_{\sigma s,l=0};m_2=1,\dots ,J_n$ are obtained by particularizing Equations (158), (159), (160), (162), (167), (168), (169) and (171) in [5] to the PERP benchmark. This procedure leads to the following expression:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial s_j\partial n_{m_2}}\right)}_{\left(s={\sigma }_{s,l=0},n=N\right)}=-\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\frac{{\partial }^2{{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial s_j\partial n_{m_2}}\hfill \\ -\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\psi }_{1,i}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,i}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial n_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial n_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial Q^g\left(\mathbf{q};r,\mathsf{\Omega }\right)}{\partial n_{m_2}}\\ -\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\theta }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{{\partial }^2{\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial s_j\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial Q^g\left(\mathbf{q};r,\mathsf{\Omega }\right)}{\partial n_{m_2}},for j=1,\dots ,J_{\sigma s,l=0};m_2=1,\dots ,J{}_n.\end{array}$$

(93)

In Equation (93), the adjoint functions ${\psi }_{1,j}^{(2),g}$ and ${\psi }_{2,j}^{(2),g},j=1,\dots ,J_{\sigma s,l=0};\hspace{0.33em}g=1,\dots ,G$ are the solutions of the 2nd-Level Adjoint Sensitivity System presented in Equations (30), (32), (36) and (37) of Part II [2], which are reproduced below for convenient reference:

$$B^g\left({\alpha }^0\right){\psi }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right)=-{\delta }_{{g^{\prime }}_{j}g}{N}_{i_j,m_j}{\phi }^g\left(r,\mathsf{\Omega }\right),\hspace{0.33em}j=1,\dots ,J_{\sigma s,l=0};\hspace{0.33em}\hspace{0.17em}g=1,\dots ,G,$$

(94)

$${\psi }_{1,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n<0;\hspace{0.33em}j=1,\dots ,J_{\sigma s,l=0};\hspace{0.33em}g=1,\dots ,G,$$

(95)

$$A^{(1),g}\left({\alpha }^0\right){\psi }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right)=-{\delta }_{{g^{\prime }}_{j}g}{N}_{i_j,m_j}{\psi }^{(1),g}\left(r,\mathsf{\Omega }\right),\hspace{0.33em}j=1,\dots ,J_{\sigma s,l=0};\hspace{0.33em}g=1,\dots ,G,$$

(96)

$${\psi }_{2,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n>0;\hspace{0.33em}j=1,\dots ,J_{\sigma s,l=0};\hspace{0.17em}\hspace{0.33em}g=1,\dots ,G.$$

(97)

The 2nd-level adjoint functions, ${\theta }_{1,j}^{(2),g}$ and ${\theta }_{2,j}^{(2),g},j=1,\dots ,J_{\sigma s,l=0};\hspace{0.33em}g=1,\dots ,G,$ in Equation (93) are solutions of the 2nd-Level Adjoint Sensitivity System presented in Equations (46), (48), (51) and (52) of Part II [2], which are reproduced below for convenient reference:

$$B^g\left({\alpha }^0\right){\theta }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right)={\delta }_{g_{j}g}{N}_{i_j,m_j}\left(2l_j+1\right)P_{l_j}\left(\mathsf{\Omega }\right){\varphi }_l^{{g^{\prime }}_j}\left(r\right),\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_{\sigma s};\hspace{0.33em}g=1,\dots ,G;\hspace{0.17em}\hspace{0.17em}l=0,\dots ,ISCT,$$

(98)

$${\theta }_{1,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n<0;\hspace{0.33em}j=1,\dots ,J_{\sigma s};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(99)

$$A^{(1),g}\left({\alpha }^0\right){\theta }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right)={\delta }_{{g^{\prime }}_{j}g}{N}_{i_j,m_j}\left(2l_j+1\right)P_{l_j}\left(\mathsf{\Omega }\right){\xi }_{l_j}^{(1),g_j}\left(r\right),j=1,\dots ,J_{\sigma s};\hspace{0.33em}g=1,\dots ,G;\hspace{0.17em}\hspace{0.17em}l=0,\dots ,ISCT,$$

(100)

$${\theta }_{2,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n>0;\hspace{0.33em}j=1,\dots ,J_{\sigma s};\hspace{0.33em}g=1,\dots ,G.$$

(101)

In Equation (93), the parameters $s_j$ and $n_{m_2}$ correspond to the 0th-order scattering cross sections and the isotopic number densities, denoted as $s_j\equiv {\sigma }_{s,l_j=0,i_j}^{{g^{\prime }}_j\to g_j}$ and $n_{m_2}\equiv N_{i_{m_2},m_{m_2}}$, respectively. Note that:

$$\begin{array}{c}\frac{{\partial }^2{{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial s_j\partial n_{m_2}}=\frac{{\partial }^2{{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial {\sigma }_{s,l_j=0,i_j}^{{g^{\prime }}_j\to g_j}\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial \left\{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{N}_{i,m}\left({\sigma }_{f,i}^g+{\sigma }_{c,i}^g+{\displaystyle \sum _{g^{\prime }=1}^G}{\sigma }_{s,l=0,i}^{g\to g^{\prime }}\right)\right]/\partial N_{i_{m_2},m_{m_2}}\right\}}{\partial {\sigma }_{s,l_j=0,i_j}^{{g^{\prime }}_j\to g_j}}\\ =\frac{\partial \left\{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{\displaystyle \sum _{g^{\prime }=1}^G}{N}_{i,m}{\sigma }_{s,l=0,i}^{g\to g^{\prime }}\right]/\partial N_{i_{m_2},m_{m_2}}\right\}}{\partial {\sigma }_{s,l_j=0,i_j}^{{g^{\prime }}_j\to g_j}}=\frac{\partial \left\{{\displaystyle \sum _{g^{\prime }=1}^G}{\sigma }_{s,l=0,i_{m_2}}^{g\to g^{\prime }}\right\}}{\partial {\sigma }_{s,l_j=0,i_j}^{{g^{\prime }}_j\to g_j}}={\delta }_{i_j{i}_{m_2}}{\delta }_{{g^{\prime }}_{j}g},\end{array}$$

(102)

and

$$\begin{array}{c}\frac{{\partial }^2{\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial s_j\partial n_{m_2}}=\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial {\sigma }_{s,l_j=0,i_j}^{{g^{\prime }}_j\to g_j}\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial \left\{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{N}_{i,m}{\sigma }_{s,i}^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })\right]/\partial N_{i_{m_2},m_{m_2}}\right\}}{\partial {\sigma }_{s,l_j=0,i_j}^{{g^{\prime }}_j\to g_j}}\\ =\frac{\partial \left\{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{\displaystyle \sum _{l=0}^{ISCT}}{N}_{i,m}\left(2l+1\right){\sigma }_{s,l,i}^{g^{\prime }\to g}{P}_l\left({\mathsf{\Omega }}^{\prime }·\mathsf{\Omega }\right)\right]/\partial N_{i_{m_2},m_{m_2}}\right\}}{\partial {\sigma }_{s,l_j=0,i_j}^{{g^{\prime }}_j\to g_j}}=\frac{\partial \left\{{\displaystyle \sum _{l=0}^{ISCT}}\left(2l+1\right){\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{P}_l\left({\mathsf{\Omega }}^{\prime }·\mathsf{\Omega }\right)\right\}}{\partial {\sigma }_{s,l_j=0,i_j}^{{g^{\prime }}_j\to g_j}}\\ ={\delta }_{i_j{i}_{m_2}}{\delta }_{{g^{\prime }}_j{g}^{\prime }}{\delta }_{g_{j}g}\left(2l_j+1\right)P_{l_j}\left({\mathsf{\Omega }}^{\prime }·\mathsf{\Omega }\right).\end{array}$$

(103)

Inserting the results obtained in Equations (47)–(52), (102) and (103) into Equation (93) yields the following expression for Equation (93):

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial s_j\partial n_{m_2}}\right)}_{\left(s={\sigma }_{s,l=0},n=N\right)}=-{\delta }_{i_j{i}_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),{g^{\prime }}_j}\left(r,\mathsf{\Omega }\right){\phi }^{{g^{\prime }}_j}\left(r,\mathsf{\Omega }\right)\\ -\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\psi }_{1,i}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,i}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]{\sigma }_{t,i_{m_2}}^g\\ +\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{\xi }_{1,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{\left(1\right),g^{\prime }}\left(r\right)+\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{\xi }_{2,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\chi }^g{\xi }_{2,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\nu }_{i_{m_2}}^{g^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)+\sum _{g=1}^G{\int }_{V}dV{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{\xi }_{1,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)\\ +\frac{1}{n_{m_2}}\sum _{g=1}^G{\int }_{V}dV{\xi }_{2,j;0}^{\left(2\right),g}\left(r\right)Q_{SF,i_{m_2}}^g-\sum _{g=1}^G{\int }_{V}dV{\sigma }_{t,i_{m_2}}^g{\int }_{4\pi }d\mathsf{\Omega }\left[{\theta }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\\ +{\delta }_{i_j{i}_{m_2}}\left(2l_j+1\right){\int }_{V}dV{\xi }_{l_j}^{\left(1\right),g_j}\left(r\right){\phi }_{l_j}^{{g^{\prime }}_j}\left(r\right)+\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{\mathrm{\Theta }}_{1,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{\left(1\right),g^{\prime }}\left(r\right)\\ +\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{\mathrm{\Theta }}_{2,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)+\sum _{g=1}^G{\int }_{V}dV{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{\mathrm{\Theta }}_{1,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\chi }^g{\mathrm{\Theta }}_{2,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\nu }_{i_{m_2}}^{g^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)+\frac{1}{n_{m_2}}\sum _{g=1}^G{\int }_{V}dV{\mathrm{\Theta }}_{2,j;0}^{\left(2\right),g}\left(r\right)Q_{SF,i_{m_2}}^g,\\ for j=1,\dots ,J_{\sigma s,l=0};m_2=1,\dots ,J{}_n.\end{array}$$

(104)

3.2.2. Second-Order Sensitivities ${\left({\partial }^{2}L/\partial s_j\partial n_{m_2}\right)}_{\left(s={\sigma }_{s,l\ge 1},n=N\right)},j=1,\dots ,{\sigma }_{s,l\ge 1};m_2=1,\dots ,J_n$

For this case, the parameters $s_j$ correspond to the $l^{th}$-order ($l\ge 1$) scattering cross sections, denoted as $s_j\equiv {\sigma }_{s,l_j,i_j}^{g_j{}^{\prime }\to g_j}$, and the parameters $n_{m_2}$ correspond to the isotopic number densities, denoted as $n_{m_2}\equiv N_{i_{m_2},m_{m_2}}$. Since the $l^{th}$-order ($l\ge 1$) scattering cross sections are not part of the total cross sections, the expression of ${\left({\partial }^{2}L/\partial s_j\partial n_{m_2}\right)}_{\left(s={\sigma }_{s,l\ge 1},n=N\right)}$ is obtained by particularizing Equations (167), (168), (169) and (171) in [5] to the PERP benchmark, which yields,

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial s_j\partial n_{m_2}}\right)}_{\left(s={\sigma }_{s,l\ge 1},n=N\right)}=-\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\theta }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial n_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{{\partial }^2{\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial s_j\partial n_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial n_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial n_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial Q^g\left(\mathbf{q};r,\mathsf{\Omega }\right)}{\partial n_{m_2}},for j=1,\dots ,J_{s,l\ge 1};m_2=1,\dots ,J{}_n,\hfill \end{array}$$

(105)

where the 2nd-level adjoint functions, ${\theta }_{1,j}^{(2),g}$ and ${\theta }_{2,j}^{(2),g},j=1,\dots ,J_{s,l\ge 1};\hspace{0.33em}g=1,\dots ,G,$ are the solutions of the 2nd-Level Adjoint Sensitivity System presented in Equations (46), (48), (51) and (52) of Part II [2], which have been reproduced, for convenience, in Equations (98)–(101). Inserting the results obtained in Equations (47)–(52) and (103) into Equation (105), using the addition theorem for spherical harmonics in one-dimensional geometry and performing the respective angular integrations yields the following expression:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial s_j\partial n_{m_2}}\right)}_{\left(s={\sigma }_{s,l\ge 1},n=N\right)}=-\sum _{g=1}^G{\int }_{V}dV{\sigma }_{t,i_{m_2}}^g{\int }_{4\pi }d\mathsf{\Omega }\left[{\theta }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\\ +{\delta }_{i_j{i}_{m_2}}\left(2l_j+1\right){\int }_{V}dV{\xi }_{l_j}^{\left(1\right),g_j}\left(r\right){\phi }_{l_j}^{{g^{\prime }}_j}\left(r\right)+\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{\mathrm{\Theta }}_{1,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{\left(1\right),g^{\prime }}\left(r\right)\\ +\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{\mathrm{\Theta }}_{2,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)+\sum _{g=1}^G{\int }_{V}dV{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{\mathrm{\Theta }}_{1,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\chi }^g{\mathrm{\Theta }}_{2,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\nu }_{i_{m_2}}^{g^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)+\frac{1}{n_{m_2}}\sum _{g=1}^G{\int }_{V}dV{\mathrm{\Theta }}_{2,j;0}^{\left(2\right),g}\left(r\right)Q_{SF,i_{m_2}}^g,\\ for j=1,\dots ,J_{s,l\ge 1};m_2=1,\dots ,J{}_n.\end{array}$$

(106)

3.3. Numerical Results for ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_s$

The second-order absolute sensitivities, ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_s$, of the leakage response with respect to the isotopic number densities and the scattering cross sections for all isotopes of the PERP benchmark have been computed using Equations (85) and (87), and have been independently verified by computing the symmetric expression ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_s\partial N$ using Equations (104) and (106). For the PERP benchmark, computing the second-order absolute sensitivities, ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_s$, using Equations (85) and (87) requires 16 forward and adjoint PARTISN [9] computations to obtain all the required adjoint functions. On the other hand, computing the alternative expression ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_s\partial N$ using Equations (104) and (106), requires 7101 forward and adjoint PARTISN [9] computations to obtain the needed second level adjoint functions. As have been discussed in Part III [3], the reason for needing “only” 7101, rather than 21,600, PARTISN [9] computations is that all of the up-scattering and some of the down-scattering cross sections are zero for the PERP benchmark. Thus, computing ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_s$ using Equations (85) and (87) is about 450 (≈7101/16) times more efficient than computing ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_s\partial N$ by using Equations (104) and (106).

The dimensions of the matrix ${\partial }^{2}L/\partial n_j\partial s_{m_2},\hspace{0.33em}j=1,\dots ,J{}_n;\hspace{0.33em}{m}_2=1,\dots ,J{}_{\sigma s}$ is $J_n\times J_{\sigma s}\hspace{0.33em}(=6\times 21600)$, where $J_{\sigma s}=G\times G\times (ISCT+1)\times I=30\times 30\times 4\times 6=21600$. The matrix of 2nd-order relative sensitivities corresponding to ${\partial }^{2}L/\partial n_j\partial s_{m_2},\hspace{0.33em}j=1,\dots ,J{}_n;\hspace{0.33em}{m}_2=1,\dots ,J{}_{\sigma s}$, denoted as $S^{(2)}\left(N_{i,m},{\sigma }_{s,l,k}^{g^{\prime }\to g}\right)$, is defined as follows:

$$S^{(2)}\left(N_{i,m},{\sigma }_{s,l,k}^{g^{\prime }\to g}\right)\triangleq \frac{{\partial }^{2}L}{\partial N_{i,m}\partial {\sigma }_{s,l,k}^{g^{\prime }\to g}}\left(\frac{N_{i,m}{\sigma }_{s,l,k}^{g^{\prime }\to g}}{L}\right),\hspace{0.17em}\hspace{0.33em}l=0,\dots ,3;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.33em}m=1,2;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{g}^{\prime },g=1,\dots ,30.$$

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To facilitate the presentation and interpretation of the numerical results, the $J_n\times J_{\sigma s}\hspace{0.33em}(=6\times 21600)$ matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l,k}^{g^{\prime }\to g}\right)$ has first been partitioned into 4 submatrices, namely, $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=0,k}^{g^{\prime }\to g}\right)$, $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=1,k}^{g^{\prime }\to g}\right)$, $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=2,k}^{g^{\prime }\to g}\right)$ and $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=3,k}^{g^{\prime }\to g}\right)$, corresponding to the scattering orders $l=0$, $l=1$, $l=2$, and $l=3$, respectively. Subsequently, each of the 4 submatrices is further partitioned into $I\times I=6\times 6$ smaller submatrices, each of dimensions $1\times (G\hspace{0.33em}\cdot G)=1\times 900$. The results are summarized in Section 3.3.1, Section 3.3.2, Section 3.3.3 and Section 3.3.4, below.

3.3.1. Results for the Relative Sensitivities $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=0,k}^{g^{\prime }\to g}\right)$

Table 12. Summary presentation of the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=0,k}^{g^{\prime }\to g}\right)$, for 2nd-order relative sensitivities of the leakage response with respect to the isotopic number densities and the 0th-order $\left(l=0\right)$ scattering cross sections for all isotopes.

	$$\begin{gathered} \mathit{k}=1 \\ {(}^{239}\mathbf{Pu}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=2 \\ {(}^{240}\mathbf{Pu}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=3 \\ \hspace{0.33em}{(}^{69}\mathbf{Ga}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=4 \\ \hspace{0.33em}{(}^{71}\mathbf{Ga}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=5 \\ \hspace{0.33em}\left(\mathbf{C}\right) \end{gathered}$$	$$\begin{gathered} \mathit{k}=6 \\ \hspace{0.33em}{(}^1\mathbf{H}) \end{gathered}$$
$i=1$ (239Pu)	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=0,1}^{g^{\prime }\to g}\end{array}\right)$ 8 elements with absolute values >1.0	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=0,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.18 × 10−1 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=0,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 6.80 × 10−3 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=0,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 4.36 × 10−3 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=0,5}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 8.70 × 10−1 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=0,6}^{g^{\prime }\to g}\end{array}\right)$ 7 elements with absolute values >1.0
$i=2$ (240Pu)	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=0,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 2.02 × 10−1 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=0,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 2.23 × 10−2 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=0,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 7.74 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=0,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 4.96 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=0,5}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 9.89 × 10−2 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=0,6}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.78 × 10−1 g’ = 16, g = 17
$i=3$ (69Ga)	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=0,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 6.82 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=0,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 4.52 × 10−5 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=0,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 5.17 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=0,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.67 × 10−6 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=0,5}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 3.01 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=0,6}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 5.47 × 10−4 g’ = 16, g = 17
$i=4$ (71Ga)	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=0,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 4.25 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=0,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 2.82 × 10−5 g′ = 13, g = 13	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=0,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.62 × 10−6 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=0,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 3.31 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=0,5}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.77 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=0,6}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 2.93 × 10−4 g’ = 16, g = 17
$i=5$ (C)	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=0,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.70 × 10−1 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=0,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.13 × 10−2 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=0,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 6.52 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=0,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 4.19 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=0,5}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.59 × 10−1 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=0,6}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.55 × 10−1 g’ = 16, g = 17
$i=6$ (1H)	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=0,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 2.72 × 10−1 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=0,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.80 × 10−2 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=0,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.04 × 10−3 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=0,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 6.68 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=0,5}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.48 × 10−1 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=0,6}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 3.81 × 10−1 g’ = 16, g = 17

presents the summary of the results for the components of the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=0,k}^{g^{\prime }\to g}\right)\triangleq \left({\partial }^{2}L/\partial N_{i,m}\partial {\sigma }_{s,l=0,k}^{g^{\prime }\to g}\right)\left(N_{i,m}{\sigma }_{s,l=0,k}^{g^{\prime }\to g}/L\right)$,$\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.33em}m=1,2;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{g}^{\prime },g=1,\dots ,30$, for the 2^nd-order relative sensitivities of the leakage response with respect to the isotopic number densities and the 0th-order scattering cross sections for all isotopes. Among the $J_n\times \hspace{0.33em}{J}_{\sigma s,l=0}=6\times 5400=32400$ elements in the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=0,k}^{g^{\prime }\to g}\right)$, 8844 elements have positive values and 2142 elements have negative values, while the remaining elements are zero. Most of these relative sensitivities are very small. However, 15 elements in the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=0,k}^{g^{\prime }\to g}\right)$ have relative sensitivities with absolute values greater than 1.0, as shown in the shaded cells in Table 12. These 15 large values reside in the sub-matrices $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=0,1}^{g^{\prime }\to g}\right)$ and $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=0,6}^{g^{\prime }\to g}\right)$. The value of the largest element of each of the other sub-matrices is positive, involving the 0th-order self-scattering cross sections for the 12th energy group of isotopes ²³⁹Pu, ²⁴⁰Pu, ⁶⁹Ga, ⁷¹G, and C, or (occasionally) the 0th-order out-scattering cross section ${\sigma }_{s,l=0,k=6}^{16\to 17}$ for isotope ¹H. The overall largest value in the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=0,k}^{g^{\prime }\to g}\right)$ is $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=0,k=1}^{12\to 12}\right)=1.912$.

Additional information regarding the two submatrices in Table 12 that have elements with absolute values greater than 1.0 is as follows:

• The submatrix $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=0,1}^{g^{\prime }\to g}\right),\hspace{0.33em}{g}^{\prime },g=1,\dots ,30$, comprises the 2nd-order sensitivities of the leakage response with respect to the isotopic number density and to the 0th-order scattering cross sections of ²³⁹Pu.

  Table 13. Elements of $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=0,k=1}^{g^{\prime }\to g}\right),\hspace{0.33em}{g}^{\prime },g=1,\dots ,30$ with absolute values greater than 1.0.

  Group	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 7\to 7\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 8\to 8\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 9\to 9\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 10\to 10\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 11\to 11\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 12\to 12\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 13\to 13\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 14\to 14\end{array}$
  values	1.461	1.155	1.206	1.147	1.036	1.912	1.660	1.235

  presents the 8 relative sensitivities in this submatrix that have values greater than 1.0. All of these sensitivities involve the 0th-order self-scattering cross sections for energy groups $g=7,\dots ,14$ of isotope ²³⁹Pu. The largest value in this submatrix is $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=0,k=1}^{12\to 12}\right)=1.912$, which involves the 0th-order self-scattering cross sections for the 12th energy group of ²³⁹Pu.
• The submatrix $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=0,k=6}^{g^{\prime }\to g}\right),\hspace{0.33em}{g}^{\prime },g=1,\dots ,30$, comprising the 2nd-order sensitivities of the leakage response with respect to the isotopic number density of ²³⁹Pu and to the 0th-order scattering cross sections of ¹H, includes 7 elements that have values greater than 1.0, as listed in

  Table 14. Elements of $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=0,k=6}^{g^{\prime }\to g}\right),\hspace{0.33em}{g}^{\prime },g=1,\dots ,30$ with absolute values greater than 1.0.

  Group	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 12\to 13\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 13\to 14\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 14\to 15\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 14\to 16\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 15\to 16\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 16\to 16\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 16\to 17\end{array}$
  values	1.386	1.300	1.110	1.146	1.430	1.289	1.585

  . Most of these 7 relative sensitivities are with respect to the 0th-order in-scattering or out-scattering cross sections. The largest value in this submatrix is $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=0,k=1}^{16\to 17}\right)=1.585$, involving the 0th-order out-scattering cross sections for energy groups $g^{\prime }=16\to g=17$ of isotope ²³⁹Pu.

3.3.2. Results for the Relative Sensitivities $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=1,k}^{g^{\prime }\to g}\right)$

Table 15. Summary presentation of the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=1,k}^{g^{\prime }\to g}\right)$, comprising the 2nd-order relative sensitivities of the leakage response with respect to the isotopic number densities and the 1st-order $\left(l=1\right)$ scattering cross sections for all isotopes.

	$$\begin{gathered} \mathit{k}=1 \\ {(}^{239}\mathbf{Pu}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=2 \\ {(}^{240}\mathbf{Pu}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=3 \\ \hspace{0.33em}{(}^{69}\mathbf{Ga}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=4 \\ \hspace{0.33em}{(}^{71}\mathbf{Ga}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=5 \\ \hspace{0.33em}\left(\mathbf{C}\right) \end{gathered}$$	$$\begin{gathered} \mathit{k}=6 \\ \hspace{0.33em}{(}^1\mathbf{H}) \end{gathered}$$
$i=1\hspace{0.33em}$ (239Pu)	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=1,1}^{g^{\prime }\to g}\end{array}\right)$1 element with absolute value >1.0	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=1,2}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −6.96 × 10−2 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=1,3}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −2.34 × 10−3 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=1,4}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.42 × 10−3 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=1,5}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −3.51 × 10−1 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=1,6}^{g^{\prime }\to g}\end{array}\right)$ 6 elements with absolute values >1.0
$i=2$ (240Pu)	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=1,1}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.30 × 10−1 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=1,2}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.32 × 10−2 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=1,3}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −2.65 × 10−4 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=1,4}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.61 × 10−4 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=1,5}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −4.00 × 10−2 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=1,6}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.50 × 10−1 g’ = 12, g = 13
$i=3$ (69Ga)	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=1,1}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −3.42 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=1,2}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −2.14 × 10−5 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=1,3}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.77 × 10−4 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=1,4}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −4.06 × 10−7 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=1,5}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.22 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=1,6}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −4.56 × 10−4 g’ = 12, g = 13
$i=4$ (71Ga)	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=1,1}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −2.13 × 10−4 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=1,2}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.33 × 10−5 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=1,3}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −4.40 × 10−7 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=1,4}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.07 × 10−4 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=1,5}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −7.23 × 10−5 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=1,6}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −2.68 × 10−4 g’ = 12, g = 13
$i=5$ (C)	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=1,1}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.12 × 10−1 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=1,2}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −6.76 × 10−3 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=1,3}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −2.27 × 10−4 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=1,4}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.38 × 10−4 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=1,5}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −6.24 × 10−2 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=1,6}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.31 × 10−1 g’ = 12, g = 13
$i=6$ (1H)	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=1,1}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.77 × 10−1 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=1,2}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.06 × 10−2 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=1,3}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −3.57 × 10−4 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=1,4}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −2.17 × 10−4 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=1,5}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −5.73 × 10−2 g’ = 12, g = 12	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=1,6}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −3.19 × 10−1 g’ = 12, g = 13

summarizes the results obtained for the elements of the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=1,k}^{g^{\prime }\to g}\right)\triangleq \left({\partial }^{2}L/\partial N_{i,m}\partial {\sigma }_{s,l=1,k}^{g^{\prime }\to g}\right)\left(N_{i,m}{\sigma }_{s,l=1,k}^{g^{\prime }\to g}/L\right)$,$\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.33em}m=1,2;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{g}^{\prime },g=1,\dots ,30$, which comprises the 2nd-order mixed relative sensitivities of the leakage response with respect to the isotopic number densities and the 1st-order scattering cross sections for all isotopes. Most of these 2nd-order mixed sensitivities are zero, and the non-zero ones are mostly negative. Specifically, the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=1,k}^{g^{\prime }\to g}\right)$, having dimensions $J_n\times \hspace{0.33em}{J}_{\sigma s,l=1}=6\times 5400=32400$, comprises 7772 elements with negative values, 2764 elements with positive values, while the remaining elements are zero. Most of the relative 2nd-order mixed relative sensitivities are very small. Only 8 components have large relative sensitivities with absolute values greater than 1.0, as shown in the shaded sub-matrices $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=1,1}^{g^{\prime }\to g}\right)$ and $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=1,6}^{g^{\prime }\to g}\right)$ in Table 15. Also, in the submatrices which have all their elements with absolute values less than 1.0, the value of the largest element of the respective submatrix is negative and involves the 1st-order self-scattering cross sections for the 7th energy group of isotopes ²³⁹Pu, ²⁴⁰Pu, ⁶⁹Ga and ⁷¹Ga, or the 12th energy group of isotope C, or the 1st-order out-scattering cross section ${\sigma }_{s,l=1,k=6}^{12\to 13}$ of isotope ¹H. The overall most negative element in the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=1,k}^{g^{\prime }\to g}\right)$ is $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=1,k=6}^{12\to 12}\right)=-1.386$.

Detailed information regarding the two submatrices in Table 15 comprising elements having absolute values greater than 1.0 is as follows:

• The sensitivity matrix $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=1,1}^{g^{\prime }\to g}\right),\hspace{0.17em}\hspace{0.17em}{g}^{\prime },g=1,\dots ,30,$ comprising the 2nd-order mixed sensitivities of the leakage response with respect to the isotopic number density and the 1st-order scattering cross sections of ²³⁹Pu, includes only one element, namely $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=1,1}^{7\to 7}\right)=-1.245$, which has an absolute value greater than 1.0. This element involves the 1st-order self-scattering cross section for the 7th energy group of ²³⁹Pu.
• The sensitivity matrix $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=1,6}^{g^{\prime }\to g}\right),\hspace{0.17em}\hspace{0.17em}{g}^{\prime },g=1,\dots ,30,$ comprising the 2nd-order sensitivities of the leakage response with respect to the isotopic number density of ²³⁹Pu and to the 1st-order scattering cross sections of ¹H, includes 6 elements that have values greater than 1.0, as listed in

  Table 16. Elements of $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=1,k=6}^{g^{\prime }\to g}\right),\hspace{0.33em}{g}^{\prime },g=1,\dots ,30$ with absolute values greater than 1.0.

  Groups	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 12\to 12\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 12\to 13\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 13\to 13\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 13\to 14\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 15\to 16\end{array}$	$\begin{array}{c}{\mathit{g}}^{\prime }\to \mathit{g}\\ 16\to 16\end{array}$
  values	−1.103	−1.327	−1.014	−1.162	−1.027	−1.210

  . These 6 large relative sensitivities involve the 1st-order self-scattering or out-scattering cross sections for energy groups $g^{\prime },g=12,\dots ,16$ of isotope ¹H, respectively.

3.3.3. Results for the Relative Sensitivities $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=2,k}^{g^{\prime }\to g}\right)$

The sensitivity results in the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=2,k}^{g^{\prime }\to g}\right)\triangleq \left({\partial }^{2}L/\partial N_{i,m}\partial {\sigma }_{s,l=2,k}^{g^{\prime }\to g}\right)\left(N_{i,m}{\sigma }_{s,l=2,k}^{g^{\prime }\to g}/L\right)$, $\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.33em}m=1,2;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{g}^{\prime },g=1,\dots ,30$, comprising the 2nd-order mixed relative sensitivities of the leakage response with respect to the isotopic number densities and the 2nd-order scattering cross sections for all isotopes, are summarized in

Table 17. Summary presentation of the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=2,k}^{g^{\prime }\to g}\right)$, for 2nd-order relative sensitivities of the leakage response with respect to the isotopic number densities and the 2nd-order $\left(l=2\right)$ scattering cross sections for all isotopes.

	$\mathit{k}=1$ (239Pu)	$\mathit{k}=2$ (240Pu)	$\mathit{k}=3$ (69Ga)	$\mathit{k}=4$ (71Pu)	$\mathit{k}=5$ (C)	$\mathit{k}=6$ (1H)
$i=1$ (239Pu)	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=2,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 7.13 × 10−2 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=2,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 4.07 × 10−3 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=2,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.22 × 10−4 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=2,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 7.57 × 10−5 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=2,5}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 9.48 × 10−2 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=2,6}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 3.50 × 10−1 g’ = 12, g = 12
$i=2$ (240Pu)	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=2,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 7.43 × 10−3 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=2,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 7.52 × 10−4 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=2,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.36 × 10−5 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=2,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 8.46 × 10−6 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=2,5}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.08 × 10−2 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=2,6}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 4.02 × 10−2 g’ = 12, g = 12
$i=3$ (69Ga)	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=2,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.34 × 10−5 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=2,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 8.17 × 10−7 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=2,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 8.93 × 10−6 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=2,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.52 × 10−8 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=2,5}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 2.60 × 10−5 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=2,6}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.24 × 10−4 g’ = 12, g = 12
$i=4$ (71Ga)	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=2,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 7.50 × 10−6 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=2,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 4.59 × 10−7 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=2,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.38 × 10−8 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=2,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 5.54 × 10−6 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=2,5}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.51 × 10−5 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=2,6}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 7.58 × 10−5 g’ = 12, g = 12
$i=5$ (C)	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=2,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 5.70 × 10−3 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=2,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 3.49 × 10−4 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=2,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.05 × 10−5 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=2,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 6.50 × 10−6 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=2,5}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.60 × 10−2 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=2,6}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 2.80 × 10−2 g’ = 12, g = 12
$i=6$ (1H)	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=2,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 7.72 × 10−3 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=2,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 4.72 × 10−4 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=2,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.42 × 10−5 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=2,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 8.79 × 10−6 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=2,5}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.41 × 10−2 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=2,6}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 6.98 × 10−2 g’ = 12, g = 12

. All of the values in this matrix are smaller than 1.0. This is expected for the 2nd-order sensitivities with respect to higher order scattering cross sections. Of the $J_n\times \hspace{0.33em}{J}_{\sigma s,l=2}=6\times 5400=32400$ components of $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=1,k}^{g^{\prime }\to g}\right)$, 6164 elements are positive, 4426 elements are negative, and the remaining elements are zero. As shown in Table 17, most of the largest absolute values in respective submatrices involve either the 2nd-order self-scattering cross sections for the 7th energy group of isotopes ²³⁹Pu, ²⁴⁰Pu, ⁶⁹Ga and ⁷¹G and C, or the 12th energy group of ¹H. Also, as shown in Table 17, the largest elements in the respective sub-matrix are all positive, and the vast majority of them are very small. The overall largest element in the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=2,k}^{g^{\prime }\to g}\right)$ is $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=2,k=6}^{12\to 12}\right)=3.50\times {10}^{-1}$.

3.3.4. Results for the Relative Sensitivities $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=3,k}^{g^{\prime }\to g}\right)$

Table 18. Summary presentation of the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=3,k}^{g^{\prime }\to g}\right)$, comprising the 2nd-order relative sensitivities of the leakage response with respect to the isotopic number densities and the 3rd-order $\left(l=3\right)$ scattering cross sections for all isotopes.

	$\mathit{k}=1$ (239Pu)	$\mathit{k}=2$ (240Pu)	$\mathit{k}=3$ (69Ga)	$\mathit{k}=4$ (71Ga)	$\mathit{k}=5$ (C)	$\mathit{k}=6$ (1H)
$i=1$ (239Pu)	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=3,1}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −8.98 × 10−5 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=3,2}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −5.43 × 10−6 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=3,3}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.54 × 10−7 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=3,4}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −9.66 × 10−8 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=3,5}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −2.38 × 10−2 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{1,1},\\ {\sigma }_{s,l=3,6}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −7.00 × 10−2 g’ = 12, g = 12
$i=2$ (240Pu)	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=3,1}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −6.38 × 10−6 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=3,2}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −4.90 × 10−7 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=3,3}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.11 × 10−8 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=3,4}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −7.62 × 10−9 g’ = 6, g=6	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=3,5}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −2.72 × 10−3 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{2,1},\\ {\sigma }_{s,l=3,6}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −8.07 × 10−3 g’ = 12, g = 12
$i=3$ (69Ga)	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=3,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.24 × 10−8 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=3,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 7.65 × 10−10 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=3,3}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −5.25 × 10−9 g’ = 6, g = 6	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=3,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.36 × 10−11 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=3,5}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −6.36 × 10−6 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{3,1},\\ {\sigma }_{s,l=3,6}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −2.51 × 10−5 g’ = 12, g = 12
$i=4$ (71Ga)	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=3,1}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −8.45 × 10−9 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=3,2}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −5.20 × 10−10 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=3,3}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.48 × 10−11 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=3,4}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −3.61 × 10−9 g’ = 6, g = 6	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=3,5}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −3.69 × 10−6 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{4,1},\\ {\sigma }_{s,l=3,6}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −1.58 × 10−5 g’ = 12, g = 12
$i=5$ (C)	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=3,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 6.63 × 10−6 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=3,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 4.08 × 10−7 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=3,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.16 × 10−8 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=3,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 7.27 × 10−9 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=3,5}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −3.63 × 10−3 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{5,2},\\ {\sigma }_{s,l=3,6}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −4.15 × 10−3 g’ = 12, g = 12
$i=6$ (1H)	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=3,1}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.41 × 10−5 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=3,2}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 8.65 × 10−7 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=3,3}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 2.46 × 10−8 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=3,4}^{g^{\prime }\to g}\end{array}\right)$ Max. value = 1.54 × 10−8 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=3,5}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −2.73 × 10−3 g’ = 7, g = 7	$S^{(2)}\left(\begin{array}{l}{N}_{6,2},\\ {\sigma }_{s,l=3,6}^{g^{\prime }\to g}\end{array}\right)$ Min. value = −9.49 × 10−3 g’ = 12, g = 12

reports the summary of the results for the 2nd-order mixed relative sensitivities $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=3,k}^{g^{\prime }\to g}\right)\triangleq \left({\partial }^{2}L/\partial N_{i,m}\partial {\sigma }_{s,l=3,k}^{g^{\prime }\to g}\right)\left(N_{i,m}{\sigma }_{s,l=3,k}^{g^{\prime }\to g}/L\right)$,$\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.33em}m=1,2;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{g}^{\prime },g=1,\dots ,30$, of the leakage response with respect to the isotopic number densities and the 3rd-order scattering cross sections for all isotopes of the PERP benchmark. Of the $J_n\times \hspace{0.33em}{J}_{\sigma s,l=3}=6\times 5400=32400$ components of the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=3,k}^{g^{\prime }\to g}\right)$, 5311 elements have negative values and 5183 elements have positive values, while the remaining elements are zero. As in Table 17, most of the largest absolute values in respective submatrices shown in Table 18 involve the 3^rd-order self-scattering cross sections for the 7th energy group of isotopes ²³⁹Pu, ²⁴⁰Pu, ⁶⁹Ga and ⁷¹G and C, or the 12th energy group of ¹H. All of the values presented in Table 18 are very small; the overall largest element in the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{s,l=3,k}^{g^{\prime }\to g}\right)$ is $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=3,6}^{12\to 12}\right)=-7.00\times {10}^{-2}$.

4. Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Parameters Underlying the Benchmark’s Isotopic Number Densities and Fission Cross Sections

This Section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_f$ of the leakage response with respect to the isotopic number densities and group-averaged fission microscopic cross sections of all isotopes of the PERP benchmark. Due to symmetry, these 2nd-order mixed sensitivities can also be computed by using the alternative expression ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_f\partial N$. These two alternative paths are presented in Section 4.1 and Section 4.2, respectively. The numerical results for the 2nd-order mixed sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_f$ have been verified with the results obtained for ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_f\partial N$, and are presented in Section 4.3.

4.1. Computing the Second-Order Sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_f$

The equations needed for deriving the expressions of the 2nd-order sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_f$ are obtained by particularizing Equations (158), (160), (167), (169), (177), (179), (204) and (206) in [5] to the PERP benchmark. Specifically, using Equation (158) in [5] in conjunction with the relations $\frac{{\partial }^{2}L}{\partial t_j\partial t_{m_2}}\frac{\partial t_j}{\partial n_j}\frac{\partial t_{m_2}}{\partial f_{m_2}}=\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}$, $\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial t_{m_2}}\frac{\partial t_{m_2}}{\partial f_{m_2}}=\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial f_{m_2}}$ and $\frac{{\partial }^2{\Sigma }_t{}^g\left(t\right)}{\partial t_j\partial t_{m_2}}\frac{\partial t_j}{\partial n_j}\frac{\partial t_{m_2}}{\partial f_{m_2}}=\frac{{\partial }^2{\Sigma }_t{}^g\left(t\right)}{\partial n_j\partial f_{m_2}}$ yields the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{(1)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{\psi }^{(1),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\frac{{\partial }^2{\Sigma }_t{}^g\left(t\right)}{\partial n_j\partial f_{m_2}}}\\ \hspace{0.17em}-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[{\psi }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial f_{m_2}}},\\ for\hspace{0.33em}j=1,\dots ,J_n\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma f}\hspace{0.17em}.\end{array}$$

(108)

The 2nd-level adjoint functions ${\psi }_{1,j}^{(2),g}$ and ${\psi }_{2,j}^{(2),g},j=1,\dots ,J_{\sigma f};\hspace{0.33em}g=1,\dots ,G$ in Equation (108) are the solutions of the 2nd-LASS presented in Equations (11)–(14). In Equation (108), the parameters $n_j$ and $f_{m_2}$ correspond to the isotopic number densities and microscopic fission cross sections, respectively, and are denoted as $n_j\equiv N_{i_j,m_j}$ and $f_{m_2}\equiv {\sigma }_{f,i_{m_2}}^{g_{m_2}}$, where the subscripts $i_{m_2}$ and $g_{m_2}$ refer to the isotope and energy group associated with the parameter $f_{m_2}$, respectively. Noting that

$$\begin{array}{c}\frac{{\partial }^2{{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial n_j\partial f_{m_2}}=\frac{{\partial }^2{{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial N_{i_j,m_j}\partial {\sigma }_{f,i_{m_2}}^{g_{m_2}}}=\frac{\partial \left\{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{N}_{i,m}\left({\sigma }_{f,i}^g+{\sigma }_{c,i}^g+{\displaystyle \sum _{g^{\prime }=1}^G}{\sigma }_{s,l=0,i}^{g\to g^{\prime }}\right)\right]/\partial N_{i_j,m_j}\right\}}{\partial {\sigma }_{f,i_{m_2}}^{g_{m_2}}}\\ =\frac{\partial \left\{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{N}_{i,m}{\sigma }_{f,i}^g\right]/\partial N_{i_j,m_j}\right\}}{\partial {\sigma }_{f,i_{m_2}}^{g_{m_2}}}=\frac{\partial \left\{{\sigma }_{f,i_j}^g\right\}}{\partial {\sigma }_{f,i_{m_2}}^{g_{m_2}}}={\delta }_{i_j{i}_{m_2}}{\delta }_{g_{m_2}g},\end{array}$$

(109)

$$\frac{\partial {\Sigma }_t^g}{\partial f_{m_2}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\sigma }_{t,i}^g\right]}{\partial {\sigma }_{f,i_{m_2}}^{g_{m_2}}}={\delta }_{g_{m_2}g}{N}_{i_{m_2},m_{m_2}},$$

(110)

and inserting the results obtained in Equations (109) and (110) into Equation (108) yields the following expression for Equation (108):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{(1)}=-{\delta }_{i_j{i}_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{\psi }^{(1),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\\ -N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[{\psi }_{1,i}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g_{m_2}}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,i}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\right],\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.33em}{m}_2=1,\dots ,J_{\sigma f}.\end{array}$$

(111)

The contributions stemming from Equation (160) in [5] takes on the following particular form:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{\left(2\right)}=\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial f_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\right]}{\partial f_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right),\\ for j=1,\dots ,J_n;m_2=1,\dots ,J_{\sigma f}.\end{array}$$

(112)

Noting that

$$\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\right]}{\partial f_{m_2}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\left(\nu {\sigma }_f\right)}_i^g\right]}{\partial {\sigma }_{f,i_{m_2}}^{g_{m_2}}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\nu }_i^g{\sigma }_{f,i}^g\right]}{\partial {\sigma }_{f,i_{m_2}}^{g_{m_2}}}={\delta }_{g_{m_2}g}{N}_{i_{m_2},m_{m_2}}{\nu }_{i_{m_2}}^g,$$

(113)

$$\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial f_{m_2}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\left(\nu {\sigma }_f\right)}_i^{g^{\prime }}\right]}{\partial {\sigma }_{f,i_{m_2}}^{g_{m_2}}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\nu }_i^{g^{\prime }}{\sigma }_{f,i}^{g^{\prime }}\right]}{\partial {\sigma }_{f,i_{m_2}}^{g_{m_2}}}={\delta }_{g_{m_2}{g}^{\prime }}{N}_{i_{m_2},m_{m_2}}{\nu }_{i_{m_2}}^{g^{\prime }},$$

(114)

and inserting the results obtained in Equations (113) and (114) into Equation (112) yields the following expression for Equation (114):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{(2)}=N_{i_{m_2},m_{m_2}}{\nu }_{i_{m_2}}^{g_{m_2}}{\displaystyle {\int }_{V}dV}\left[{\xi }_{1,j;0}^{(2),g_{m_2}}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)}+{\phi }_0^{g_{m_2}}(r){\displaystyle \sum _{g=1}^G{\chi }^g{\xi }_{2,j;0}^{(2),g}(r)}\right],\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma f}\hspace{0.17em}.\end{array}$$

(115)

Using Equation (167) in [5] in conjunction with the relations $\frac{{\partial }^{2}L}{\partial s_j\partial t_{m_2}}\frac{\partial s_j}{\partial n_j}\frac{\partial t_{m_2}}{\partial f_{m_2}}=\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}$ and $\frac{\partial {\Sigma }_t{}^g}{\partial t_{m_2}}\frac{\partial t_{m_2}}{\partial f_{m_2}}=\frac{\partial {\Sigma }_t{}^g}{\partial f_{m_2}}$ yields the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{(3)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[{\theta }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {\Sigma }_t{}^g}{\partial f_{m_2}}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma f},\end{array}$$

(116)

where the 2nd-level adjoint functions ${\theta }_{1,j}^{(2),g}$, and ${\theta }_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G$, are the solutions of the 2nd-Level Adjoint Sensitivity System presented in Equations (19)–(22). Inserting the results obtained in Equation (110) into Equation (116) yields the following relation:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{(3)}=-N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[{\theta }_{1,j}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g_{m_2}}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\right],\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma f}.\end{array}$$

(117)

The contributions stemming from Equation (169) in [5] to ${\partial }^{2}L/\partial n_j\partial f_{m_2}$ are obtained in the following form:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{\left(4\right)}=\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\right]}{\partial f_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial f_{m_2}},\\ for j=1,\dots ,J_n;m_2=1,\dots ,J{}_{\sigma f}.\end{array}$$

(118)

Inserting Equations (113) and (114) into Equation (118) yields the following expression for Equation (118):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{(4)}=N_{i_{m_2},m_{m_2}}{\nu }_{i_{m_2}}^{g_{m_2}}{\displaystyle {\int }_{V}dV}\left[{\mathrm{\Theta }}_{1,j;0}^{(2),g_{m_2}}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)}+{\phi }_0^{g_{m_2}}(r){\displaystyle \sum _{g=1}^G{\chi }^g{\mathrm{\Theta }}_{2,j;0}^{(2),g}(r)}\right],\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;m_2=1,\dots ,J{}_{\sigma f}.\end{array}$$

(119)

Further contributions stem from Equation (177) in [5] in conjunction with the relations $\frac{{\partial }^{2}L}{\partial f_j\partial t_{m_2}}\frac{\partial f_j}{\partial n_j}\frac{\partial t_{m_2}}{\partial f_{m_2}}=\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}$ and $\frac{\partial {\Sigma }_t{}^g}{\partial t_{m_2}}\frac{\partial t_{m_2}}{\partial f_{m_2}}=\frac{\partial {\Sigma }_t{}^g}{\partial f_{m_2}}$, as follows:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{(5)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[u_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {\Sigma }_t{}^g}{\partial f_{m_2}}},\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma f},\end{array}$$

(120)

where the 2nd-level adjoint functions $u_{1,j}^{(2),g}$, and $u_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G$, are the solutions of the 2nd-Level Adjoint Sensitivity System comprising Equations (27)–(30). Replacing the result obtained in Equation (110) into Equation (120) yields the following relation:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{(5)}=-N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}\left[u_{1,j}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g_{m_2}}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\right],\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma f}.\end{array}$$

(121)

Contributions stemming from Equation (179) in [5] to ${\partial }^{2}L/\partial n_j\partial f_{m_2}$ are given by the following relation:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{\left(6\right)}=\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial n_j\partial f_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\right]}{\partial f_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\hfill \\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial f_{m_2}},\\ forj=1,\dots ,J_n;m_2=1,\dots ,J_{\sigma f},\end{array}$$

(122)

where

$$\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial n_j\partial f_{m_2}}=\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial N_{i_j,m_j}\partial {\sigma }_{f,i_{m_2}}^{g_{m_2}}}=\frac{\partial \left[\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\nu }_i^{g^{\prime }}{\sigma }_{f,i}^{g^{\prime }}\right]/\partial N_{i_j,m_j}\right]}{\partial {\sigma }_{f,i_{m_2}}^{g_{m_2}}}=\frac{\partial \left\{{\nu }_{i_j}^{g^{\prime }}{\sigma }_{f,i_j}^{g^{\prime }}\right\}}{\partial {\sigma }_{f,i_{m_2}}^{g_{m_2}}}={\delta }_{i_j{i}_{m_2}}{\delta }_{g_{m_2}{g}^{\prime }}{\nu }_{i_{m_2}}^{g^{\prime }}.$$

(123)

Inserting the results obtained in Equations (113), (114) and (123) into Equation (122) yields the following expression for Equation (122):

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{\left(6\right)}={\delta }_{i_j{i}_{m_2}}{\nu }_{i_{m_2}}^{g_{m_2}}{\int }_{V}dV{\phi }_0^{g_{m_2}}\left(r\right)\sum _{g=1}^G{\chi }^g{\xi }_0^{\left(1\right),g}\left(r\right)\\ +N_{i_{m_2},m_{m_2}}{\nu }_{i_{m_2}}^{g_{m_2}}{\int }_{V}dV\left[U_{1,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)+{\phi }_0^{g_{m_2}}\left(r\right)\sum _{g=1}^G{\chi }^g{U}_{2,j;0}^{\left(2\right),g}\left(r\right)\right],\\ forj=1,\dots ,J_n;m_2=1,\dots ,J_{\sigma f}.\end{array}$$

(124)

Additional contributions stemming from the sources are computed by particularizing Equations (204) and (206) in [5] to the PERP benchmark. The expression obtained by particularizing Equation (204) in [5], in conjunction with the relations $\frac{{\partial }^{2}L}{\partial q_j\partial t_{m_2}}\frac{\partial q_j}{\partial n_j}\frac{\partial t_{m_2}}{\partial f_{m_2}}=\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}$ and $\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial t_{m_2}}\frac{\partial t_{m_2}}{\partial f_{m_2}}=\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial f_{m_2}}$ yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{(7)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{g}_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g}\left(r,\mathsf{\Omega }\right)\frac{\partial {\Sigma }_t{}^g\left(t\right)}{\partial f_{m_2}}},\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma f},\end{array}$$

(125)

where the 2nd-level adjoint functions $g_{1,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G$, are the solutions of the 2nd-LASS presented previously in Equations (35) and (36). Inserting the result obtained in Equation (110) into Equation (125) yields the following result:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{(7)}=-N_{i_{m_2},m_{m_2}}{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{g}_{1,j}^{(2),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{(1),g_{m_2}}\left(r,\mathsf{\Omega }\right),\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma f}.\end{array}$$

(126)

Finally, using Equation (206) in [5] to the PERP benchmark yields the following contributions:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{(8)}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{g}_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(f\right)\right]}{\partial f_{m_2}}{\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }}{\chi }^{g^{\prime }}{\psi }^{(1),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma f}.\end{array}$$

(127)

Using the result obtained in Equation (113) into Equation (127) yields the following expression for Equation (127):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{(8)}=N_{i_{m_2},m_{m_2}}{\nu }_{i_{m_2}}^{g_{m_2}}{\displaystyle {\int }_{V}dV}{G}_{1,j;0}^{(2),g_{m_2}}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)},\hspace{0.33em}\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=1,\dots ,J_{\sigma f}.\end{array}$$

(128)

Collecting the partial contributions obtained in Equations (111), (115), (117), (119), (121), (124), (126) and (128), yields the following result:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}=\sum _{i=1}^8{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f={\sigma }_f\right)}^{\left(i\right)}\\ =-{\delta }_{i_j{i}_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\psi }_{1,i}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g_{m_2}}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,i}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\right]\\ +N_{i_{m_2},m_{m_2}}{\nu }_{i_{m_2}}^{g_{m_2}}{\int }_{V}dV\left[{\xi }_{1,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)+{\phi }_0^{g_{m_2}}\left(r\right)\sum _{g=1}^G{\chi }^g{\xi }_{2,j;0}^{\left(2\right),g}\left(r\right)\right]\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\theta }_{1,j}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g_{m_2}}\left(r,\mathsf{\Omega }\right)+{\theta }_{2,j}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\right]\\ +N_{i_{m_2},m_{m_2}}{\nu }_{i_{m_2}}^{g_{m_2}}{\int }_{V}dV\left[{\mathrm{\Theta }}_{1,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)+{\phi }_0^{g_{m_2}}\left(r\right)\sum _{g=1}^G{\chi }^g{\mathrm{\Theta }}_{2,j;0}^{\left(2\right),g}\left(r\right)\right]\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[u_{1,j}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g_{m_2}}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\phi }^{g_{m_2}}\left(r,\mathsf{\Omega }\right)\right]\\ +{\delta }_{i_j{i}_{m_2}}{\nu }_{i_{m_2}}^{g_{m_2}}{\int }_{V}dV{\phi }_0^{g_{m_2}}\left(r\right)\sum _{g=1}^G{\chi }^g{\xi }_0^{\left(1\right),g}\left(r\right)\\ +N_{i_{m_2},m_{m_2}}{\nu }_{i_{m_2}}^{g_{m_2}}{\int }_{V}dV\left[U_{1,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)+{\phi }_0^{g_{m_2}}\left(r\right)\sum _{g=1}^G{\chi }^g{U}_{2,j;0}^{\left(2\right),g}\left(r\right)\right]\\ -N_{i_{m_2},m_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{g}_{1,j}^{\left(2\right),g_{m_2}}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g_{m_2}}\left(r,\mathsf{\Omega }\right)\\ +N_{i_{m_2},m_{m_2}}{\nu }_{i_{m_2}}^{g_{m_2}}{\int }_{V}dV{G}_{1,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right),forj=1,\dots ,J_n;m_2=1,\dots ,J_{\sigma f}.\end{array}$$

(129)

4.2. Alternative Path: Computing the Second-Order Sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_f\partial N$

Due to symmetry of the mixed 2nd-order sensitivities, the results computed using the expression for ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_f$ obtained in Equation (129) can be verified by obtaining and using the expressions for ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_f\partial N$. The equations needed for deriving the expression of the 2nd-order mixed sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_f\partial N$ are obtained by particularizing Equations (158), (159), (160), (162), (177), (178), (179) and (181) in [5] to the PERP benchmark, which yields the following relation:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial f_j\partial n_{m_2}}\right)}_{\left(f={\sigma }_f,n=N\right)}=-\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\frac{{\partial }^2{{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial f_j\partial n_{m_2}}\\ -\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\psi }_{1,i}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,i}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial Q^g\left(\mathbf{q};r,\mathsf{\Omega }\right)}{\partial n_{m_2}}\\ -\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[u_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial f_j\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\right]}{\partial n_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial Q^g\left(\mathbf{q};r,\mathsf{\Omega }\right)}{\partial n_{m_2}},for j=1,\dots ,J_{\sigma f};m_2=1,\dots ,J{}_n.\end{array}$$

(130)

In Equation (130), the adjoint functions ${\psi }_{1,j}^{(2),g}$ and ${\psi }_{2,j}^{(2),g},j=1,\dots ,J_{\sigma f};\hspace{0.33em}g=1,\dots ,G$ are the solutions of the 2nd-Level Adjoint Sensitivity System presented in Equations (33), (35), (39) and (40) of Part III [3], which are reproduced below for convenient reference:

$$B^g\left({\alpha }^0\right){\psi }_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right)=-{\delta }_{g_{j}g}{N}_{i_j,m_j}{\phi }^g\left(r,\mathsf{\Omega }\right),\hspace{0.33em}j=1,\dots ,J_{\sigma f};\hspace{0.33em}\hspace{0.17em}g=1,\dots ,G,$$

(131)

$${\psi }_{1,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n<0;\hspace{0.33em}j=1,\dots ,J_{\sigma f};\hspace{0.33em}g=1,\dots ,G,$$

(132)

$$A^{(1),g}\left({\alpha }^0\right){\psi }_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right)=-{\delta }_{g_{j}g}{N}_{i_j,m_j}{\psi }^{(1),g}\left(r,\mathsf{\Omega }\right),\hspace{0.33em}j=1,\dots ,J_{\sigma f};\hspace{0.33em}g=1,\dots ,G,$$

(133)

$${\psi }_{2,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n>0;\hspace{0.33em}j=1,\dots ,J_{\sigma f};\hspace{0.17em}\hspace{0.33em}g=1,\dots ,G.$$

(134)

Furthermore, the 2nd-level adjoint functions, $u_{1,j}^{(2),g}$ and $u_{2,j}^{(2),g},j=1,\dots ,J_{\sigma f};\hspace{0.33em}g=1,\dots ,G,$ which appear in Equation (130) are the solutions of the 2nd-Level Adjoint Sensitivity System presented in Equations (19), (21), (29) and (30) of Part III [3], which are reproduced below for convenient reference:

$$B^g\left({\alpha }^0\right)u_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right)=N_{i_j,m_j}{\nu }_{i_j}^{g_j}{\chi }^g{\phi }_0^{g_j}\left(r\right),\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_{\sigma f};\hspace{0.33em}g=1,\dots ,G,$$

(135)

$$u_{1,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n<0;\hspace{0.33em}j=1,\dots ,J_{\sigma f};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(136)

$$A^{(1),g}\left({\alpha }^0\right)u_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right)={\delta }_{g_{j}g}{N}_{i_j,m_j}{\nu }_{i_j}^g{\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_{\sigma f};\hspace{0.33em}g=1,\dots ,G,$$

(137)

$$u_{2,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n>0;\hspace{0.33em}j=1,\dots ,J_{\sigma f};\hspace{0.33em}g=1,\dots ,G.$$

(138)

In Equation (130), the parameters $f_j$ and $n_{m_2}$ correspond to the fission cross sections and the isotopic number densities, denoted as $f_j\equiv {\sigma }_{f,i_j}^{g_j}$ and $n_{m_2}\equiv N_{i_{m_2},m_{m_2}}$, respectively. The following results will be used in subsequent derivations:

$$\begin{array}{c}\frac{{\partial }^2{{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial f_j\partial n_{m_2}}=\frac{{\partial }^2{{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial {\sigma }_{f,i_j}^{g_j}\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial \left\{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{N}_{i,m}\left({\sigma }_{f,i}^g+{\sigma }_{c,i}^g+{\displaystyle \sum _{g^{\prime }=1}^G}{\sigma }_{s,l=0,i}^{g\to g^{\prime }}\right)\right]/\partial {\sigma }_{f,i_j}^{g_j}\right\}}{\partial N_{i_{m_2},m_{m_2}}}\\ =\frac{\partial \left\{\partial \left[{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{i=1}^I}{N}_{i,m}{\sigma }_{f,i}^g\right]/\partial {\sigma }_{f,i_j}^{g_j}\right\}}{\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial \left({\delta }_{g_{j}g}{N}_{i_j,m_j}\right)}{\partial N_{i_{m_2},m_{m_2}}}={\delta }_{i_j{i}_{m_2}}{\delta }_{g_{j}g},\end{array}$$

(139)

$$\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial f_j\partial n_{m_2}}=\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial {\sigma }_{f,i_j}^{g_j}\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial \left[\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\nu }_i^{g^{\prime }}{\sigma }_{f,i}^{g^{\prime }}\right]/\partial {\sigma }_{f,i_j}^{g_j}\right]}{\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial \left({\delta }_{g_j{g}^{\prime }}{N}_{i_j,m_j}{\nu }_{i_j}^{g^{\prime }}\right)}{\partial N_{i_{m_2},m_{m_2}}}={\delta }_{i_j{i}_{m_2}}{\delta }_{g_j{g}^{\prime }}{\nu }_{i_{m_2}}^{g^{\prime }}.$$

(140)

Inserting the results obtained in Equations (47)–(52), (139) and (140) into Equation (130) and performing the respective angular integrations yields the following expression for Equation (130):

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial f_j\partial n_{m_2}}\right)}_{\left(f={\sigma }_f,n=N\right)}=-{\delta }_{i_j{i}_{m_2}}{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g_j}\left(r,\mathsf{\Omega }\right){\phi }^{g_j}\left(r,\mathsf{\Omega }\right)\\ -\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[{\psi }_{1,i}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+{\psi }_{2,i}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]{\sigma }_{t,i_{m_2}}^g\\ +\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{\xi }_{1,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{\left(1\right),g^{\prime }}\left(r\right)+\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{\xi }_{2,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\chi }^g{\xi }_{2,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\nu }_{i_{m_2}}^{g^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)+\sum _{g=1}^G{\int }_{V}dV{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{\xi }_{1,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)\\ +\frac{1}{n_{m_2}}\sum _{g=1}^G{\int }_{V}dV{\xi }_{2,j;0}^{\left(2\right),g}\left(r\right)Q_{SF,i_{m_2}}^g-\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[u_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]{\sigma }_{t,i_{m_2}}^g\\ +\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{U}_{1,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{\left(1\right),g^{\prime }}\left(r\right)+\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{U}_{2,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)\\ +{\delta }_{i_j{i}_{m_2}}{\nu }_{i_{m_2}}^{g_j}{\int }_{V}dV{\phi }_0^{g_j}\left(r\right)\sum _{g=1}^G{\chi }^g{\xi }_0^{\left(1\right),g}\left(r\right)+\sum _{g=1}^G{\int }_{V}dV{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{U}_{1,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\chi }^g{U}_{2,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\nu }_{i_{m_2}}^{g^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)+\frac{1}{n_{m_2}}\sum _{g=1}^G{\int }_{V}dV{U}_{2,j;0}^{\left(2\right),g}\left(r\right)Q_{SF,i_{m_2}}^g,\\ for j=1,\dots ,J_{\sigma f};m_2=1,\dots ,J{}_n.\end{array}$$

(141)

4.3. Numerical Results for ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_f$

The second-order absolute sensitivities, ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_f$, of the leakage response with respect to the isotopic number densities and the fission cross sections for all isotopes of the PERP benchmark have been computed using Equation (129) and have been independently verified by computing ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_f\partial N$ using Equation (141). For the PERP benchmark, computing the second-order absolute sensitivities, ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_f$, using Equation (129) requires 16 forward and adjoint PARTISN computations to obtain all the adjoint functions required in Equation (129). On the other hand, computing the alternative expression ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_f\partial N$ using Equation (141), requires 120 forward and adjoint PARTISN computations to obtain the needed second level adjoint functions required in Equation (141). Thus, computing ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_f$ using Equation (129) is about 8 (≈ 120/16) times more efficient than computing ${\partial }^{2}L\left(\alpha \right)/\partial {\sigma }_f\partial N$ by using Equation (141).

The matrix ${\partial }^{2}L/\partial n_j\partial f_{m_2},\hspace{0.33em}j=1,\dots ,J{}_n;\hspace{0.33em}{m}_2=1,\dots ,J{}_{\sigma f}$ has dimensions $J_n\times J_{\sigma f}\hspace{0.33em}(=6\times 60)$, where $J_{\sigma f}=G\times N_f=30\times 2$, 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 ${\partial }^{2}L/\partial n_j\partial f_{m_2},\hspace{0.33em}j=1,\dots ,J{}_n;\hspace{0.33em}{m}_2=1,\dots ,J{}_{\sigma f}$, is denoted as $S^{(2)}\left(N_{i,m},{\sigma }_{f,k}^g\right)$ and is defined as follows:

$$S^{(2)}\left(N_{i,m},{\sigma }_{f,k}^g\right)\triangleq \frac{{\partial }^{2}L}{\partial N_{i,m}\partial {\sigma }_{f,k}^g}\left(\frac{N_{i,m}{\sigma }_{f,k}^g}{L}\right),\hspace{0.17em}\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\dots ,6;\hspace{0.33em}m=1,2;\hspace{0.33em}k=1,2;\hspace{0.17em}g=1,\dots ,30.$$

(142)

Table 19. Summary presentation of the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{f,k}^g\right)\hspace{0.17em},\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}i=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}k,m=1,2;\hspace{0.33em}g=1,\dots ,30$

	$$\begin{gathered} \mathit{k}=1 \\ {(}^{239}\mathbf{Pu}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=2 \\ {(}^{240}\mathbf{Pu}) \end{gathered}$$
$i=1\hspace{0.33em}$ (239Pu)	$S^{(2)}\left(N_{1,1},{\sigma }_{f,1}^g\right)$ 12 elements with absolute values >1.0	$S^{(2)}\left(N_{1,1},{\sigma }_{f,2}^g\right)$ Max. value = 5.62 × 10−1 at g = 12
$i=2$ (240Pu)	$S^{(2)}\left(N_{2,1},{\sigma }_{f,1}^g\right)$1 element with absolute value >1.0	$S^{(2)}\left(N_{2,1},{\sigma }_{f,2}^g\right)$ Max. value = 1.12 × 10−1 at g = 12
$i=3$ (69Ga)	$S^{(2)}\left(N_{3,1},{\sigma }_{f,1}^g\right)$ Max. value = 4.72 × 10−3 at g = 12	$S^{(2)}\left(N_{3,1},{\sigma }_{f,2}^g\right)$ Max. value = 2.44 × 10−4 at g = 12
$i=4$ (71Ga)	$S^{(2)}\left(N_{4,1},{\sigma }_{f,1}^g\right)$ Max. value = 2.98 × 10−3 at g = 12	$S^{(2)}\left(N_{4,1},{\sigma }_{f,2}^g\right)$ Max. value = 1.54 × 10−4 at g = 12
$i=5$ (C)	$S^{(2)}\left(N_{5,2},{\sigma }_{f,1}^g\right)$1 element with absolute value >1.0	$S^{(2)}\left(N_{5,2},{\sigma }_{f,2}^g\right)$ Max. value = 6.13 × 10−2 at g = 12
$i=6$ (1H)	$S^{(2)}\left(N_{6,2},{\sigma }_{f,1}^g\right)$ 7 elements with absolute values >1.0	$S^{(2)}\left(N_{6,2},{\sigma }_{f,2}^g\right)$ Max. value = 9.73 × 10−2 at g = 12

summarizes the results for the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{f,k}^g\right)$,$\hspace{0.17em}i=1,\dots ,6;\hspace{0.33em}k,m=1,2;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$, which comprises the 2nd-order relative sensitivities of the leakage response with respect to the isotopic number densities and the fission cross sections, for all isotopes. To facilitate the presentation of the numerical results, the $J_n\times J_{\sigma f}\hspace{0.33em}(=6\times 60)$ matrix $S^{(2)}\left(N_{i,m},{\sigma }_{f,k}^g\right)$ has been partitioned into $J_n\times N_f\hspace{0.33em}(=6\times 2)$ submatrices, each of dimensions $1\times G=1\times 30$. The computational results are as follows: (i) all 360 elements of the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{f,k}^g\right)$ have positive values, and (ii) of the 360 elements, 21 elements have very large relative sensitivities, with absolute values greater than 1.0, as shown in shaded cells in the table. All of these large sensitivities involve the fission cross sections of ²³⁹Pu and most of them relate to the isotopic number densities of ²³⁹Pu or ¹H. Of the sensitivities summarized in Table 19, the single largest relative value is $S^{(2)}\left(N_{1,1},{\sigma }_{f,1}^{12}\right)=11.735$. The results in Table 19 also indicate that, when the 2nd-order mixed relative sensitivities $S^{(2)}\left(N_{i,m},{\sigma }_{f,k}^g\right)$ involve the isotopic number densities of isotopes ⁶⁹Ga and ⁷¹Ga or the microscopic fission cross sections of isotope ²⁴⁰Pu, their absolute values are all smaller than 1.0. The element with the largest value in the respective submatrix is related to the microscopic fission cross sections for the 12th energy group of isotopes ²³⁹Pu and ²⁴⁰Pu.

4.3.1. Second-Order Relative Sensitivities $S^{(2)}\left(N_{1,1},{\sigma }_{f,1}^g\right),g=1,\dots ,30$

The submatrix $S^{(2)}\left(N_{1,1},{\sigma }_{f,1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$, comprises the 2^nd-order sensitivities of the leakage response with respect to the isotopic number density and the fission cross sections of ²³⁹Pu. The 12 elements of this matrix which have values greater than 1.0 are presented in bold in

Table 20. Second-Order Relative Sensitivities $S^{(2)}\left(N_{1,1},{\sigma }_{f,1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$.

g	Relative Sensitivities	g	Relative Sensitivities
1	0.005	16	2.654
2	0.011	17	0.979
3	0.032	18	0.536
4	0.144	19	0.443
5	0.683	20	0.424
6	1.735	21	0.380
7	7.787	22	0.335
8	6.470	23	0.286
9	7.761	24	0.267
10	8.073	25	0.228
11	7.521	26	0.212
12	11.735	27	0.194
13	8.197	28	0.115
14	5.313	29	0.154
15	3.007	30	1.467

. These 12 large 2nd-order mixed relative sensitivities involve the fission cross sections of isotope ²³⁹Pu for the energy groups $g=6,\dots ,16$ and $g=30$, respectively. The element having the largest value in this submatrix is $S^{(2)}\left(N_{1,1},{\sigma }_{f,1}^{12}\right)=11.735$.

4.3.2. Second-Order Relative Sensitivities $S^{(2)}\left(N_{2,1},{\sigma }_{f,1}^g\right),g=1,\dots ,30$

The submatrix $S^{(2)}\left(N_{2,1},{\sigma }_{f,1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$, comprising the 2nd-order sensitivities of the leakage response with respect to the isotopic number density of isotope 2 (²⁴⁰Pu) and the fission cross sections of isotope 1 (²³⁹Pu), contains a single large element that has an absolute value greater than 1.0, which is $S^{(2)}\left(N_{2,1},{\sigma }_{f,1}^{g=12}\right)=1.290$.

4.3.3. Second-Order Relative Sensitivities $S^{(2)}\left(N_{5,2},{\sigma }_{f,1}^g\right),g=1,\dots ,30$

The submatrix $S^{(2)}\left(N_{5,2},{\sigma }_{f,1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$, for the 2nd-order sensitivities of the leakage response with respect to the isotopic number density of isotope 5 (C) and the fission cross sections of isotope 1 (²³⁹Pu), also contains a single large element that has an absolute value greater than 1.0, namely, $S^{(2)}\left(N_{5,2},{\sigma }_{f,1}^{g=12}\right)=1.184$.

4.3.4. Second-Order Relative Sensitivities $S^{(2)}\left(N_{6,2},{\sigma }_{f,1}^g\right),g=1,\dots ,30$

The submatrix $S^{(2)}\left(N_{6,2},{\sigma }_{f,1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$, comprising the 2nd-order sensitivities of the leakage response with respect to the isotopic number density of isotope 6 (¹H) and the fission cross sections of isotope 1 (²³⁹Pu), includes 7 elements that have values greater than 1.0, as listed in

Table 21. Elements of $S^{(2)}\left(N_{6,2},{\sigma }_{f,1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$ with absolute values greater than 1.0.

Group	g = 7	g = 8	g = 9	g = 10	g = 11	g = 12	g = 13
values	1.279	1.061	1.266	1.312	1.217	1.879	1.294

. These 7 relative sensitivities are concentrated in the energy groups $g=7,\dots ,13$ of the fission cross sections for isotope ²³⁹Pu.

5. Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Parameters Underlying the Benchmark’s Isotopic Number Densities and Average Number of Neutrons per Fission

This Section presents the computation and analysis of the numerical results for the 2nd-order mixed sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial \nu$ of the leakage response with respect to the isotopic number densities and the average number of neutrons per fission of all isotopes of the PERP benchmark. These 2nd-order mixed sensitivities can also be computed using the alternative expression ${\partial }^{2}L\left(\alpha \right)/\partial \nu \partial N$. These two alternative paths are presented in Section 5.1 and Section 5.2, respectively.

5.1. Computing the Second-Order Sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial \nu$

The equations needed for deriving the expressions of the 2nd-order sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial \nu$ are obtained by particularizing Equations (160), (169), (179) and (206) in [5] to the PERP benchmark. Specifically, Equation (160) in [5] takes on the following particular form for the PERP benchmark:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f=\nu \right)}^{\left(1\right)}=\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial f_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\right]}{\partial f_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right),\\ for j=1,\dots ,J_n;m_2=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu },\end{array}$$

(143)

where the 2nd-level adjoint functions ${\psi }_{1,j}^{(2),g}$ and ${\psi }_{2,j}^{(2),g},j=1,\dots ,J_{\sigma f};\hspace{0.33em}g=1,\dots ,G$ are the solutions of the 2nd-Level Adjoint Sensitivity System presented previously in Equations (11)–(14). In Equation (143), the parameters $n_j$ correspond to the isotopic number densities, denoted as $n_j\equiv N_{i_j,m_j}$, and the parameters $f_{m_2},\hspace{0.33em}{m}_2=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu }$, denoted as $f_{m_2}\equiv {\nu }_{i_{m_2}}^{g_{m_2}}$, correspond to the respective parameter for average number of neutrons per fission in the vector $\nu \triangleq \hspace{0.17em}\hspace{0.17em}{\left[f_{J_{\sigma f}+1},\dots ,f_{J_{\sigma f}+J_{\nu }}\right]}^{†}\triangleq \hspace{0.17em}{\left[{\nu }_{i=1}^1,{\nu }_{i=1}^2,\dots ,{\nu }_{i=1}^G,\dots ,{\nu }_i^g,\dots ,{\nu }_{i=N_f}^1,\dots ,{\nu }_{i=N_f}^G\right]}^{†}$, for $i=1,\dots ,N_f;\hspace{0.17em}g=1,\dots ,G;\hspace{0.33em}{J}_{\nu }=G\times N_f$, as shown in Part I [1] and Appendix A. Noting that

$$\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\right]}{\partial f_{m_2}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\left(\nu {\sigma }_f\right)}_i^g\right]}{\partial {\nu }_{i_{m_2}}^{g_{m_2}}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\nu }_i^g{\sigma }_{f,i}^g\right]}{\partial {\nu }_{i_{m_2}}^{g_{m_2}}}={\delta }_{g_{m_2}g}{N}_{i_{m_2},m_{m_2}}{\sigma }_{f,i_{m_2}}^g,$$

(144)

$$\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial f_{m_2}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\left(\nu {\sigma }_f\right)}_i^{g^{\prime }}\right]}{\partial {\nu }_{i_{m_2}}^{g_{m_2}}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\nu }_i^{g^{\prime }}{\sigma }_{f,i}^{g^{\prime }}\right]}{\partial {\nu }_{i_{m_2}}^{g_{m_2}}}={\delta }_{g_{m_2}{g}^{\prime }}{N}_{i_{m_2},m_{m_2}}{\sigma }_{f,i_{m_2}}^{g^{\prime }},$$

(145)

and inserting the results obtained in Equations (144) and (145) into Equation (143), yields the following expression for Equation (143):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f=\nu \right)}^{(1)}=N_{i_{m_2},m_{m_2}}{\sigma }_{f,i_{m_2}}^{g_{m_2}}{\displaystyle {\int }_{V}dV}\left[{\xi }_{1,j;0}^{(2),g_{m_2}}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)}+{\phi }_0^{g_{m_2}}(r){\displaystyle \sum _{g=1}^G{\chi }^g{\xi }_{2,j;0}^{(2),g}(r)}\right],\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu }\hspace{0.17em}.\end{array}$$

(146)

The contributions stemming from Equation (169) in [5] to ${\partial }^{2}L/\partial n_j\partial f_{m_2}$ are:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f=\nu \right)}^{\left(2\right)}=\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\right]}{\partial f_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\theta }_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial f_{m_2}},\\ for j=1,\dots ,J_n;m_2=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu },\end{array}$$

(147)

where the 2nd-level adjoint functions ${\theta }_{1,j}^{(2),g}$, and ${\theta }_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G$, are the solutions of the 2nd-Level Adjoint Sensitivity System presented in Equations (19)–(22). Inserting Equations (144) and (145) into Equation (147), yields the following expression for Equation (147):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f=\nu \right)}^{(2)}=N_{i_{m_2},m_{m_2}}{\sigma }_{f,i_{m_2}}^{g_{m_2}}{\displaystyle {\int }_{V}dV}\left[{\mathrm{\Theta }}_{1,j;0}^{(2),g_{m_2}}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)}+{\phi }_0^{g_{m_2}}(r){\displaystyle \sum _{g=1}^G{\chi }^g{\mathrm{\Theta }}_{2,j;0}^{(2),g}(r)}\right],\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu }\hspace{0.17em}.\end{array}$$

(148)

The contributions stemming from Equation (179) in [5] to ${\partial }^{2}L/\partial n_j\partial f_{m_2}$ are given by the following expression:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f=\nu \right)}^{\left(3\right)}=\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial n_j\partial f_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\right]}{\partial f_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial f_{m_2}},\\ for j=1,\dots ,J_n;m_2=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu },\end{array}$$

(149)

where the 2nd-level adjoint functions $u_{1,j}^{(2),g}$, and $u_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.33em}g=1,\dots ,G$, are the solutions of the 2nd-Level Adjoint Sensitivity System presented in Equations (27)–(30). Note that the following relations hold:

$$\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial n_j\partial f_{m_2}}=\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial N_{i_j,m_j}\partial {\nu }_{i_{m_2}}^{g_{m_2}}}=\frac{\partial \left[\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\nu }_i^{g^{\prime }}{\sigma }_{f,i}^{g^{\prime }}\right]/\partial N_{i_j,m_j}\right]}{\partial {\nu }_{i_{m_2}}^{g_{m_2}}}=\frac{\partial \left\{{\nu }_{i_j}^{g^{\prime }}{\sigma }_{f,i_j}^{g^{\prime }}\right\}}{\partial {\nu }_{i_{m_2}}^{g_{m_2}}}={\delta }_{i_j{i}_{m_2}}{\delta }_{g_{m_2}{g}^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}.$$

(150)

Inserting the results obtained in Equations (144), (145) and (150) into Equation (149), yields the following expression for Equation (149):

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f=\nu \right)}^{\left(3\right)}={\delta }_{i_j{i}_{m_2}}{\sigma }_{f,i_{m_2}}^{g_{m_2}}{\int }_{V}dV{\phi }_0^{g_{m_2}}\left(r\right)\sum _{g=1}^G{\chi }^g{\xi }_0^{\left(1\right),g}\left(r\right)\\ +N_{i_{m_2},m_{m_2}}{\sigma }_{f,i_{m_2}}^{g_{m_2}}{\int }_{V}dV\left[U_{1,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)+{\phi }_0^{g_{m_2}}\left(r\right)\sum _{g=1}^G{\chi }^g{U}_{2,j;0}^{\left(2\right),g}\left(r\right)\right],\\ for j=1,\dots ,J_n;m_2=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu }.\end{array}$$

(151)

Additional contributions stemming from the sources are computed by particularizing Equation (206) in [5] to the PERP benchmark, which yields the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f=\nu \right)}^{(4)}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathsf{\Omega }}\hspace{0.33em}{g}_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(f\right)\right]}{\partial f_{m_2}}{\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }}{\chi }^{g^{\prime }}{\psi }^{(1),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)}},\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu }\hspace{0.17em},\end{array}$$

(152)

where the 2nd-level adjoint functions $g_{1,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G$, are the solutions of the 2nd-LASS presented previously in Equations (35) and (36). Inserting the results obtained in Equations (144) into Equation (152) yields the following expression for Equation (152):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f=\nu \right)}^{(4)}=N_{i_{m_2},m_{m_2}}{\sigma }_{f,i_{m_2}}^{g_{m_2}}{\displaystyle {\int }_{V}dV}{G}_{1,j;0}^{(2),g_{m_2}}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)},\hspace{0.33em}\\ for\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_2=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu }\hspace{0.17em}.\end{array}$$

(153)

Collecting the partial contributions obtained in Equations (146), (148), (151) and (153), yields the following result:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f=\nu \right)}=\sum _{i=1}^4{\left(\frac{{\partial }^{2}L}{\partial n_j\partial f_{m_2}}\right)}_{\left(n=N,f=\nu \right)}^{\left(i\right)}\\ =N_{i_{m_2},m_{m_2}}{\sigma }_{f,i_{m_2}}^{g_{m_2}}{\int }_{V}dV\left[{\xi }_{1,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)+{\phi }_0^{g_{m_2}}\left(r\right)\sum _{g=1}^G{\chi }^g{\xi }_{2,j;0}^{\left(2\right),g}\left(r\right)\right]\\ +N_{i_{m_2},m_{m_2}}{\sigma }_{f,i_{m_2}}^{g_{m_2}}{\int }_{V}dV\left[{\mathrm{\Theta }}_{1,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)+{\phi }_0^{g_{m_2}}\left(r\right)\sum _{g=1}^G{\chi }^g{\mathrm{\Theta }}_{2,j;0}^{\left(2\right),g}\left(r\right)\right]\\ +{\delta }_{i_j{i}_{m_2}}{\sigma }_{f,i_{m_2}}^{g_{m_2}}{\int }_{V}dV{\phi }_0^{g_{m_2}}\left(r\right)\sum _{g=1}^G{\chi }^g{\xi }_0^{\left(1\right),g}\left(r\right)\\ +N_{i_{m_2},m_{m_2}}{\sigma }_{f,i_{m_2}}^{g_{m_2}}{\int }_{V}dV\left[U_{1,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)+{\phi }_0^{g_{m_2}}\left(r\right)\sum _{g=1}^G{\chi }^g{U}_{2,j;0}^{\left(2\right),g}\left(r\right)\right]\\ +N_{i_{m_2},m_{m_2}}{\sigma }_{f,i_{m_2}}^{g_{m_2}}{\int }_{V}dV{G}_{1,j;0}^{\left(2\right),g_{m_2}}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right),for j=1,\dots ,J_n;m_2=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu }.\end{array}$$

(154)

5.2. Alternative Path: Computing the Second-Order Sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial \nu \partial N$

Due to symmetry of the mixed 2nd-order sensitivities, the results to be computed using the expressions for ${\partial }^{2}L\left(\alpha \right)/\partial N\partial \nu$ obtained in Equation (154) can be verified by obtaining and using the expressions for ${\partial }^{2}L\left(\alpha \right)/\partial \nu \partial N$. The equations needed for deriving the expression of the 2nd-order mixed sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial \nu \partial N$ are obtained by particularizing Equations (177), (178), (179) and (181) in [5] to the PERP benchmark, which yields:

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial f_j\partial n_{m_2}}\right)}_{\left(f=\nu ,n=N\right)}=-\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[u_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]\frac{\partial {{\Sigma }_t}^g\left(\mathbf{t}\right)}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime })}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathsf{\Omega }}^{\prime }\to \mathsf{\Omega })}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial f_j\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\right]}{\partial n_{m_2}}\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\chi }^{g^{\prime }}{\psi }^{\left(1\right),g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\sum _{g^{\prime }=1}^G{\int }_{4\pi }d{\mathsf{\Omega }}^{\prime }{\phi }^{g^{\prime }}\left(r,{\mathsf{\Omega }}^{\prime }\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial n_{m_2}}\\ +\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }{u}_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right)\frac{\partial Q^g\left(\mathbf{q};r,\mathsf{\Omega }\right)}{\partial n_{m_2}},for j=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu };m_2=1,\dots ,J{}_n,\end{array}$$

(155)

where the 2nd-level adjoint functions, $u_{1,j}^{(2),g}$ and $u_{2,j}^{(2),g},j=1,\dots ,J_{\sigma f};\hspace{0.33em}g=1,\dots ,G,$ are the solutions of the 2nd-Level Adjoint Sensitivity System presented in Equations (116), (118), (124) and (125) of Part III [3], which are reproduced below for convenient reference:

$$B^g\left({\alpha }^0\right)u_{1,j}^{(2),g}\left(r,\mathsf{\Omega }\right)=N_{i_j,m_j}{\sigma }_{f,i_j}^{g_j}{\chi }^g{\phi }_0^{g_j}\left(r\right),\hspace{0.17em}\hspace{0.17em}j=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu };\hspace{0.33em}g=1,\dots ,G,$$

(156)

$$u_{1,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n<0;\hspace{0.33em}j=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu };\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(157)

$$A^{(1),g}\left({\alpha }^0\right)u_{2,j}^{(2),g}\left(r,\mathsf{\Omega }\right)={\delta }_{g_{j}g}{N}_{i_j,m_j}{\sigma }_{f,i_j}^{g_j}{\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu };\hspace{0.33em}g=1,\dots ,G,$$

(158)

$$u_{2,j}^{(2),g}\left(r_d,\mathsf{\Omega }\right)=0,\hspace{0.33em}\mathsf{\Omega }\cdot n>0;\hspace{0.33em}j=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu };\hspace{0.33em}g=1,\dots ,G.$$

(159)

In Equation (155), the parameters $f_j$ and $n_{m_2}$ correspond to the average number of neutrons per fission and the isotopic number densities, denoted as $f_j\equiv {\nu }_{i_j}^{g_j}$ and $n_{m_2}\equiv N_{i_{m_2},m_{m_2}}$, respectively. The following relations hold:

$$\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial f_j\partial n_{m_2}}=\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\right]}{\partial {\nu }_{i_j}^{g_j}\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial \left[\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\nu }_i^{g^{\prime }}{\sigma }_{f,i}^{g^{\prime }}\right]/\partial {\nu }_{i_j}^{g_j}\right]}{\partial N_{i_{m_2},m_{m_2}}}=\frac{\partial \left({\delta }_{g_j{g}^{\prime }}{N}_{i_j,m_j}{\sigma }_{f,i_j}^{g^{\prime }}\right)}{\partial N_{i_{m_2},m_{m_2}}}={\delta }_{i_j{i}_{m_2}}{\delta }_{g_j{g}^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}.$$

(160)

Inserting the results obtained in Equations (47)–(52) and (160) into Equation (155) and performing the respective angular integrations yields the following expression for Equation (155):

$$\begin{array}{c}{\left(\frac{{\partial }^{2}L}{\partial f_j\partial n_{m_2}}\right)}_{\left(f=\nu ,n=N\right)}=-\sum _{g=1}^G{\int }_{V}dV{\int }_{4\pi }d\mathsf{\Omega }\left[u_{1,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\psi }^{\left(1\right),g}\left(r,\mathsf{\Omega }\right)+u_{2,j}^{\left(2\right),g}\left(r,\mathsf{\Omega }\right){\phi }^g\left(r,\mathsf{\Omega }\right)\right]{\sigma }_{t,i_{m_2}}^g\\ +\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{U}_{1,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{\left(1\right),g^{\prime }}\left(r\right)+\sum _{g=1}^G\sum _{l=0}^{ISCT}\left(2l+1\right){\int }_{V}dV{U}_{2,j;l}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)\\ +{\delta }_{i_j{i}_{m_2}}{\sigma }_{f,i_{m_2}}^{g_j}{\int }_{V}dV{\phi }_0^{g_j}\left(r\right)\sum _{g=1}^G{\chi }^g{\xi }_0^{\left(1\right),g}\left(r\right)+\sum _{g=1}^G{\int }_{V}dV{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{U}_{1,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\chi }^{g^{\prime }}{\xi }_0^{\left(1\right),g^{\prime }}\left(r\right)\\ +\sum _{g=1}^G{\int }_{V}dV{\chi }^g{U}_{2,j;0}^{\left(2\right),g}\left(r\right)\sum _{g^{\prime }=1}^G{\nu }_{i_{m_2}}^{g^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)+\frac{1}{n_{m_2}}\sum _{g=1}^G{\int }_{V}dV{U}_{2,j;0}^{\left(2\right),g}\left(r\right)Q_{SF,i_{m_2}}^g,\\ for j=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu };m_2=1,\dots ,J{}_n.\end{array}$$

(161)

5.3. Numerical Results for ${\partial }^{2}L\left(\alpha \right)/\partial N\partial \nu$

The second-order absolute sensitivities, ${\partial }^{2}L\left(\alpha \right)/\partial N\partial \nu$, of the leakage response with respect to the isotopic number densities and the average number of neutrons per fission for all isotopes of the PERP benchmark have been computed using Equation (154) and have been independently verified by computing ${\partial }^{2}L\left(\alpha \right)/\partial \nu \partial N$ using Equation (161). Computing the second-order absolute sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial \nu$ using Equation (154) requires 16 forward and adjoint PARTISN computations to obtain all of the required 2nd-level adjoint functions. On the other hand, computing the alternative expression ${\partial }^{2}L\left(\alpha \right)/\partial \nu \partial N$ using Equation (161) requires 60 forward and adjoint PARTISN computations to obtain the second-level adjoint functions required in Equation (161). Thus, computing ${\partial }^{2}L\left(\alpha \right)/\partial N\partial \nu$ using Equation (154) is about 4 times more efficient than computing ${\partial }^{2}L\left(\alpha \right)/\partial \nu \partial N$ by using Equation (161).

The matrix ${\partial }^{2}L/\partial n_j\partial f_{m_2},\hspace{0.33em}j=1,\dots ,J{}_n;\hspace{0.33em}{m}_2=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu }\hspace{0.17em}$ has dimensions $J_n\times J_{\nu }\hspace{0.33em}(=6\times 60)$, where $J_{\nu }=G\times N_f=30\times 2$. The matrix of 2nd-order relative sensitivities corresponding to ${\partial }^{2}L/\partial n_j\partial f_{m_2},\hspace{0.33em}j=1,\dots ,J{}_n;\hspace{0.33em}{m}_2=J_{\sigma f}+1,\dots ,J_{\sigma f}+J_{\nu }\hspace{0.17em}$, is denoted as $S^{(2)}\left(N_{i,m},{\nu }_k^g\right)$ and is defined as follows:

$$S^{(2)}\left(N_{i,m},{\nu }_k^g\right)\triangleq \frac{{\partial }^{2}L}{\partial N_{i,m}\partial {\nu }_k^g}\left(\frac{N_{i,m}{\nu }_k^g}{L}\right),\hspace{0.17em}\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\dots ,6;\hspace{0.33em}m=1,2;\hspace{0.33em}k=1,2;\hspace{0.17em}g=1,\dots ,30.$$

(162)

Table 22. Summary presentation of the matrix $S^{(2)}\left(N_{i,m},{\nu }_k^g\right)\hspace{0.17em},\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}i=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}k,m=1,2;\hspace{0.33em}g=1,\dots ,30$.

	$$\begin{gathered} \mathit{k}=1 \\ {(}^{239}\mathbf{Pu}) \end{gathered}$$	$$\begin{gathered} \mathit{k}=2 \\ {(}^{240}\mathbf{Pu}) \end{gathered}$$
$i=1\hspace{0.33em}$ (239Pu)	$S^{(2)}\left(N_{1,1},{\nu }_{k=1}^g\right)$ 13 elements with absolute values >1.0	$S^{(2)}\left(N_{1,1},{\nu }_{k=2}^g\right)$ Max. value = 7.72 × 10−1 at g = 12
$i=2$ (240Pu)	$S^{(2)}\left(N_{2,1},{\nu }_{k=1}^g\right)$ 6 elements with absolute values >1.0	$S^{(2)}\left(N_{2,1},{\nu }_{k=2}^g\right)$ Max. value = 1.55 × 10−1 at g = 12
$i=3$ (69Ga)	$S^{(2)}\left(N_{3,1},{\nu }_{k=1}^g\right)$ Max. value = 6.52 × 10−3 at g = 12	$S^{(2)}\left(N_{3,1},{\nu }_{k=2}^g\right)$ Max. value = 3.39 × 10−4 at g = 12
$i=4$ (71Ga)	$S^{(2)}\left(N_{4,1},{\nu }_{k=1}^g\right)$ Max. value = 4.11 × 10−3 at g = 12	$S^{(2)}\left(N_{4,1},{\nu }_{k=2}^g\right)$ Max. value = 2.14 × 10−4 at g = 12
$i=5$ (C)	$S^{(2)}\left(N_{5,2},{\nu }_{k=1}^g\right)$ 6 elements with absolute values >1.0	$S^{(2)}\left(N_{5,2},{\nu }_{k=2}^g\right)$ Max. value = 8.52 × 10−2 at g = 12
$i=6$ (1H)	$S^{(2)}\left(N_{6,2},{\nu }_{k=1}^g\right)$ 9 elements with absolute values >1.0	$S^{(2)}\left(N_{6,2},{\nu }_{k=2}^g\right)$ Max. value = 1.35 × 10−1 at g = 12

summarizes the results obtained for the elements of the matrix $S^{(2)}\left(N_{i,m},{\nu }_k^g\right)$,$\hspace{0.17em}i=1,\dots ,6;\hspace{0.33em}k,m=1,2;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$, for the 2nd-order relative sensitivities of the leakage response with respect to the isotopic number densities and the average number of neutrons per fission for all isotopes. To facilitate the presentation of the numerical results, the $J_n\times J_{\nu }\hspace{0.33em}(=6\times 60)$ matrix $S^{(2)}\left(N_{i,m},{\nu }_k^g\right)$ has been partitioned into $J_n\times N_f\hspace{0.33em}(=6\times 2)$ submatrices, each of dimensions $1\times G=1\times 30$. The computational results have shown that the majority (358 out of 360) of the elements in the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{f,k}^g\right)$ have positive values; only 2 elements have very small negative values, of the order of 10⁻⁴ and less. As shown in shaded cells in Table 22, 34 among the 360 components of $S^{(2)}\left(N_{i,m},{\nu }_k^g\right)$ have very large relative sensitivities, with absolute values greater than 1.0. All of these large sensitivities involve the average number of neutrons per fission of isotope ²³⁹Pu and relate to the isotopic number densities of isotopes ²³⁹Pu, ²⁴⁰Pu, C or ¹H, respectively. The overall largest relative value in the matrix $S^{(2)}\left(N_{i,m},{\nu }_k^g\right)$ is $S^{(2)}\left(N_{1,1},{\nu }_1^{12}\right)=16.06$. The computed results have also shown that the 2^nd-order mixed relative sensitivities $S^{(2)}\left(N_{i,m},{\nu }_k^g\right)$ involving the isotopic number densities of isotopes ⁶⁹Ga and ⁷¹Ga or the average number of neutrons per fission of ²⁴⁰Pu have absolute values smaller than 1.0; the element with the largest value in the respective submatrix is related to the average number of neutrons per fission for the 12th energy group of isotopes ²³⁹Pu and ²⁴⁰Pu.

5.3.1. Second-Order Relative Sensitivities $S^{(2)}\left(N_{1,1},{\nu }_{k=1}^g\right),g=1,\dots ,30$

The submatrix $S^{(2)}\left(N_{1,1},{\nu }_{k=1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$ comprises the 2nd-order sensitivities of the leakage response with respect to the isotopic number density and the average number of neutrons per fission of ²³⁹Pu.

Table 23. Elements of $S^{(2)}\left(N_{1,1},{\nu }_{k=1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$, having absolute values greater than 1.0.

Group	g = 6	7	8	9	10	11	12	13	14	15	16	17	30
values	2.267	10.10	8.675	10.53	11.07	10.34	16.06	11.30	7.458	4.330	3.987	1.535	5.217

presents the 13 elements of this submatrix that have values greater than 1.0; these large 2nd-order mixed relative sensitivities are concentrated in energy groups $g=6,\dots ,17$ of the average number of neutrons per fission of isotope ²³⁹Pu. The largest value in this submatrix is $S^{(2)}\left(N_{1,1},{\nu }_1^{12}\right)=16.06$.

5.3.2. Second-Order Relative Sensitivities $S^{(2)}\left(N_{2,1},{\nu }_{k=1}^g\right),g=1,\dots ,30$

The submatrix $S^{(2)}\left(N_{2,1},{\nu }_{k=1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$, comprising the 2nd-order sensitivities of the leakage response with respect to the isotopic number density of isotope 2 (²⁴⁰Pu) and the average number of neutrons per fission of isotope 1 (²³⁹Pu), contains 6 large elements that have values greater than 1.0, as listed in

Table 24. Elements of $S^{(2)}\left(N_{2,1},{\nu }_{k=1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$ with absolute values greater than 1.0.

Group	g = 7	g = 9	g = 10	g = 11	g = 12	g = 13
values	1.107	1.162	1.221	1.142	1.772	1.248

.

5.3.3. Second-Order Relative Sensitivities $S^{(2)}\left(N_{5,2},{\nu }_{k=1}^g\right),g=1,\dots ,30$

The submatrix $S^{(2)}\left(N_{5,2},{\sigma }_{f,1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$ comprises the 2nd-order sensitivities of the leakage response with respect to the isotopic number density of isotope C and the average number of neutrons per fission of isotope ²³⁹Pu.

Table 25. Elements of $S^{(2)}\left(N_{5,2},{\nu }_{k=1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$ with absolute values greater than 1.0.

Group	g = 7	g = 9	g = 10	g = 11	g = 12	g = 13
values	1.016	1.083	1.135	1.056	1.638	1.159

presents the 6 elements of this submatrix which have values greater than 1.0.

5.3.4. Second-Order Relative Sensitivities $S^{(2)}\left(N_{6,2},{\nu }_{k=1}^g\right),g=1,\dots ,30$

The submatrix $S^{(2)}\left(N_{6,2},{\nu }_{k=1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$, comprising the 2^nd-order sensitivities of the leakage response with respect to the isotopic number density of ¹H and to the average number of neutrons per fission of isotope ²³⁹Pu, includes 9 elements that have values greater than 1.0, as listed in

Table 26. Elements of $S^{(2)}\left(N_{6,2},{\nu }_{k=1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$ with absolute values greater than 1.0.

Group	g = 7	g = 8	g = 9	g = 10	g = 11	g = 12	g = 13	g = 14	g = 30
values	1.660	1.424	1.723	1.809	1.687	2.605	1.815	1.188	1.930

. These 9 relative sensitivities are concentrated in the energy groups $g=7,\dots ,14$ and $g=30$ of the average number of neutrons per fission of isotope ²³⁹Pu.

6. Discussion and Conclusions

The following conclusions can be drawn from the results for the mixed 2nd-order sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_t$, ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_s$, ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_f$ and ${\partial }^{2}L\left(\alpha \right)/\partial N\partial \nu$ reported in this work:

The 2nd-order mixed sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_t$ are mostly negative. Almost all, namely 1072 out of the $J_n\times J_{\sigma t}\hspace{0.33em}(=1080)$ elements in the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{t,k}^g\right)\hspace{0.17em},\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}m=1,2;\hspace{0.33em}g=1,\dots ,30$ of 2nd-order mixed sensitivities have negative values; only 8 elements have very small positive values (e.g., in the order of 10⁻⁴ or less). Among the 1080 elements in the matrix, 125 elements have very large relative sensitivities, with absolute values greater than 1.0. Majority of those large sensitivities involve the isotopic number densities of isotopes ²³⁹Pu or ¹H (namely, $N_{1,1}$ and $N_{6,2}$) and/or the microscopic total cross sections of isotopes ²³⁹Pu or ¹H (namely, ${\sigma }_{t,1}^g$ and ${\sigma }_{t,6}^g$). In the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{t,k}^g\right)\hspace{0.17em}$, the single largest relative value is $S^{(2)}\left(N_{1,1},{\sigma }_{t,6}^{30}\right)=-94.91$. Moreover, the element with the most negative value in each of the submatrices mostly involves the microscopic total cross sections for the 12th energy group or the 30th energy group (i.e., ${\sigma }_{t,k}^{12},k=1,\dots ,4$ or ${\sigma }_{t,k}^{30},\hspace{0.33em}k=5,6$) of the respective isotopes.

The 2nd-order mixed relative sensitivities corresponding to the $J_n\times J_{\sigma s}=6\times 21600$-dimensional matrix ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_s$ are generally very small, with a few exceptions. Specifically, 25 of the $J_n\times J_{\sigma s}=6\times 21600$ elements have relative sensitivities with absolute values greater than 1.0. These 25 large elements belong to the submatrices $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=0,1}^{g^{\prime }\to g}\right)$, $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=0,6}^{g^{\prime }\to g}\right)$, $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=1,1}^{g^{\prime }\to g}\right)$ and $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=1,6}^{g^{\prime }\to g}\right)$, respectively, and involve the isotopic number density $N_{1,1}$ of ²³⁹Pu and the 0th-order or 1st-order scattering cross sections (namely, ${\sigma }_{s,l=0,k=1}^{g^{\prime }\to g},{\sigma }_{s,l=0,k=6}^{g^{\prime }\to g},{\sigma }_{s,l=1,k=1}^{g^{\prime }\to g},{\sigma }_{s,l=1,k=6}^{g^{\prime }\to g}$) of ²³⁹Pu or ¹H. These large sensitivities are positive when involving even-order $\left(l=0,2\right)$ scattering cross sections but are negative when involving odd-order $\left(l=1,3\right)$ scattering cross sections. Furthermore, the larger the Legendre expansion order $\left(l=0,\dots ,3\right)$, the smaller the absolute values of the corresponding mixed 2nd-order relative sensitivities. Noteworthy for the 2nd-order mixed sensitivities ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_s$ is also the observation that for the scattering order $l=0$, most of the largest absolute values of the respective submatrix occur at the 0th-order self-scattering cross sections for the 12th energy group for isotopes ²³⁹Pu, ²⁴⁰Pu, ⁶⁹Ga, ⁷¹Ga and C (i.e., ${\sigma }_{s,l=0,k}^{12\to 12},\hspace{0.33em}k=1,\dots ,5$), or the 0th-order out-scattering cross section between energy groups $g^{\prime }=12\to g=13$ of isotope ¹H (namely, ${\sigma }_{s,l=0,6}^{12\to 13}$). On the other hand, when the scattering order $l=1,2,3$, these large sensitivities mostly involve the $l$^th-order self-scattering cross sections for the 7th energy group of isotopes ²³⁹Pu, ²⁴⁰Pu, ⁶⁹Ga, ⁷¹Ga and C (namely, ${\sigma }_{s,l,k}^{7\to 7},\hspace{0.33em}l=1,2,3;\hspace{0.33em}k=1,\dots ,5$), or 12th energy group of isotope ¹H (namely, ${\sigma }_{s,l,k=6}^{12\to 12},\hspace{0.33em}l=1,2,3$). The overall largest 2nd-order mixed relative sensitivity involving an isotopic number density and a scattering cross section is $S^{(2)}\left(N_{1,1},{\sigma }_{s,l=0,k=1}^{12\to 12}\right)=1.912$.

All values of the 2nd-order mixed relative sensitivities $S^{(2)}\left(N_{i,m},{\sigma }_{f,k}^g\right)\hspace{0.17em},\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\dots ,6;\hspace{0.33em}k,m=1,2;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$ corresponding to the $J_n\times J_{\sigma f}\hspace{0.33em}(=6\times 60)$ elements of the matrix ${\partial }^{2}L\left(\alpha \right)/\partial N\partial {\sigma }_f$ are positive. The matrix $S^{(2)}\left(N_{i,m},{\sigma }_{f,k}^g\right)$ comprises 21 elements which have values greater than 1.0. Most of these elements belong to the submatrices $S^{(2)}\left(N_{1,1},{\sigma }_{f,1}^g\right)$ and $S^{(2)}\left(N_{6,2},{\sigma }_{f,1}^g\right)$, involving the fission cross sections of ²³⁹Pu and the isotopic number densities of ²³⁹Pu or ¹H. The 2nd-order mixed relative sensitivities involving the isotopic number densities of ²⁴⁰Pu, ⁶⁹Ga, ⁷¹Ga and C (i.e.,$N_{2,1},N_{3,1},\hspace{0.33em}{N}_{4,1},\hspace{0.33em}{N}_{5,2}$) or the microscopic fission cross sections ${\sigma }_{f,\hspace{0.33em}2}^g,\hspace{0.33em}g=1,\dots ,30$ of ²⁴⁰Pu are generally smaller than 1.0. The largest element of the matrix $S^{(2)}\left(N_{i,m},{\sigma }_{f,k}^g\right)$ is $S^{(2)}\left(N_{1,1},{\sigma }_{f,1}^{12}\right)=11.735$.

The majority of the elements belonging to the $J_n\times J_{\nu }\hspace{0.33em}(=6\times 60)$-dimensional matrix $S^{(2)}\left(N_{i,m},{\nu }_k^g\right)\hspace{0.17em},\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\dots ,6;\hspace{0.33em}k,m=1,2;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30,$ of 2nd-order mixed relative sensitivities corresponding to the matrix ${\partial }^{2}L\left(\alpha \right)/\partial N\partial \nu$ have positive values. The matrix $S^{(2)}\left(N_{i,m},{\nu }_k^g\right)$ comprises 34 elements that have relative sensitivities greater than 1.0. These large sensitivities occur in the submatrices $S^{(2)}\left(N_{1,1},{\nu }_{k=1}^g\right)$, $S^{(2)}\left(N_{2,1},{\nu }_{k=1}^g\right)$, $S^{(2)}\left(N_{5,2},{\nu }_{k=1}^g\right)$ and $S^{(2)}\left(N_{6,2},{\nu }_{k=1}^g\right)$, which involve the average number of neutrons per fission of isotope ²³⁹Pu (i.e., ${\nu }_{k=1}^g$) and the isotopic number densities of isotopes ²³⁹Pu, ²⁴⁰Pu, C or ¹H (i.e., $N_{1,1},\hspace{0.33em}{N}_{2,1},N_{5,2},N_{6,2}$), respectively. The remaining 2^nd-order mixed relative sensitivities in the matrix $S^{(2)}\left(N_{i,m},{\nu }_k^g\right)$ are all smaller than 1.0. The largest values among these smaller sensitivities involve the average number of neutrons per fission, ${\nu }_k^{12},\hspace{0.33em}k=1,2$, for the 12th energy group of ²³⁹Pu and ²⁴⁰Pu. The overall largest 2nd-order mixed relative sensitivities comprised in the matrix $S^{(2)}\left(N_{i,m},{\nu }_k^g\right)$ is $S^{(2)}\left(N_{1,1},{\nu }_1^{12}\right)=16.06$.

Subsequent work will report the values and analyze the effects of the 1st-order and unmixed 2nd-order sensitivities of the PERP’s leakage response with respect to the imprecisely known isotopic number densities [13], and the 1st-order sensitivities of the leakage response with respect to the imprecisely known fission spectrum parameters [13]. The overall impact of 1st- and 2nd-order sensitivities on the response uncertainties will also be highlighted [13].