Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark. VI: Overall Impact of 1st- and 2nd-Order Sensitivities on Response Uncertainties

Abstract

This work applies the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to compute the 1st-order and unmixed 2nd-order sensitivities of a polyethylene-reflected plutonium (PERP) benchmark’s leakage response with respect to the benchmark’s imprecisely known isotopic number densities. The numerical results obtained for these sensitivities indicate that the 1st-order relative sensitivity to the isotopic number densities for the two fissionable isotopes have large values, which are comparable to, or larger than, the corresponding sensitivities for the total cross sections. Furthermore, several 2nd-order unmixed sensitivities for the isotopic number densities are significantly larger than the corresponding 1st-order ones. This work also presents results for the first-order sensitivities of the PERP benchmark’s leakage response with respect to the fission spectrum parameters of the two fissionable isotopes, which have very small values. Finally, this work presents the overall summary and conclusions stemming from the research findings for the total of 21,976 first-order sensitivities and 482,944,576 second-order sensitivities with respect to all model parameters of the PERP benchmark, as presented in the sequence of publications in the Special Issue of Energies dedicated to “Sensitivity Analysis, Uncertainty Quantification and Predictive Modeling of Nuclear Energy Systems”.

1. Introduction

In Parts I−V [1,2,3,4,5], which are precursors of this work, the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) conceived by Cacuci [6,7,8,9] has been successfully applied to the subcritical polyethylene-reflected plutonium (acronym: PERP) metal fundamental physics benchmark [10] to compute the exact values of the sensitivities of the PERP’s benchmark leakage response with respect to the PERP model parameters, as follows:

(i)

The 1st- and 2nd-order unmixed and mixed response sensitivities to the 180 group-averaged total microscopic cross sections [1];

(ii)

The 1st- and 2nd-order unmixed and mixed response sensitivities to the 21,600 group-averaged scattering microscopic cross sections [2];

(iii)

The 1st- and 2nd-order unmixed and mixed response sensitivities to the 120 fission process parameters [3];

(iv)

The 1st- and 2nd-order unmixed and mixed response sensitivities to the 10 source parameters [4]; and

(v)

The 2nd-order mixed response sensitivities involving the isotopic number densities and the other parameters of the PERP benchmark [5].

Although the geometrical dimensions and material composition of the PERP benchmark have been detailed in Part I [1], they are summarized for convenient reference in

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

.

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 total, scattering and fission macroscopic cross sections, as well as the source term of the neutron transport equation. The 1st-order and 2nd-order sensitivities of the leakage response with respect to the isotopic number densities will be computed by specializing the general expressions derived by Cacuci [6,7,8] to the PERP benchmark, and summarizing the respective contributions stemming from the total, scattering and fission macroscopic cross sections, and the sources, respectively. In addition, this work also presents results for the 1st-order sensitivities of the leakage response with respect to the fission spectrum parameters of the two fissionable isotopes, ²³⁹Pu and ²⁴⁰Pu, contained in the PERP benchmark.

The 2nd-order sensitivity analysis of the PERP benchmark is completed and concluded in this work, which is structured as follows: Section 2 presents the computational results for the six first-order sensitivities of the leakage response with respect to the benchmark’s isotopic number densities. Section 3 presents the 6 × 6 second-order unmixed and mixed sensitivities of the leakage response solely with respect to the benchmark’s isotopic number densities. Section 4 illustrates the impact of the 1st- and 2nd-order sensitivities on the uncertainties induced in the leakage response by the imprecisely known isotopic number densities. Section 5 reports the numerical results for the 60 first-order sensitivities to the fission spectrum. Finally, Section 6 highlights the overall conclusions arising from the pioneering computations presented in [1,2,3,4,5] and in this work of the 21,976 first-order sensitivities and 482,944,576 second-order sensitivities of the PERP benchmark’s leakage response with respect to the imprecisely known model parameters of the PERP benchmark.

2. Computation of 1st-Order Sensitivities of the PERP Leakage Response to Isotopic Number Densities

As described in Part I [1], the neutron flux is computed by solving numerically the neutron transport equation using the PARTISN [11] multigroup discrete ordinates transport code with a spontaneous fission source provided by the code SOURCES4C [12] and employing the MENDF71X [13] 618-group cross sections collapsed to $G=30$ energy groups, as detailed in [1]. For the PERP benchmark under consideration, PARTISN [11] solves the following multi-group approximation of the neutron transport equation:

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

(1)

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

(2)

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

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

(3)

$$Q^g\left(r\right)\triangleq {\displaystyle \sum _{k=1}^{N_f}{\lambda }_k{N}_{k,1}{F}_k^{SF}{\nu }_k^{SF}\left(\frac{2}{\sqrt{\pi a_k{}^3{b}_k}}{e}^{-\frac{a_k{b}_k}{4}}\right){\displaystyle {\int }_{E^{g+1}}^{E^g}dE\hspace{0.17em}{e}^{-E/a_k}\sinh \sqrt{b_{k}E}}}.$$

(4)

The vector of model parameters $\mathsf{\alpha }\triangleq {\left[{\mathsf{\sigma }}_t;{\mathsf{\sigma }}_s;{\mathsf{\sigma }}_f;\mathsf{\nu };\mathbf{p};\mathbf{q};\mathbf{N}\right]}^{†}$, which appears in Equation (1), has been defined in previous works [1,2,3,4,5] and is reproduced, for convenience, in Appendix A. The total neutron leakage from the PERP sphere is denoted [1,2,3,4,5] as $L\left(\mathsf{\alpha }\right)$ and is defined [1,2,3,4,5] as follows:

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

(5)

where $S_b$ is the external surface area of the PERP ball.

Of course, L(α) depends on the imprecisely known model parameters through the neutron flux. For convenient reference, the histogram plot of the leakage for each energy group for the PERP benchmark is reproduced [1,2,3,4] in Figure 1. The value of the total leakage computed using Equation (5) for the PERP benchmark is 1.7648 × 10⁶ neutrons/sec.

The PERP benchmark comprises six isotopes ($I=6$) and two materials ($M=2$). Since the respective isotopes are all distinct materials, as specified in Table 1, it follows that only the following isotopic number densities exist for this benchmark: $N_{1,1},N_{2,1},N_{3,1},N_{4,1},N_{5,2},N_{6,2}$, which implies that:

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

(6)

The isotopic number densities are components of the vectors $\mathbf{t},\hspace{0.17em}\mathbf{s},\hspace{0.17em}\mathbf{f}$ and $\mathbf{q}$, respectively, as defined in Appendix A. Therefore, the vector of first-order sensitivity of the PERP leakage response to the isotopic number densities, which will be denoted as $\partial L\left(\mathsf{\alpha }\right)/\partial \mathbf{N}$, comprises the following four components:

$$\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial n_j}={\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial t_{J_{\sigma t}+j}}\right]}_{t=N}+{\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial s_{J_{\sigma s}+j}}\right]}_{s=N}+{\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial f_{J_{\sigma f}+J_{\nu }+j}}\right]}_{f=N}+{\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial q_{J_q+j}}\right]}_{q=N},\hspace{0.17em}j=1,\dots ,J_n.$$

(7)

The contributions arising from the macroscopic total cross sections are computed using the following particular form of Equation (150) derived in [7]:

$${\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial n_j}\right]}^{(1)}={\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial t_{J_{\sigma t}+j}}\right]}_{t=N}=-{\displaystyle \sum _{g=1}^G{\displaystyle \int dV}{\displaystyle \underset{4\pi }{\int }d\mathbf{\Omega }\hspace{0.17em}}{\psi }^{\left(1\right),g}\left(r,\mathbf{\Omega }\right){\phi }^g\left(r,\mathbf{\Omega }\right)\frac{\partial {\Sigma }_t^g\left(\mathbf{t}\right)}{\partial n_j}\hspace{0.17em}\hspace{0.17em}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n.$$

(8)

The multigroup adjoint fluxes ${\psi }^{\left(1\right),g}\left(r,\mathbf{\Omega }\right),\hspace{0.17em}g=1,\dots ,G$, appearing in Equation (8) are the solutions of the following 1st-Level Adjoint Sensitivity System (1st-LASS) presented in Equations (156) and (157) of [7]:

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

(9)

$${\psi }^{\left(1\right),g}\left(r_d,\mathbf{\Omega }\right)=0,\mathbf{\Omega }\cdot \mathbf{n}>0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(10)

where the adjoint operator $A^{\left(1\right),g}\left(\mathsf{\alpha }\right)$ takes on the following particular form of Equation (149) in [7]:

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

(11)

For the PERP benchmark, the parameter $n_j,\hspace{0.17em}j=1,\dots ,J_n$ corresponds to the isotopic number density, i.e., $n_j\equiv N_{i_j,m_j}$, where the subscripts $i_j$ and $m_j$ denote, respectively, the isotope and material associated with the parameter $n_j$. Hence, the following relation holds:

$$\frac{\partial {\Sigma }_t^g\left(\mathbf{t}\right)}{\partial n_j}=\frac{\partial {\Sigma }_t^g\left(\mathbf{t}\right)}{\partial N_{i_j,m_j}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\sigma }_{t,i}^g\right]}{\partial N_{i_j,m_j}}={\sigma }_{t,i_j}^g.$$

(12)

Inserting Equation (12) into Equation (8) yields:

$${\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial n_j}\right]}^{(1)}={\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial t_{J_{\sigma t}+j}}\right]}_{t=N}=-{\displaystyle \sum _{g=1}^G{\displaystyle \int dV}{\sigma }_{t,i_j}^g{\displaystyle \underset{4\pi }{\int }d\mathbf{\Omega }\hspace{0.17em}}{\psi }^{\left(1\right),g}\left(r,\mathbf{\Omega }\right){\phi }^g\left(r,\mathbf{\Omega }\right)\hspace{0.17em}\hspace{0.17em}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n.$$

(13)

The contributions resulting from the macroscopic scattering cross sections are computed using the following particular form of Equation (151) in [7]:

$$\begin{array}{l}{\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial n_j}\right]}^{(2)}={\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial s_{J_{\sigma s}+j}}\right]}_{s=N}={\displaystyle \sum _{g=1}^G{\displaystyle \int dV}{\displaystyle \underset{4\pi }{\int }d\mathbf{\Omega }}\hspace{0.17em}\hspace{0.17em}{\psi }^{\left(1\right),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g^{\prime }=1}^G{\displaystyle \underset{4\pi }{\int }d{\mathbf{\Omega }}^{\mathbf{\prime }}}\hspace{0.17em}\frac{\partial {\Sigma }_s^{g^{\prime }\to g}\left(\mathbf{s};{\mathbf{\Omega }}^{\mathbf{\prime }}\to \mathbf{\Omega }\right)}{\partial n_j}{\phi }^{g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)}}\hspace{0.17em},\hspace{0.17em}\\ for\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n.\end{array}$$

(14)

Noting that:

$$\begin{array}{l}\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathbf{\Omega }}^{\mathbf{\prime }}\to \mathbf{\Omega })}{\partial n_j}=\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathbf{\Omega }}^{\mathbf{\prime }}\to \mathbf{\Omega })}{\partial N_{i_j,m_j}}=\frac{\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}{\sigma }_{s,i}^{g^{\prime }\to g}(\mathbf{s};{\mathbf{\Omega }}^{\mathbf{\prime }}\to \mathbf{\Omega })}}\right]}{\partial N_{i_j,m_j}}\\ =\frac{\partial \left[{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{l=0}^{ISCT}{N}_{i,m}\left(2l+1\right){\sigma }_{s,l,i}^{g^{\prime }\to g}{P}_l\left({\mathbf{\Omega }}^{\mathbf{\prime }}\cdot \mathbf{\Omega }\right)}}}\right]}{\partial N_{i_j,m_j}}={\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right){\sigma }_{s,l,i_j}^{g^{\prime }\to g}{P}_l\left({\mathbf{\Omega }}^{\mathbf{\prime }}\cdot \mathbf{\Omega }\right)}.\end{array}$$

(15)

Inserting Equation (15) into Equation (14), using the addition theorem for spherical harmonics in one-dimensional geometry, and performing the respective angular integrations yields:

$${\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial n_j}\right]}^{(2)}={\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial s_{J_{\sigma s}+j}}\right]}_{s=N}={\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle \int dV}{\xi }_l^{(1),g}\left(r\right){\displaystyle \sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_j}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n,$$

(16)

where:

$${\phi }_l^g\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{P}_l(\mathbf{\Omega }){\phi }^g\left(r,\mathbf{\Omega }\right),$$

(17)

$${\xi }_l^{(1),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{P}_l(\mathbf{\Omega }){\psi }^{(1),g}\left(r,\mathbf{\Omega }\right).$$

(18)

The contributions stemming from the macroscopic fission cross sections are computed using the following particular form of Equation (152) in [7]:

$$\begin{array}{l}{\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial n_j}\right]}^{(3)}={\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial f_{J_{\sigma f}+J_{\nu }+j}}\right]}_{f=N}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }^{(1),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial n_j}{\chi }^g{\phi }^{g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)}}\hspace{0.17em},\hspace{0.17em}\\ for\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n.\end{array}$$

(19)

Inserting the result below:

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

(20)

into Equation (19) and performing the respective angular integrations yields:

$${\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial n_j}\right]}^{(3)}={\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial f_{J_{\sigma f}+J_{\nu }+j}}\right]}_{f=N}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\xi }_0^{(1),g}\left(r\right){\chi }^g{\displaystyle \sum _{g’=1}^G{\nu }_{i_j}^{g^{\prime }}{\sigma }_{f,i_j}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)}}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n,$$

(21)

where:

$${\phi }_0^g\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\phi }^g\left(r,\mathbf{\Omega }\right),$$

(22)

$${\xi }_0^{(1),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }^{(1),g}\left(r,\mathbf{\Omega }\right).$$

(23)

The contributions arising from the source are computed using the following particular form of Equation (154) in [7]:

$${\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial n_j}\right]}^{(4)}={\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial q_{J_q+j}}\right]}_{q=N}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }^{(1),g}\left(r,\mathbf{\Omega }\right)\frac{\partial Q^g\left(\mathbf{q};r,\mathbf{\Omega }\right)}{\partial n_j}}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n,$$

(24)

where the source $Q^g$ is the sum of all sources of neutrons in in group g. Since the PERP benchmark contains just spontaneous fission sources, it follows that:

$$Q^g=Q_{SF}^g,$$

(25)

where $Q_{SF}^g$ denotes the total spontaneous fission source rate density in group $g$. The spontaneous fission source rate density $Q_{SF,k}^g$ in group $g$ for isotope $k$ for the PERP benchmark is modeled by the following expression:

$$Q_{SF,k}^g={\lambda }_k{N}_{k,m}{\chi }_{SF,k}^g={\lambda }_k{N}_{k,m}{F}_k^{SF}{\mathsf{\nu }}_k^{SF}\left(\frac{2}{\sqrt{\pi a_k{}^3{b}_k}}{e}^{-\frac{a_k{b}_k}{4}}\right){\displaystyle {\int }_{E^{g+1}}^{E^g}dE\hspace{0.17em}{e}^{-E/a_k}\sinh \sqrt{b_{k}E}},$$

(26)

where ${\lambda }_k$ is the decay constant for isotope $k$, and ${\chi }_{SF,k}^g$ includes the spontaneous fission branch ratio and the spontaneous fission neutron spectra, which are approximated by a Watt’s fission spectrum using two evaluated parameters ($a_k$ and $b_k$). The total spontaneous fission source rate density in group $g$, $Q_{SF}^g$, is the sum over all fissionable isotopes, i.e.,:

$$Q_{SF}^g={\displaystyle \sum _{m=1}^M{\displaystyle \sum _{k=1}^{N_f}{Q}_{SF,k}^g}}={\displaystyle \sum _{m=1}^M{\displaystyle \sum _{k=1}^{N_f}{\lambda }_k{N}_{k,m}{\chi }_{SF,k}^g}}.$$

(27)

Since:

$$\frac{\partial Q^g\left(\mathbf{q};r,\mathbf{\Omega }\right)}{\partial n_j}=\frac{\partial Q_{SF}^g}{\partial N_{i_j,m_j}}=\frac{\partial {\displaystyle \sum _{m=1}^M{\displaystyle \sum _{k=1}^{N_f}{\lambda }_k{N}_{k,m}{\chi }_{SF,k}^g}}}{\partial N_{i_j,m_j}}={\lambda }_{i_j}{\chi }_{SF,i_j}^g=\frac{Q_{SF,i_j}^g}{N_{i_j,m_j}}=\frac{Q_{SF,i_j}^g}{n_j},$$

(28)

the last term on the right side of Equation (28) is inserted into Equation (24), which enables performing the respective angular integrations to obtain:

$${\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial n_j}\right]}^{(4)}={\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial q_{J_q+j}}\right]}_{q=N}=\frac{1}{n_j}{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\xi }_0^{(1),g}\left(r\right)Q_{SF,i_j}^g}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n.$$

(29)

Adding the partial contributions from Equations (13), (16), (21) and (29) yields the following complete expression for $\partial L\left(\mathsf{\alpha }\right)/\partial n_j$ in terms of the isotopic number density parameter $n_j\equiv N_{i_j,m_j}$:

$$\begin{array}{l}\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial n_j}={\displaystyle \sum _{i=1}^4{\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial n_j}\right]}^{(i)}}\\ =-{\displaystyle \sum _{g=1}^G{\displaystyle \int dV}{\sigma }_{t,i_j}^g{\displaystyle \underset{4\pi }{\int }d\mathbf{\Omega }\hspace{0.17em}}{\psi }^{\left(1\right),g}\left(r,\mathbf{\Omega }\right){\phi }^g\left(r,\mathbf{\Omega }\right)\hspace{0.17em}\hspace{0.17em}}+{\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle \int dV}{\xi }_l^{(1),g}\left(r\right){\displaystyle \sum _{g^{\prime }=1}^G{\sigma }_{s,l,i_j}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\xi }_0^{(1),g}\left(r\right){\chi }^g{\displaystyle \sum _{g’=1}^G{\nu }_j^{g^{\prime }}{\sigma }_{f,i_j}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)}}+\frac{1}{n_j}{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\xi }_0^{(1),g}\left(r\right)Q_{SF,i_j}^g},for\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n.\end{array}$$

(30)

The 1st-order relative sensitivities corresponding to $\partial L\left(\mathsf{\alpha }\right)/\partial n_j$ are defined as $S^{(1)}\left(N_{i,m}\right)\triangleq \left(\partial L/\partial N_{j,m}\right)\left(N_{j,m}/L\right),\hspace{0.17em}\hspace{0.17em}i=1,\dots ,I;m=1,\dots ,M$. The numerical values of the 1st-order relative sensitivities $S^{(1)}\left(N_{i,m}\right)$ of leakage response with respect to the isotope number densities for the six isotopes contained in the PERP benchmark will be presented in Section 3, below, in the table that comparing these values with the numerical values of the corresponding 2nd-order unmixed relative sensitivities $S^{(2)}\left(N_{i,m},N_{i,m}\right)\triangleq \left({\partial }^{2}L/\partial N_{i,m}\partial N_{i,m}\right)\left(N_{i,m}{N}_{i,m}/L\right),\hspace{0.17em}\hspace{0.17em}j=1,\dots ,I;\hspace{0.17em}\hspace{0.17em}m=1,\dots ,M.$

3. Computation of the 2nd-Order Mixed and Unmixed Sensitivities of the PERP Leakage Response with Respect to the Isotopic Number Densities

The complete expression of the 2nd-order sensitivities ${\partial }^{2}L\left(\mathsf{\alpha }\right)/\partial \mathbf{N}\partial \mathbf{N}$ comprises 16 components, which stem from the neutron sources and from the total, scattering and fission macroscopic cross sections. The expression of ${\partial }^{2}L\left(\mathsf{\alpha }\right)/\partial \mathbf{N}\partial \mathbf{N}$ is as follows:

$$\begin{array}{l}\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}=\hspace{0.17em}\hspace{0.17em}{\left[\frac{{\partial }^{2}L}{\partial t_{J_{\sigma t}+j}\partial t_{J_{\sigma t}+m_2}}\right]}_{t=N,t=N}+{\left[\frac{{\partial }^{2}L}{\partial t_{J_{\sigma t}+j}\partial s_{J_{\sigma s}+m_2}}\right]}_{t=N,s=N}+{\left[\frac{{\partial }^{2}L}{\partial t_{J_{\sigma t}+j}\partial f_{J_{\sigma f}+J_{\nu }+m_2}}\right]}_{t=N,f=N}\\ +{\left[\frac{{\partial }^{2}L}{\partial t_{J_{\sigma t}+j}\partial q_{J_q+m_2}}\right]}_{t=N,q=N}+{\left[\frac{{\partial }^{2}L}{\partial s_{J_{\sigma s}+j}\partial t_{J_{\sigma t}+m_2}}\right]}_{s=N,t=N}+{\left[\frac{{\partial }^{2}L}{\partial s_{J_{\sigma s}+j}\partial s_{J_{\sigma s}+m_2}}\right]}_{s=N,s=N}\\ +{\left[\frac{{\partial }^{2}L}{\partial s_{J_{\sigma s}+j}\partial f_{J_{\sigma f}+J_{\nu }+m_2}}\right]}_{s=N,f=N}+{\left[\frac{{\partial }^{2}L}{\partial s_{J_{\sigma s}+j}\partial q_{J_q+m_2}}\right]}_{s=N,q=N}+{\left[\frac{{\partial }^{2}L}{\partial f_{J_{\sigma f}+J_{\nu }+j}\partial t_{J_{\sigma t}+m_2}}\right]}_{f=N,t=N}\\ +{\left[\frac{{\partial }^{2}L}{\partial f_{J_{\sigma f}+J_{\nu }+j}\partial s_{J_{\sigma s}+m_2}}\right]}_{f=N,s=N}+{\left[\frac{{\partial }^{2}L}{\partial f_{J_{\sigma f}+J_{\nu }+j}\partial f_{J_{\sigma f}+J_{\nu }+m_2}}\right]}_{f=N,f=N}+{\left[\frac{{\partial }^{2}L}{\partial f_{J_{\sigma f}+J_{\nu }+j}\partial q_{J_q+m_2}}\right]}_{f=N,q=N}\\ +{\left[\frac{{\partial }^{2}L}{\partial q_{J_q+j}\partial t_{J_{\sigma t}+m_2}}\right]}_{q=N,t=N}+{\left[\frac{{\partial }^{2}L}{\partial q_{J_q+j}\partial s_{J_{\sigma s}+m_2}}\right]}_{q=N,s=N}+{\left[\frac{{\partial }^{2}L}{\partial q_{J_q+j}\partial f_{J_{\sigma f}+J_{\nu }+m_2}}\right]}_{q=N,f=N}\\ +{\left[\frac{{\partial }^{2}L}{\partial q_{J_q+j}\partial q_{J_q+m_2}}\right]}_{q=N,q=N},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(31)

The 16 components shown on the right-side of Equation (31) are obtained by particularizing Equations (158), (159), (160), (162), (167), (168), (169), (171), (177), (178), (179), (181), (204), (205), (206) and (208) in [7] to the PERP benchmark. Thus, the contribution stemming from the macroscopic total cross sections is obtained by particularizing Equation (158) in [7] to the PERP benchmark, to obtain:

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

(32)

The 2nd-level adjoint functions ${\psi }_{1,i}^{(2),g}$ and ${\psi }_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.17em}g=1,\dots ,G$, in Equation (32) are the solutions of the following 2nd-Level Adjoint Sensitivity System (2nd-LASS) presented in Equations (164)–(166) of [7]:

$$B^g\left({\mathsf{\alpha }}^0\right){\psi }_{1,i}^{(2),g}\left(r,\mathbf{\Omega }\right)=-{\phi }^g\left(r,\mathbf{\Omega }\right)\frac{\partial {\Sigma }_t{}^g\left(\mathbf{t}\right)}{\partial n_i},\hspace{0.17em}i=1,\dots ,J_n;\hspace{0.17em}g=1,\dots ,G,$$

(33)

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

(34)

$$A^{(1),g}\left({\mathsf{\alpha }}^0\right){\psi }_{2,i}^{(2),g}\left(r,\mathbf{\Omega }\right)=-{\psi }^{(1),g}\left(r,\mathbf{\Omega }\right)\frac{\partial {\Sigma }_t{}^g\left(\mathbf{t}\right)}{\partial n_j},\hspace{0.17em}\hspace{0.17em}i=1,\dots ,J_n;\hspace{0.17em}g=1,\dots ,G,$$

(35)

$${\psi }_{2,i}^{(2),g}\left(r_d,\mathbf{\Omega }\right)=0,\hspace{0.17em}\mathbf{\Omega }\cdot \mathbf{n}>0;\hspace{0.17em}\hspace{0.17em}i=1,\dots ,J_n;\hspace{0.17em}g=1,\dots ,G.$$

(36)

The parameters $n_j,\hspace{0.17em}j=1,\dots ,J_n$ and $n_{m_2},\hspace{0.17em}{m}_2=1,\dots ,J_n$ which appear in Equation (32) correspond to the following isotopic number densities: $n_j\equiv N_{i_j,m_j}$ and $n_{m_2}\equiv N_{i_{m_2},m_{m_2}}$, respectively; the subscripts $i_{m_2}$ and $m_{m_2}$ denote the isotope and material associated with the parameter $n_{m_2}$. Consequently, the following results hold:

$$\frac{{\partial }^2{\Sigma }_t{}^g\left(\mathbf{t}\right)}{\partial n_j\partial n_{m_2}}=\frac{{\partial }^2{\Sigma }_t{}^g\left(\mathbf{t}\right)}{\partial N_{i_j,m_j}\partial N_{i_{m_2},m_{m_2}}}=0,$$

(37)

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

(38)

$$\frac{\partial {\Sigma }_t{}^g\left(\mathbf{t}\right)}{\partial n_j}=\frac{\partial {\Sigma }_t{}^g\left(\mathbf{t}\right)}{\partial N_{i_j,m_j}}=\frac{\partial \left({\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}{\sigma }_{t,i}^g}}\right)}{\partial N_{i_j,m_j}}={\sigma }_{t,i_j}^g.$$

(39)

Inserting Equations (37)–(39) into Equation (32) yields:

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

(40)

where the 2nd-level adjoint functions ${\psi }_{1,j}^{(2),g}$, and ${\psi }_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G$, are the solutions of the following particular forms of Equations (33) and (35):

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

(41)

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

(42)

while being subject to the boundary conditions shown in Equations (34) and (36).

The contribution stemming from the macroscopic total and scattering cross sections is obtained by particularizing Equation (159) in [7] to the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(2)}={\left[\frac{{\partial }^{2}L}{\partial t_{J_{\sigma t}+j}\partial s_{J_{\sigma s}+m_2}}\right]}_{t=N,s=N}\\ ={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\psi }^{(1),g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)}}\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathbf{\Omega }\to {\mathbf{\Omega }}^{\mathbf{\prime }})}{\partial n_{m_2}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\phi }^{g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathbf{\Omega }}^{\mathbf{\prime }}\to \mathbf{\Omega })}{\partial n_{m_2}}}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(43)

Note the following relations:

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

(44)

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

(45)

Inserting the results obtained in Equations (44) and (45) into Equation (43) and performing the respective angular integrations yields the following expression for Equation (43):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(2)}={\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV}{\xi }_{1,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{(1),g^{\prime }}\left(r\right)}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV}{\xi }_{2,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)}},\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n,\end{array}$$

(46)

where:

$${\xi }_{1,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{P}_l(\mathbf{\Omega }){\psi }_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right),$$

(47)

$${\xi }_{2,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{P}_l(\mathbf{\Omega }){\psi }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right).$$

(48)

The contribution stemming from the macroscopic total and fission cross sections is obtained by particularizing Equation (160) in [7] to the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(3)}={\left[\frac{{\partial }^{2}L}{\partial t_{J_{\sigma t}+j}\partial f_{J_{\sigma f}+J_{\nu }+m_2}}\right]}_{t=N,f=N}\\ ={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\phi }^{g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}{\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\chi }^{g^{\prime }}{\psi }^{(1),g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(49)

Note that:

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

(50)

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

(51)

Inserting Equations (50) and (51) into Equation (49) and performing the respective angular integrations yields the following expression for Equation (49):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(3)}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\chi }^g{\xi }_{2,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\nu }_{i_{m_2}}^{g^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)}}+{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{\xi }_{1,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n,\end{array}$$

(52)

where:

$${\xi }_{1,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right),$$

(53)

$${\xi }_{2,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right).$$

(54)

The contribution stemming from the macroscopic total cross sections and the source term is obtained by particularizing Equation (162) in [7] to the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(4)}={\left[\frac{{\partial }^{2}L}{\partial t_{J_{\sigma t}+j}\partial q_{J_q+m_2}}\right]}_{t=N,q=N}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right)\frac{\partial Q^g\left(\mathbf{q};r,\mathbf{\Omega }\right)}{\partial n_{m_2}}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(55)

Note that:

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

(56)

Inserting Equation (56) into Equation (55), yields the following simplified expression for Equation (55):

$${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(4)}=\frac{1}{n_{m_2}}{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\xi }_{2,j;0}^{(2),g}\left(r\right)Q_{SF,i_{m_2}}^g},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.$$

(57)

The contribution stemming from the macroscopic scattering and total cross sections is obtained by particularizing Equation (167) in [7] to the PERP benchmark, which yields:

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

(58)

where the 2nd-level adjoint functions ${\theta }_{1,j}^{(2),g}$, and ${\theta }_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.17em}g=1,\dots ,G$, are the solutions of the following 2nd-Level Adjoint Sensitivity System presented in Equations (164)–(166) of [7]:

$$B^g\left({\mathsf{\alpha }}^0\right){\theta }_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right)={\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathbf{\Omega }}^{\mathbf{\prime }}\to \mathbf{\Omega })}{\partial n_j}{\phi }^{g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)},j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(59)

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

(60)

$$A^{(1),g}\left({\mathsf{\alpha }}^0\right){\theta }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right)={\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\psi }^{(1),g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathbf{\Omega }\to {\mathbf{\Omega }}^{\mathbf{\prime }})}{\partial n_j}},\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}g=1,\dots ,G,$$

(61)

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

(62)

For the PERP benchmark, the following relation holds:

$$\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathbf{\Omega }\to {\mathbf{\Omega }}^{\mathbf{\prime }})}{\partial n_j}=\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathbf{\Omega }\to {\mathbf{\Omega }}^{\mathbf{\prime }})}{\partial N_{i_j,m_j}}={\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right){\sigma }_{s,l,i_j}^{g\to g^{\prime }}{P}_l\left({\mathbf{\Omega }}^{\mathbf{\prime }}\cdot \mathbf{\Omega }\right)}.$$

(63)

Inserting Equations (15), (38) and (63) into Equations (59), (61) and (58) reduces the latter equation to the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(5)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\sigma }_{t,i_{m_2}}^g{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}\left[{\theta }_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\psi }^{(1),g}\left(r,\mathbf{\Omega }\right)+{\theta }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\phi }^g\left(r,\mathbf{\Omega }\right)\right]},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n,\end{array}$$

(64)

where the 2nd-level adjoint functions ${\theta }_{1,j}^{(2),g}$, and ${\theta }_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G$, are the solutions of the following particular forms of Equations (59) and (61):

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

(65)

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

(66)

while being subject to the boundary conditions shown in Equations (60) and (62). Due to symmetry, the result obtained in Equation (64) for ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(5)}={\left[\frac{{\partial }^{2}L}{\partial s_{J_{\sigma s}+j}\partial t_{J_{\sigma t}+m_2}}\right]}_{s=N,t=N}$ must equal to the result obtained in Equation (46) for ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(2)}={\left[\frac{{\partial }^{2}L}{\partial t_{J_{\sigma t}+j}\partial s_{J_{\sigma s}+m_2}}\right]}_{t=N,s=N}$.

The contribution stemming from the macroscopic scattering cross sections is obtained by particularizing Equation (168) in [7] to the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(6)}={\left[\frac{{\partial }^{2}L}{\partial s_{J_{\sigma s}+j}\partial s_{J_{\sigma s}+j}}\right]}_{s=N,s=N}\\ ={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }^{(1),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\phi }^{g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)\frac{{\partial }^2{\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathbf{\Omega }}^{\mathbf{\prime }}\to \mathbf{\Omega })}{\partial n_j\partial n_{m_2}}}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\theta }_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\psi }^{(1),g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)}}\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathbf{\Omega }\to {\mathbf{\Omega }}^{\mathbf{\prime }})}{\partial n_{m_2}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\theta }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\phi }^{g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathbf{\Omega }}^{\mathbf{\prime }}\to \mathbf{\Omega })}{\partial n_{m_2}}}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(67)

Noting that:

$$\frac{{\partial }^2{\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathbf{\Omega }}^{\mathbf{\prime }}\to \mathbf{\Omega })}{\partial n_j\partial n_{m_2}}=0,$$

(68)

and inserting Equations (44) and (45) into Equation (67) yields the following expression for the latter equation:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(6)}={\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV}{\Theta }_{1,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{(1),g^{\prime }}\left(r\right)}}+{\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV}{\Theta }_{2,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n,\end{array}$$

(69)

where:

$${\Theta }_{1,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{P}_l(\mathbf{\Omega }){\theta }_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right),$$

(70)

$${\Theta }_{2,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{P}_l(\mathbf{\Omega }){\theta }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right).$$

(71)

The contribution stemming from the macroscopic scattering and fission cross sections is obtained by particularizing Equation (169) in [7] to the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(7)}={\left[\frac{{\partial }^{2}L}{\partial s_{J_{\sigma s}+j}\partial f_{J_{\sigma f}+J_{\nu }+m_2}}\right]}_{s=N,f=N}\\ ={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\theta }_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}{\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\chi }^{g^{\prime }}{\psi }^{(1),g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\theta }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\phi }^{g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(72)

Inserting Equations (50) and (51) into Equation (72) yields the following expression for the latter equation:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(7)}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{\Theta }_{1,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)}}+{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\chi }^g{\Theta }_{2,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\nu }_{i_{m_2}}^{g^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(73)

where:

$${\Theta }_{1,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\theta }_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right),$$

(74)

$${\Theta }_{2,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\theta }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right).$$

(75)

The contribution stemming from the macroscopic scattering cross sections and the source term is obtained by particularizing Equation (171) in [7] to the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(8)}={\left[\frac{{\partial }^{2}L}{\partial s_{J_{\sigma s}+j}\partial q_{J_q+m_2}}\right]}_{s=N,q=N}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\theta }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right)\frac{\partial Q^g\left(\mathbf{q};r,\mathbf{\Omega }\right)}{\partial n_{m_2}}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(76)

Inserting Equation (56) into Equation (76) yields the following expression for Equation (76):

$${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(8)}=\frac{1}{n_{m_2}}{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\Theta }_{2,j;0}^{(2),g}\left(r\right)Q_{SF,i_{m_2}}^g},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.$$

(77)

The contribution stemming from the macroscopic fission and total cross sections is obtained by particularizing Equation (177) in [7] to the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(9)}={\left[\frac{{\partial }^{2}L}{\partial f_{J_{\sigma f}+J_{\nu }+j}\partial t_{J_{\sigma t}+m_2}}\right]}_{f=N,t=N}\\ =-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}\left[u_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\psi }^{(1),g}\left(r,\mathbf{\Omega }\right)+u_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\phi }^g\left(r,\mathbf{\Omega }\right)\right]\frac{\partial {\Sigma }_t{}^g\left(\mathbf{t}\right)}{\partial n_{m_2}}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n,\end{array}$$

(78)

where the 2nd-level adjoint functions $u_{1,j}^{(2),g}$, and $u_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.17em}g=1,\dots ,G$, are the solutions of the following 2nd-Level Adjoint Sensitivity System presented in Equations (183)–(185) of [7], namely:

$$B^g\left({\mathbf{\alpha }}^0\right)u_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right)={\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\phi }^{g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial n_j}},j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(79)

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

(80)

$$A^{(1),g}\left({\mathbf{\alpha }}^0\right)u_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right)=\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(\mathbf{f}\right)\right]}{\partial n_j}{\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\psi }^{(1),g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right){\chi }^{g^{\prime }}},\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}g=1,\dots ,G,$$

(81)

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

(82)

For the PERP benchmark, the following relation holds:

$$\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(\mathbf{f}\right)\right]}{\partial n_j}=\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(\mathbf{f}\right)\right]}{\partial N_{i_j,m_j}}=\frac{\partial {\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{N}_{i,m}}}{\nu }_i^g{\sigma }_{f,i}^g}{\partial N_{i_j,m_j}}={\nu }_j^g{\sigma }_{f,i_j}^g.$$

(83)

Inserting Equations (20), (38) and (83) into Equations (79), (81) and (78) reduces the latter equation to the following expression:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(9)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}\left[u_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\psi }^{(1),g}\left(r,\mathbf{\Omega }\right)+u_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\phi }^g\left(r,\mathbf{\Omega }\right)\right]{\sigma }_{t,i_{m_2}}^g},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n,\end{array}$$

(84)

where the 2nd-level adjoint functions $u_{1,j}^{(2),g}$, and $u_{2,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G$, are the solutions of the following particular forms of Equations (79) and (81):

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

(85)

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

(86)

while being subject to the boundary conditions shown in Equations (80) and (82). Due to symmetry, the results obtained in Equation (52) for ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(3)}={\left[\frac{{\partial }^{2}L}{\partial t_{J_{\sigma t}+j}\partial f_{J_{\sigma f}+J_{\nu }+m_2}}\right]}_{t=N,f=N}$ must equal to those obtained in Equation (84) for ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(9)}={\left[\frac{{\partial }^{2}L}{\partial f_{J_{\sigma f}+J_{\nu }+j}\partial t_{J_{\sigma t}+m_2}}\right]}_{f=N,t=N}$.

The contribution stemming from the macroscopic fission and scattering cross sections is obtained by particularizing Equation (178) in [7] to the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(10)}={\left[\frac{{\partial }^{2}L}{\partial f_{J_{\sigma f}+J_{\nu }+j}\partial s_{J_{\sigma s}+m_2}}\right]}_{f=N,s=N}\\ ={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{u}_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\psi }^{(1),g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)}}\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathbf{\Omega }\to {\mathbf{\Omega }}^{\mathbf{\prime }})}{\partial n_{m_2}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{u}_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\phi }^{g^{\prime }}\left(\mathbf{r},{\mathbf{\Omega }}^{\mathbf{\prime }}\right)\frac{\partial {\Sigma }_s^{g^{\prime }\to g}(\mathbf{s};{\mathbf{\Omega }}^{\mathbf{\prime }}\to \mathbf{\Omega })}{\partial n_{m_2}}}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(87)

Inserting Equations (44) and (45) into Equation (87) and performing the respective angular integrations yields the following simplified expression for Equation (87):

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(10)}={\left[\frac{{\partial }^{2}L}{\partial f_{J_{\sigma f}+J_{\nu }+j}\partial s_{J_{\sigma s}+m_2}}\right]}_{f=N,s=N}\\ ={\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV}\hspace{0.17em}{U}_{1,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{(1),g^{\prime }}\left(r\right)}}+{\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV}\hspace{0.17em}{U}_{2,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(88)

where:

$$U_{1,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{P}_l(\mathbf{\Omega })u_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right),$$

(89)

$$U_{2,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{P}_l(\mathbf{\Omega })u_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right).$$

(90)

The contribution stemming from the macroscopic fission cross sections is obtained by particularizing Equation (179) in [7] to the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(11)}={\left[\frac{{\partial }^{2}L}{\partial f_{J_{\sigma f}+J_{\nu }+j}\partial f_{J_{\sigma f}+J_{\nu }+m_2}}\right]}_{f=N,f=N}\\ ={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }^{(1),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d}{\mathbf{\Omega }}^{\mathbf{\prime }}{\phi }^{g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right){\chi }^g\left(r\right)\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial n_j\partial n_{m_2}}}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{u}_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}{\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\chi }^{g^{\prime }}{\psi }^{(1),g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{u}_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\phi }^{g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right){\chi }^g\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(91)

Noting that:

$$\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial n_j\partial n_{m_2}}=\frac{{\partial }^2\left[{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}\left(\mathbf{f}\right)\right]}{\partial N_{i_j,m_j}\partial N_{i_{m_2},m_{m_2}}}=0,$$

(92)

and inserting Equations (50), (51) and (92) into Equation (91) yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(11)}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{U}_{1,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)}}+{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\chi }^g{U}_{2,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\nu }_{i_{m_2}}^{g^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n,\end{array}$$

(93)

where:

$$U_{1,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{u}_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right),$$

(94)

$$U_{2,j;0}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{u}_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right).$$

(95)

The contribution stemming from the macroscopic fission cross sections and the source term is obtained by particularizing Equation (181) in [7] for the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(12)}={\left[\frac{{\partial }^{2}L}{\partial f_{J_{\sigma f}+J_{\nu }+j}\partial q_{J_q+m_2}}\right]}_{f=N,q=N}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{u}_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right)\frac{\partial Q^g\left(\mathbf{q};r,\mathbf{\Omega }\right)}{\partial n_{m_2}}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(96)

Inserting Equation (56) into Equation (96) yields the following expression:

$${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(12)}=\frac{1}{n_{m_2}}{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}\hspace{0.17em}{U}_{2,j;0}^{(2),g}\left(r\right)Q_{SF,i_{m_2}}^g},\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.$$

(97)

The contribution stemming from the source term and macroscopic total cross sections is obtained by particularizing Equation (204) in [7] for the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(13)}={\left[\frac{{\partial }^{2}L}{\partial q_{J_q+j}\partial t_{J_{\sigma t}+m_2}}\right]}_{q=N,t=N}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{g}_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\psi }^{(1),g}\left(r,\mathbf{\Omega }\right)\frac{\partial {\Sigma }_t{}^g\left(\mathbf{t}\right)}{\partial n_{m_2}}},\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n,\end{array}$$

(98)

where the 2nd-level adjoint functions, $g_{1,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.17em}g=1,\dots ,G$, are the solutions of the following 2nd-Level Adjoint Sensitivity System presented in Equations (200) and (202) of [7]:

$$B^g\left({\mathsf{\alpha }}^0\right)g_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right)=\frac{\partial Q^g\left(\mathbf{q};r,\mathbf{\Omega }\right)}{\partial n_j},j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G,$$

(99)

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

(100)

Inserting Equations (28) and (38) into Equations (99) and (98) reduces the latter equation to the following expression:

$${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(13)}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\sigma }_{t,i_{m_2}}^g{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{g}_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\psi }^{(1),g}\left(r,\mathbf{\Omega }\right)},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n,$$

(101)

where the 2nd-level adjoint functions, $g_{1,j}^{(2),g},j=1,\dots ,J_n;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G$, are the solutions of the following particular forms of Equation (99):

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

(102)

subject to the boundary conditions shown in Equation (100). Due to symmetry, the results obtained from Equation (57) for ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(4)}={\left[\frac{{\partial }^{2}L}{\partial t_{J_{\sigma t}+j}\partial q_{J_q+m_2}}\right]}_{t=N,q=N}$ must equal to that obtained from Equation (101) for ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(13)}={\left[\frac{{\partial }^{2}L}{\partial q_{J_q+j}\partial t_{J_{\sigma t}+m_2}}\right]}_{q=N,t=N}$.

The contribution stemming from the source term and macroscopic scattering cross sections is obtained by particularizing Equation (205) in [7] to the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(14)}={\left[\frac{{\partial }^{2}L}{\partial q_{J_q+j}\partial s_{J_{\sigma s}+m_2}}\right]}_{q=N,s=N}\\ ={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{g}_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\psi }^{(1),g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)}}\frac{\partial {\Sigma }_s^{g\to g^{\prime }}(\mathbf{s};\mathbf{\Omega }\to {\mathbf{\Omega }}^{\mathbf{\prime }})}{\partial n_{m_2}},\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(103)

Inserting Equation (44) into Equation (103) and performing the respective angular integrations yields the following expression for Equation (103):

$${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(14)}={\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV\hspace{0.17em}}{G}_{1,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{(1),g^{\prime }}\left(r\right)}},\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n,$$

(104)

where:

$$G_{1,j;l}^{(2),g}\left(r\right)\triangleq {\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{P}_l(\mathbf{\Omega })g_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right).$$

(105)

Due to symmetry, the result obtained in Equation (77) for ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(8)}={\left[\frac{{\partial }^{2}L}{\partial s_{J_{\sigma s}+j}\partial q_{J_q+m_2}}\right]}_{s=N,q=N}$ must equal to that obtained in Equation (104) for ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(14)}={\left[\frac{{\partial }^{2}L}{\partial q_{J_q+j}\partial s_{J_{\sigma s}+m_2}}\right]}_{q=N,s=N}$.

The contribution stemming from the source term and macroscopic fission cross sections is obtained by particularizing Equation (206) in [7] for the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(15)}={\left[\frac{{\partial }^{2}L}{\partial q_{J_q+j}\partial f_{J_{\sigma f}+J_{\nu }+m_2}}\right]}_{q=N,f=N}\\ ={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{g}_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right)\frac{\partial \left[{\left(\nu {\Sigma }_f\right)}^g\left(\mathbf{f}\right)\right]}{\partial n_{m_2}}{\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}{\chi }^{g^{\prime }}{\psi }^{(1),g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)}},\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(106)

Inserting Equation (50) into Equation (106) and performing the respective angular integrations yields the following simplified expression for Equation (106):

$${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(15)}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{G}_{1,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}}}{\xi }_0^{(1),g^{\prime }}\left(r\right),\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.$$

(107)

Due to symmetry, the result obtained in Equation (97) for ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(12)}={\left[\frac{{\partial }^{2}L}{\partial f_{J_{\sigma f}+J_{\nu }+j}\partial q_{J_q+m_2}}\right]}_{f=N,q=N}$ must equal to that obtained in Equation (107) for ${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(15)}={\left[\frac{{\partial }^{2}L}{\partial q_{J_q+j}\partial f_{J_{\sigma f}+J_{\nu }+m_2}}\right]}_{q=N,f=N}$.

Finally, the contribution stemming from the source term is obtained by particularizing Equation (208) in [7] for the PERP benchmark, which yields:

$$\begin{array}{l}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(16)}={\left[\frac{{\partial }^{2}L}{\partial q_{J_q+j}\partial q_{J_q+m_2}}\right]}_{q=N,q=N}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }^{(1),g}\left(r,\mathbf{\Omega }\right)\frac{\partial Q^g\left(\mathbf{q};r,\mathbf{\Omega }\right)}{\partial n_j\partial n_{m_2}}},\\ for\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(108)

Since:

$$\frac{\partial Q^g\left(\mathbf{q};r,\mathbf{\Omega }\right)}{\partial n_j\partial n_{m_2}}=\frac{\partial Q^g\left(\mathbf{q};r,\mathbf{\Omega }\right)}{\partial N_{j,m}\partial N_{m_2,m^{\prime }}}=0,$$

(109)

it follows that the contribution from Equation (108) vanishes, i.e.,

$${\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{(16)}=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.$$

(110)

Collecting the partial contributions obtained in Equations (40), (46), (52), (57), (64), (69), (73), (77), (84), (88), (93), (97), (101), (104), (107) and (110) yields the following result:

$$\begin{array}{l}\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}={\displaystyle \sum _{i=1}^{16}{\left(\frac{{\partial }^{2}L}{\partial n_j\partial n_{m_2}}\right)}^{\left(i\right)}}=-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}\left[{\psi }_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\psi }^{(1),g}\left(r,\mathbf{\Omega }\right)+{\psi }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\phi }^g\left(r,\mathbf{\Omega }\right)\right]{\sigma }_{t,i_{m_2}}^g}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV}{\xi }_{1,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{(1),g^{\prime }}\left(r\right)}}+{\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV}{\xi }_{2,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\chi }^g{\xi }_{2,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\nu }_{i_{m_2}}^{g^{\prime }}{\sigma }_{f,i_{m_2}}^{g^{\prime }}{\phi }_0^{g^{\prime }}\left(r\right)}}+{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{\xi }_{1,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)}}\\ +\frac{1}{n_{m_2}}{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\xi }_{2,j;0}^{(2),g}\left(r\right)Q_{SF,i_{m_2}}^g}-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\sigma }_{t,i_{m_2}}^g{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}\left[{\theta }_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\psi }^{(1),g}\left(r,\mathbf{\Omega }\right)+{\theta }_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\phi }^g\left(r,\mathbf{\Omega }\right)\right]}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV}{\Theta }_{1,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{(1),g^{\prime }}\left(r\right)}}+{\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV}{\Theta }_{2,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{\Theta }_{1,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)}}+{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\chi }^g{\Theta }_{2,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=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}}{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\Theta }_{2,j;0}^{(2),g}\left(r\right)Q_{SF,i_{m_2}}^g}-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}\left[u_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\psi }^{(1),g}\left(r,\mathbf{\Omega }\right)+u_{2,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\phi }^g\left(r,\mathbf{\Omega }\right)\right]{\sigma }_{t,i_{m_2}}^g}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV}\hspace{0.17em}{U}_{1,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{(1),g^{\prime }}\left(r\right)}}+{\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV}\hspace{0.17em}{U}_{2,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g^{\prime }\to g}{\phi }_l^{g^{\prime }}\left(r\right)}}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{U}_{1,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}{\xi }_0^{(1),g^{\prime }}\left(r\right)}}+{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\chi }^g{U}_{2,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=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}}{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}\hspace{0.17em}{U}_{2,j;0}^{(2),g}\left(r\right)Q_{SF,i_{m_2}}^g}-{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\sigma }_{t,i_{m_2}}^g{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{g}_{1,j}^{(2),g}\left(r,\mathbf{\Omega }\right){\psi }^{(1),g}\left(r,\mathbf{\Omega }\right)}\\ +{\displaystyle \sum _{g=1}^G{\displaystyle \sum _{l=0}^{ISCT}\left(2l+1\right)}{\displaystyle {\int }_{V}dV\hspace{0.17em}}{G}_{1,j;l}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\sigma }_{s,l,i_{m_2}}^{g\to g^{\prime }}{\xi }_l^{(1),g^{\prime }}\left(r\right)}}+{\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\nu }_{i_{m_2}}^g{\sigma }_{f,i_{m_2}}^g{G}_{1,j;0}^{(2),g}\left(r\right){\displaystyle \sum _{g’=1}^G{\chi }^{g^{\prime }}}}{\xi }_0^{(1),g^{\prime }}\left(r\right),\\ for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,J_n;\hspace{0.17em}{m}_2=1,\dots ,J_n.\end{array}$$

(111)

The numerical values of the 2nd-order absolute sensitivities ${\partial }^{2}L/\partial N_{i,m}\partial N_{k,m^{\prime }},\hspace{0.17em}\hspace{0.17em}i,k=1,\dots ,I;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m,m^{\prime }=1,\dots ,M$, where $I=6$ and $M=2$, of the PERP benchmark’s leakage response with respect to the isotopic number densities are computed using Equation (111). The matrix ${\partial }^{2}L/\partial N_{i,m}\partial N_{k,m^{\prime }},\hspace{0.17em}\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m,m^{\prime }=1,2$ of 2nd-order absolute sensitivities has dimensions $J_n\times J_n\hspace{0.17em}(=6\times 6)$. For convenient comparisons, the numerical results presented in this Sub-section are presented in unit-less values of the relative sensitivities that correspond to ${\partial }^{2}L/\partial N_{i,m}\partial N_{k,m^{\prime }},\hspace{0.17em}\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m,m^{\prime }=1,2$, which are denoted as ${\mathbf{S}}^{(2)}\left(N_{i,m},N_{k,m^{\prime }}\right)$ and are defined as follows:

$${\mathbf{S}}^{(2)}\left(N_{i,m},N_{k,m^{\prime }}\right)\hspace{0.17em}\triangleq \hspace{0.17em}\frac{{\partial }^{2}L}{\partial N_{i,m}\partial N_{k,m^{\prime }}}\left(\frac{N_{i,m}{N}_{k,m^{\prime }}}{L}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m,m^{\prime }=1,2.$$

(112)

The numerical results obtained for the matrix ${\mathbf{S}}^{(2)}\left(N_{i,m},N_{k,m^{\prime }}\right)\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m,m^{\prime }=1,2$ are presented in

Table 2. Second-order relative sensitivities of the leakage response with respect to the isotope number densities ${\mathbf{S}}^{(2)}\left(N_{i,m},N_{k,m^{\prime }}\right)\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m,m^{\prime }=1,2$.

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)
$i=1\hspace{0.17em}$ (239Pu)	$S^{(2)}\left(N_{1,1},N_{1,1}\right)$ $=7.380\times {10}^1$	$S^{(2)}\left(N_{1,1},N_{2,1}\right)$ $=8.751\times {10}^0$	$S^{(2)}\left(N_{1,1},N_{3,1}\right)$ $=3.034\times {10}^{-2}$	$S^{(2)}\left(N_{1,1},N_{4,1}\right)$ $=1.917\times {10}^{-2}$	$S^{(2)}\left(N_{1,1},N_{5,2}\right)$ $=8.054\times {10}^0$	$S^{(2)}\left(N_{1,1},N_{6,2}\right)$ $=1.282\times {10}^1$
$i=2$ (240Pu)	$S^{(2)}\left(N_{2,1},N_{1,1}\right)$ $=8.751\times {10}^0$	$S^{(2)}\left(N_{2,1},N_{2,1}\right)$ $=5.758\times {10}^{-1}$	$S^{(2)}\left(N_{2,1},N_{3,1}\right)$ $=3.332\times {10}^{-3}$	$S^{(2)}\left(N_{2,1},N_{4,1}\right)$ $=2.061\times {10}^{-3}$	$S^{(2)}\left(N_{2,1},N_{5,2}\right)$ $=9.111\times {10}^{-1}$	$S^{(2)}\left(N_{2,1},N_{6,2}\right)$ $=1.420\times {10}^0$
$i=3$ (69Ga)	$S^{(2)}\left(N_{3,1},N_{1,1}\right)$ $=3.035\times {10}^{-2}$	$S^{(2)}\left(N_{3,1},N_{2,1}\right)$ $=3.332\times {10}^{-3}$	$S^{(2)}\left(N_{3,1},N_{3,1}\right)$ $=1.167\times {10}^{-5}$	$S^{(2)}\left(N_{3,1},N_{4,1}\right)$ $=7.750\times {10}^{-6}$	$S^{(2)}\left(N_{3,1},N_{5,2}\right)$ $=2.707\times {10}^{-3}$	$S^{(2)}\left(N_{3,1},N_{6,2}\right)$ $=4.429\times {10}^{-3}$
$i=4$ (71Ga)	$S^{(2)}\left(N_{4,1},N_{1,1}\right)$ $=1.917\times {10}^{-2}$	$S^{(2)}\left(N_{4,1},N_{2,1}\right)$ $=2.061\times {10}^{-3}$	$S^{(2)}\left(N_{4,1},N_{3,1}\right)$ $=7.750\times {10}^{-6}$	$S^{(2)}\left(N_{4,1},N_{4,1}\right)$ $=1.154\times {10}^{-5}$	$S^{(2)}\left(N_{4,1},N_{5,2}\right)$ $=1.561\times {10}^{-3}$	$S^{(2)}\left(N_{4,1},N_{6,2}\right)$ $=2.499\times {10}^{-3}$
$i=5$ (C)	$S^{(2)}\left(N_{5,2},N_{1,1}\right)$ $=8.055\times {10}^0$	$S^{(2)}\left(N_{5,2},N_{2,1}\right)$ $=9.112\times {10}^{-1}$	$S^{(2)}\left(N_{5,2},N_{3,1}\right)$ $=2.707\times {10}^{-3}$	$S^{(2)}\left(N_{5,2},N_{4,1}\right)$ $=1.561\times {10}^{-3}$	$S^{(2)}\left(N_{5,2},N_{5,2}\right)$ $=8.212\times {10}^{-1}$	$S^{(2)}\left(N_{5,2},N_{6,2}\right)$ $=1.345\times {10}^0$
$i=6$ (1H)	$S^{(2)}\left(N_{6,2},N_{1,1}\right)$ $=1.283\times {10}^1$	$S^{(2)}\left(N_{6,2},N_{2,1}\right)$ $=1.420\times {10}^0$	$S^{(2)}\left(N_{6,2},N_{3,1}\right)$ $=4.429\times {10}^{-3}$	$S^{(2)}\left(N_{6,2},N_{4,1}\right)$ $=2.499\times {10}^{-3}$	$S^{(2)}\left(N_{6,2},N_{5,2}\right)$ $=1.345\times {10}^0$	$S^{(2)}\left(N_{6,2},N_{6,2}\right)$ $=1.912\times {10}^0$

. This matrix is symmetrical with respect to its principal diagonal.

As indicated in Table 2, all $J_n\times J_n\hspace{0.17em}=36$ elements of the matrix ${\mathbf{S}}^{(2)}\left(N_{i,m},N_{k,m^{\prime }}\right)$ are positive, and 12 of these sensitivities are large, having values greater than 1.0. These large sensitivities involve the isotopic number densities of either ²³⁹Pu or ¹H, and the isotopic number densities of ²⁴⁰Pu or C. The largest element is the unmixed 2nd-order sensitivity with respect to the isotopic number density of ²³⁹Pu, i.e., $S^{(2)}\left(N_{1,1},N_{1,1}\right)=7.380\times {10}^1$. On the other hand, the results shown in Table 2 indicate that all of the mixed 2nd-order relative sensitivities involving the isotopic number densities of ⁶⁹Ga or ⁷¹Ga have absolute values smaller than 1.0.

The 2nd-order unmixed sensitivities $S^{(2)}\left(N_{i,m},N_{i,m}\right)\triangleq \left({\partial }^{2}L/\partial N_{i,m}\partial N_{i,m}\right)\left(N_{i,m}{N}_{i,m}/L\right),$ $\hspace{0.17em}i=1,\dots ,6; m=1,2,$ which are the elements on the diagonal of the matrix ${\mathbf{S}}^{(2)}\left(N_{i,m},N_{k,m^{\prime }}\right), i,k=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m,m^{\prime }=1,2$, can be directly compared to the values of the 1st-order relative sensitivities $S^{(1)}\left(N_{i,m}\right)\triangleq \left(\partial L/\partial N_{i,m}\right)\left(N_{i,m}/L\right),\hspace{0.17em}\hspace{0.17em}i=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}m=1,2$, of the leakage response with respect to the isotopic number density parameters.

Table 3. Comparison of 1st-order relative sensitivities $\left(\partial L/\partial N_{i,m}\right)\left(N_{i,m}/L\right),\hspace{0.17em}\hspace{0.17em}i=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}m=1,2$ and unmixed 2nd-order relative sensitivities $\left({\partial }^{2}L/\partial N_{i,m}\partial N_{i,m}\right)\left(N_{i,m}{N}_{i,m}/L\right),\hspace{0.17em}\hspace{0.17em}i=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m=1,2$ for all isotopes.

Isotopes	1st-Order	2nd-Order
$i=1$ (239Pu)	5.963 × 100	7.380 × 101
$i=2$ (240Pu)	1.220 × 100	5.758 × 10−1
$i=3$ (69Ga)	2.229 × 10−3	1.167 × 10−5
$i=4$ (71Ga)	1.365 × 10−3	1.154 × 10−5
$i=5$ (C)	6.312 × 10−1	8.212 × 10−1
$i=6$ (1H)	1.001 × 100	1.912 × 100

compares the 1st- and the unmixed 2nd-order relative sensitivities for all six isotopes. This comparison indicates that the values of the 2nd-order unmixed sensitivities with respect to the isotopic number densities of isotopes ²³⁹Pu, C and ¹H, are 1100%, 30% and 90% larger than the corresponding values of the 1st-order sensitivities for the same isotope, respectively. The largest 1st-order relative sensitivity is $S^{(1)}\left(N_{1,1}\right)=5.963$ while the largest 2nd-order unmixed relative sensitivity is $S^{(2)}\left(N_{1,1},N_{1,1}\right)=73.80$; both involve the isotopic number density of ²³⁹Pu. It is noteworthy that all of the 1st-order relative sensitivities are positive, signifying that an increase in the isotopic number density will cause an increase in the total neutron leakage from the PERP sphere.

4. Quantification of Uncertainties in the PERP Leakage Response due to Uncertainties in Isotopic Number Densities

Correlations between the isotopic number densities or correlations between these isotopic number densities and other cross section parameters are not available for the PERP benchmark. When such correlations are unavailable, the maximum entropy principle [14] indicates that neglecting them minimizes the inadvertent introduction of spurious information into the computations of the various moments of the response’s distribution in parameter space. The formulas for computing the expected value, variance and skewness of the response distribution by considering the 2nd-order response sensitivities together with the standard deviations of the isotopic number densities parameter correlations are as follows:

(1)

The expected value, ${\left[E\left(L\right)\right]}_N$, of the leakage response $L\left(\mathsf{\alpha }\right)$ has the following expression:

$${\left[E\left(L\right)\right]}_N=L\left({\mathsf{\alpha }}^0\right)+{\left[E\left(L\right)\right]}_N^{\left(2,U\right)},$$

(113)

where the subscript “N” indicates contributions solely from the isotopic number densities, and where the 2nd-order contributions from uncorrelated isotopic number densities, denoted as ${\left[E\left(L\right)\right]}_N^{\left(2,U\right)}$, has the following expression:

$${\left[E\left(L\right)\right]}_N^{\left(2,U\right)}=\frac{1}{2}{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I\frac{{\partial }^{2}L\left(\mathsf{\alpha }\right)}{\partial N_{i,m}\partial N_{i,m}}{\left(s_{N_{i,m}}\right)}^2}}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}I=6,M=2.$$

(114)

The quantity $s_{N_{i,m}}$ which appears in Equation (114) denotes the standard deviation associated with the imprecisely known model parameter $N_{i,m}$.

(2)

Since correlations among parameters are unavailable, the following formulas pertain solely to uncorrelated and normally-distributed isotopic number densities, a restriction that will be indicated by using the superscript “(U, N).” Under these restrictions, the expression for computing the variance, denoted as ${\left[\mathrm{var}(L)\right]}_N^{(U,N)}$ of the PERP leakage response has the following form:

$${\left[\mathrm{var}(L)\right]}_N^{(U,N)}={\left[\mathrm{var}\hspace{0.17em}(L)\right]}_N^{\left(1,U,N\right)}+{\left[\mathrm{var}\hspace{0.17em}(L)\right]}_N^{\left(2,U,N\right)},$$

(115)

where the first-order contribution term, ${\left[\mathrm{var}\hspace{0.17em}(L)\right]}_N^{\left(1,U,N\right)}$, to the variance ${\left[\mathrm{var}(L)\right]}_N^{(U,N)}$ is defined as follows:

$${\left[\mathrm{var}\hspace{0.17em}(L)\right]}_N^{\left(1,U,N\right)}\triangleq {\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial N_{i,m}}\right]}^2{\left(s_{N_{i,m}}\right)}^2}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}I=6,\hspace{0.17em}M=2.$$

(116)

The second-order contribution term, ${\left[\mathrm{var}\hspace{0.17em}(L)\right]}_N^{\left(2,U,N\right)}$, to the variance ${\left[\mathrm{var}(L)\right]}_N^{(U,N)}$ in Equation (115) is defined as:

$${\left[\mathrm{var}\hspace{0.17em}(L)\right]}_N^{\left(2,U,N\right)}\triangleq \frac{1}{2}{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{\left[\frac{{\partial }^{2}L\left(\mathsf{\alpha }\right)}{\partial N_{i,m}\partial N_{i,m}}{\left(s_{N_{i,m}}\right)}^2\right]}^2}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}I=6,\hspace{0.17em}M=2.$$

(117)

(3)

Considering uncorrelated normally-distributed isotopic number densities, the third-order moment, ${\left[{\mu }_3(L)\right]}_N^{(U,N)}$, of the leakage response for the PERP benchmark has the expression:

$${\left[{\mu }_3(L)\right]}_N^{(U,N)}=3{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=1}^I{\left[\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial N_{i,m}}\right]}^2\frac{{\partial }^{2}L\left(\mathsf{\alpha }\right)}{\partial N_{i,m}\partial N_{i,m}}{\left(s_{N_{i,m}}\right)}^4}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}I=6,\hspace{0.17em}M=2.$$

(118)

As Equation (118) indicates, if the 2nd-order sensitivities were unavailable, the third moment ${\left[{\mu }_3(L)\right]}_N^{(U,N)}$ would vanish and the response distribution would by default be assumed to be Gaussian.

(4)

The skewness, ${\left[{\gamma }_1(L)\right]}_N^{(U,N)}$, in the leakage response which stems from the variances of isotopic number densities is defined as follows:

$${\left[{\gamma }_1(L)\right]}_N^{(U,N)}={\left[{\mu }_3(L)\right]}_N^{(U,N)}/{\left\{{\left[\mathrm{var}(L)\right]}_N^{(U,N)}\right\}}^{3/2}.$$

(119)

The effects of the first- and second-order sensitivities on the response’s expected value, variance and skewness are quantified by using Equations (114) through (119) in conjunction the sensitivities computed in Section 2 and Section 3 and using illustrative values of 1%, 5%, and 10%, respectively, for the standard deviations for the uncorrelated isotopic number densities. The results thus obtained are presented in

Table 4. Comparison of Response Moments Induced by Various Relative Standard Deviations Assumed for the Parameters $N_{i,m}$.

Relative Standard Deviation	10%	5%	1%
$L\left({\mathsf{\alpha }}^0\right)$	1.7648 × 106	1.7648 × 106	1.7648 × 106
${\left[E\left(L\right)\right]}_N^{\left(2,U\right)}$	6.8042 × 105	1.7011 × 105	6.8042 × 103
${\left[E\left(L\right)\right]}_N=L\left({\mathsf{\alpha }}^0\right)+{\left[E\left(L\right)\right]}_N^{\left(2,U\right)}$	2.4452 × 106	1.9349 × 106	1.7716 × 106
${\left[\mathrm{var}\hspace{0.17em}(L)\right]}_N^{\left(1,U,N\right)}$	1.1993 × 1012	2.9983 × 1011	1.1993 × 1010
${\left[\mathrm{var}\hspace{0.17em}(L)\right]}_N^{\left(2,U,N\right)}$	8.4892 × 1011	5.3075 × 1010	8.4892 × 107
${\left[\mathrm{var}(L)\right]}_N^{(U,N)}={\left[\mathrm{var}\hspace{0.17em}(L)\right]}_N^{\left(1,U,N\right)}+{\left[\mathrm{var}\hspace{0.17em}(L)\right]}_N^{\left(2,U,N\right)}$	2.0482 × 1012	3.5289 × 1011	1.2078 × 1010
${\left[{\mu }_3(L)\right]}_N^{(U,N)}$	4.3399 × 1018	2.7125 × 1017	4.3399 × 1014
${\left[{\gamma }_1(L)\right]}_N^{(U,N)}={\left[{\mu }_3(L)\right]}_N^{(U,N)}/{\left\{{\left[\mathrm{var}(L)\right]}_N^{(U,N)}\right\}}^{3/2}$	1.481	1.294	0.327

.

The relative effects of uncertainties in the isotopic number densities can be compared to the corresponding effects stemming from the total and, respectively, fission cross sections, by considering standard deviations of 10% for all of these cross sections and by comparing the corresponding results shown in Table 4 with the corresponding results presented in Table 25 of Part I [1] and Table 27 of Part III [3]. This comparison reveals that the following relations hold:

${\left[E\left(L\right)\right]}_f^{\left(2,U\right)}=3.7191\times {10}^4\ll {\left[E\left(L\right)\right]}_N^{\left(2,U\right)}=6.8042\times {10}^5\ll {\left[E\left(L\right)\right]}_t^{\left(2,U\right)}=4.5980\times {10}^6$,

${\left[\mathrm{var}\hspace{0.17em}(L)\right]}_f^{\left(1,U,N\right)}=9.5932\times {10}^{10}\ll {\left[\mathrm{var}\hspace{0.17em}(L)\right]}_N^{\left(1,U,N\right)}=1.1993\times {10}^{12}<{\left[\mathrm{var}\hspace{0.17em}(L)\right]}_t^{\left(1,U,N\right)}=3.4196\times {10}^{12}$,

${\left[\mathrm{var}\hspace{0.17em}(L)\right]}_f^{\left(2\right)}=5.4830\times {10}^8\ll {\left[\mathrm{var}\hspace{0.17em}(L)\right]}_N^{\left(2\right)}=8.4892\times {10}^{11}\ll {\left[\mathrm{var}\hspace{0.17em}(L)\right]}_t^{\left(2\right)}=2.8789\times {10}^{13}$,

${\left[{\gamma }_1(L)\right]}_f^{(U,N)}=0.1172<{\left[{\gamma }_1(L)\right]}_t^{(U,N)}=0.3407\ll {\left[{\gamma }_1(L)\right]}_N^{(U,N)}=1.481$.

The above relations indicate that the contributions to the expected value and variance stemming from the uncorrelated isotopic number densities are much smaller than the corresponding contributions stemming from the group-averaged uncorrelated microscopic total cross sections, but are much greater than the corresponding contributions stemming from the group-averaged uncorrelated microscopic fission cross sections. However, the contributions to the skewness stemming from the uncorrelated isotopic number densities are much larger than the corresponding contributions stemming from the group-averaged uncorrelated microscopic total and fission cross sections.

It is important to note that the results presented in Table 4 consider only the standard deviations of the isotopic number densities, since correlations between these parameters are unavailable. In the absence of parameter correlations, the possible contributions stemming from the mixed 2nd-order sensitivities involving the isotopic number densities cannot be accounted for. Recall that the results presented in Section 3 of this work and Ref. [5] indicated that a significant number of mixed 2nd-order sensitivities involving the isotopic number densities have absolute values larger than 1.0, including:

(a)

12 elements of the matrix ${\mathbf{S}}^{(2)}\left(N_{i,m},N_{k,m^{\prime }}\right)\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m,m^{\prime }=1,2$;

(b)

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

(c)

15 elements of the matrix ${\mathbf{S}}^{(2)}\left(N_{i,m},{\sigma }_{s,l=0,k}^{g^{\prime }\to g}\right)$,$\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.17em}m=1,2;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{g}^{\prime },g=1,\dots ,30$;

(d)

7 elements of the matrix ${\mathbf{S}}^{(2)}\left(N_{i,m},{\sigma }_{s,l=1,k}^{g^{\prime }\to g}\right)$,$\hspace{0.17em}i,k=1,\dots ,6;\hspace{0.17em}m=1,2;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{g}^{\prime },g=1,\dots ,30$;

(e)

21 elements of the matrix ${\mathbf{S}}^{(2)}\left(N_{i,m},{\sigma }_{f,k}^g\right)\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\dots ,6;\hspace{0.17em}\hspace{0.17em}k,m=1,2;\hspace{0.17em}g=1,\dots ,30$;

(f)

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

The effects of the large sensitivities mentioned above on the uncertainties in the response distribution cannot be considered until the corresponding correlations among the various model parameters become available.

5. Computation of 1st-Order Sensitivities of the PERP Leakage Response with Respect to Fission Spectrum Parameters

The fission spectrum is considered to depend on the vector of parameters $\mathbf{p}$, which were defined in Part I [1] and are reproduced below:

$$\mathbf{p}\triangleq {\left[p_1,\dots ,p_{J_p}\right]}^{†}\triangleq {\left[{\chi }_{i=1}^{g=1},{\chi }_{i=1}^{g=2},\dots ,{\chi }_{i=1}^G,\dots ,{\chi }_i^g,\dots ,{\chi }_{N_f}^G\right]}^{†},\hspace{0.17em}i=1,\dots ,N_f;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,G;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{J}_p=G\times N_f.$$

(120)

In Equation (120), the quantity ${\chi }_i^g$ denotes the fission spectrum of isotope $\hspace{0.17em}i$ in group $\hspace{0.17em}g$. The first-order sensitivities $\partial L\left(\mathsf{\alpha }\right)/\partial p_j,\hspace{0.17em}j=1,\dots ,J_p$, of the PERP leakage response to the fission spectrum parameters are computed using the following particular form of Equation (153) in [7]:

$$\frac{\partial L\left(\mathsf{\alpha }\right)}{\partial p_j}={\displaystyle \sum _{g=1}^G{\displaystyle {\int }_{V}dV}{\displaystyle {\int }_{4\pi }d\mathbf{\Omega }}\hspace{0.17em}{\psi }^{(1),g}\left(r,\mathbf{\Omega }\right){\displaystyle \sum _{g’=1}^G{\displaystyle {\int }_{4\pi }d{\mathbf{\Omega }}^{\mathbf{\prime }}}\frac{\partial {\chi }^g}{\partial p_j}{\left(\nu {\Sigma }_f\right)}^{g^{\prime }}{\phi }^{g^{\prime }}\left(r,{\mathbf{\Omega }}^{\mathbf{\prime }}\right)}}\hspace{0.17em},\hspace{0.17em}j=1,\dots ,J_p,$$

(121)

where the quantity $f_i^g$ denotes the corresponding spectrum weighting function and where ${\chi }^g$ denotes the material fission spectrum in energy group $\hspace{0.17em}g$, as has been defined in Part I [1], i.e.,

$${\chi }^g\triangleq \frac{{\displaystyle \sum _{i=1}^{N_f}{\chi }_i^g{N}_{i,m}}{\displaystyle \sum _{g^{\prime }=1}^G{\left(\nu {\sigma }_f\right)}_i^{g^{\prime }}{f}_i^{g^{\prime }}}}{{\displaystyle \sum _{i=1}^{N_f}{N}_{i,m}}{\displaystyle \sum _{g^{\prime }=1}^G{\left(\nu {\sigma }_f\right)}_i^{g^{\prime }}{f}_i^{g^{\prime }}}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}with\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{g=1}^G{\chi }_i^g}=1.$$

(122)

The numerical values of the 1st-order relative sensitivities,${\mathbf{S}}^{(1)}\left({\chi }_i^g\right)\triangleq \left(\partial L/\partial {\chi }_i^g\right)\left({\chi }_i^g/L\right),\hspace{0.17em}\hspace{0.17em}i=1,2;g=1,\dots ,30$, of the leakage response with respect to the fission spectrum parameters for the two fissionable isotopes (namely, ²³⁹Pu and ²⁴⁰Pu) contained in the PERP benchmark are presented in

Table 5. First-Order Relative Sensitivities ${\mathbf{S}}^{(1)}\left({\chi }_{i=1}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$.

g	Relative Sensitivities	g	Relative Sensitivities
1	2.539 × 10−4	16	7.557 × 10−3
2	4.642 × 10−4	17	3.309 × 10−3
3	1.220 × 10−3	18	8.557 × 10−4
4	4.836 × 10−3	19	2.332 × 10−4
5	1.920 × 10−2	20	1.219 × 10−4
6	2.601 × 10−2	21	5.140 × 10−5
7	−5.541 × 10−3	22	1.180 × 10−5
8	2.059 × 10−2	23	4.199 × 10−6
9	1.743 × 10−2	24	3.010 × 10−7
10	2.005 × 10−2	25	2.974 × 10−7
11	5.346 × 10−3	26	6.408 × 10−8
12	−5.005 × 10−2	27	−2.804 × 10−10
13	−4.730 × 10−2	28	−3.975 × 10−9
14	−2.116 × 10−2	29	2.057 × 10−9
15	−3.506 × 10−3	30	1.672 × 10−9

and

Table 6. First-Order Relative Sensitivities ${\mathbf{S}}^{(1)}\left({\chi }_{i=2}^g\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=1,\dots ,30$.

g	Relative Sensitivities	g	Relative Sensitivities
1	1.671 × 10−8	16	4.627 × 10−7
2	3.022 × 10−8	17	2.023 × 10−7
3	7.797 × 10−8	18	5.245 × 10−8
4	3.037 × 10−7	19	1.425 × 10−8
5	1.187 × 10−6	20	7.480 × 10−9
6	1.595 × 10−6	21	3.174 × 10−9
7	−3.390 × 10−7	22	7.361 × 10−10
8	1.253 × 10−6	23	2.657 × 10−10
9	1.061 × 10−6	24	1.941 × 10−11
10	1.222 × 10−6	25	1.966 × 10−11
11	3.254 × 10−7	26	4.419 × 10−12
12	−3.062 × 10−6	27	−2.031 × 10−14
13	−2.900 × 10−6	28	−2.953 × 10−13
14	−1.299 × 10−6	29	1.545 × 10−13
15	−2.154 × 10−7	30	1.260 × 10−13

, respectively.

The values presented in Table 5 for the 1st-order relative sensitivities of the leakage response with respect to the fission spectrum parameters of ²³⁹Pu are all very small, the largest being of the order of 10⁻². Similarly, as shown in Table 6, the 1st-order relative sensitivities of the leakage response with respect to the fission spectrum of ²⁴⁰Pu are even smaller, largest being of the order of 10⁻⁶. Based on the results that have been presented in Part I−Part III [1,2,3] for the 1st-order and 1st-order relative sensitivities of the leakage response with respect to the parameters underlying the total, scattering, fission microscopic cross sections and average number of neutrons per fission, and in view of the small values of the 1st-order relative sensitivities presented in Table 5 and Table 6, it can be inferred that the 2nd-order relative sensitivities of the leakage response with respect to the fission spectrum parameters of isotopes ²³⁹Pu and ²⁴⁰Pu would be very small. This inference has been confirmed by sample computations of 2nd-order sensitivities of the PERP leakage response with respect to fission spectrum parameters. For the sake of brevity, these computations will not be presented herein.

6. Overall Conclusions: Ranking and Impact of the First- and Second-Order Sensitivities of the PERP Benchmark’s Leakage Response with Respect to the Benchmark’s 21976 Imprecisely Known Parameters

This Section presents the overall summary and conclusions based on the comprehensive results that have been detailed in Parts I−V [1,2,3,4,5] and in the present work for the 1st- and 2nd-order sensitivities of the PERP benchmark’s leakage response with respect to (all 21976 of) the PERP model’s imprecisely known parameters. To begin with,

Table 7. Summary of the large 1st-order relative sensitivities for the PERP benchmark.

${\mathbf{S}}^{(1)}\left({\mathsf{\sigma }}_{\mathit{t}}\right)$	${\mathbf{S}}^{(1)}\left({\mathsf{\sigma }}_{\mathit{s}}\right)$	${\mathbf{S}}^{(1)}\left({\mathsf{\sigma }}_{\mathit{f}}\right)$	${\mathbf{S}}^{(1)}\left(\mathsf{\nu }\right)$	${\mathbf{S}}^{(1)}\left(\mathbf{q}\right)$	${\mathbf{S}}^{(1)}\left(\mathbf{p}\right)$	${\mathbf{S}}^{(1)}\left(\mathbf{N}\right)$
$n_{1-10}$= 8, $n_{>10.0}$= 0	$n_{1-10}$= 0, $n_{>10.0}$= 0	$n_{1-10}$= 0, $n_{>10.0}$= 0	$n_{1-10}$= 1, $n_{>10.0}$= 0	$n_{1-10}$= 1, $n_{>10.0}$= 0	$n_{1-10}$= 0, $n_{>10.0}$= 0	$n_{1-10}$= 3, $n_{>10.0}$= 0

Note: (1) $n_{1-10}$ denotes the total number of elements with absolute values between [1.0,10.0]; (2) $n_{>10.0}$ denotes the total number of elements with absolute values greater than 10.0.

presents a summary of the magnitudes attained by the 1st-order relative sensitivities of the PERP benchmark’s leakage response, which indicates that the largest 13 of these sensitivities attain values between 1.0 and 10.0; all others are smaller than 1.0. Of these 13 large sensitivities, 8 involve the total microscopic group cross sections, 3 involve the isotopic number densities (of ²³⁹Pu, ²⁴⁰Pu, and ¹H; see Table 3 in this work), 1 involves the average number of neutrons produced per fission of ²³⁹Pu, and 1 involves a source parameter. Additional details regarding these large 1st-order relative sensitivities (which involve nuclear data pertaining to ²³⁹Pu, ²⁴⁰Pu, and ¹H) of the PERP benchmark’s leakage response are further discussed below in comparison to the corresponding 2nd-order unmixed sensitivities.

Table 8. Summary of the large 2nd-order relative sensitivities for the PERP benchmark.

Parameters	${\mathsf{\sigma }}_{\mathit{t}}$	${\mathsf{\sigma }}_{\mathit{s}}$	${\mathsf{\sigma }}_{\mathit{f}}$	$\mathsf{\nu }$	$\mathbf{q}$	$\mathbf{N}$
${\mathsf{\sigma }}_t$	${\mathbf{S}}^{(2)}\left({\mathsf{\sigma }}_t,{\mathsf{\sigma }}_t\right)$ $n_{1-10}$= 665 $n_{>10.0}$= 55
${\mathsf{\sigma }}_s$	${\mathbf{S}}^{(2)}\left({\mathsf{\sigma }}_s,{\mathsf{\sigma }}_t\right)$ $n_{1-10}$= 52 $n_{>10.0}$= 1	${\mathbf{S}}^{(2)}\left({\mathsf{\sigma }}_s,{\mathsf{\sigma }}_s\right)$ $n_{1-10}$= 0 $n_{>10.0}$= 0
${\mathsf{\sigma }}_f$	${\mathbf{S}}^{(2)}\left({\mathsf{\sigma }}_f,{\mathsf{\sigma }}_t\right)$ $n_{1-10}$= 83 $n_{>10.0}$= 1	${\mathbf{S}}^{(2)}\left({\mathsf{\sigma }}_f,{\mathsf{\sigma }}_s\right)$ $n_{1-10}$= 0 $n_{>10.0}$= 0	${\mathbf{S}}^{(2)}\left({\mathsf{\sigma }}_f,{\mathsf{\sigma }}_f\right)$ $n_{1-10}$= 11 $n_{>10.0}$= 0
$\mathsf{\nu }$	${\mathbf{S}}^{(2)}\left(\mathsf{\nu },{\mathsf{\sigma }}_t\right)$ $n_{1-10}$= 171 $n_{>10.0}$= 8	${\mathbf{S}}^{(2)}\left(\mathsf{\nu },{\mathsf{\sigma }}_s\right)$ $n_{1-10}$= 0 $n_{>10.0}$= 0	${\mathbf{S}}^{(2)}\left(\mathsf{\nu },{\mathsf{\sigma }}_f\right)$ $n_{1-10}$= 28 $n_{>10.0}$= 0	${\mathbf{S}}^{(2)}\left(\mathsf{\nu },\mathsf{\nu }\right)$ $n_{1-10}$= 52 $n_{>10.0}$= 0
$\mathbf{q}$	${\mathbf{S}}^{(2)}\left(\mathbf{q},{\mathsf{\sigma }}_t\right)$ $n_{1-10}$= 32 $n_{>10.0}$= 0	${\mathbf{S}}^{(2)}\left(\mathbf{q},{\mathsf{\sigma }}_s\right)$ $n_{1-10}$= 0 $n_{>10.0}$= 0	${\mathbf{S}}^{(2)}\left(\mathbf{q},{\mathsf{\sigma }}_f\right)$ $n_{1-10}$= 0 $n_{>10.0}$= 0	${\mathbf{S}}^{(2)}\left(\mathbf{q},\mathsf{\nu }\right)$ $n_{1-10}$= 4 $n_{>10.0}$= 0	${\mathbf{S}}^{(2)}\left(\mathbf{q},\mathbf{q}\right)$ $n_{1-10}$= 0 $n_{>10.0}$= 0
$\mathbf{N}$	${\mathbf{S}}^{(2)}\left(\mathbf{N},{\mathsf{\sigma }}_t\right)$ $n_{1-10}$= 108 $n_{>10.0}$= 17	${\mathbf{S}}^{(2)}\left(\mathbf{N},{\mathsf{\sigma }}_s\right)$ $n_{1-10}$= 23 $n_{>10.0}$= 0	${\mathbf{S}}^{(2)}\left(\mathbf{N},{\mathsf{\sigma }}_f\right)$ $n_{1-10}$= 20 $n_{>10.0}$= 1	${\mathbf{S}}^{(2)}\left(\mathbf{N},\mathsf{\nu }\right)$ $n_{1-10}$= 28 $n_{>10.0}$= 6	${\mathbf{S}}^{(2)}\left(\mathbf{N},\mathbf{q}\right)$ $n_{1-10}$= 9 $n_{>10.0}$= 0	${\mathbf{S}}^{(2)}\left(\mathbf{N},\mathbf{N}\right)$ $n_{1-10}$= 9 $n_{>10.0}$= 3

Note: (1) $n_{1-10}$ denotes the total number of elements with absolute values between [1.0,10.0]; (2) $n_{>10.0}$ denotes the total number of elements with absolute values greater than 10.0.

presents the summary of the 2nd-order mixed relative sensitivities of the PERP benchmark’s leakage response having absolute values greater than 1.0. Since Table 8 is symmetrical, the upper triangular submatrices are not shown in the table; of course, each of the upper triangular submatrices has the same number of large sensitivities as its symmetrical counterpart. Table 8 indicates that 126 2nd-order relative sensitivities have values greater than 10.0, and 1853 2nd-order relative sensitivities have values between 1.0 and 10.0. Evidently, the number of 2nd-order relative sensitivities having values greater than 1.0 is far larger than the number of 1st-order relative sensitivities that have values greater than 1.0.

The results detailed in Parts I−V [1,2,3,4,5] and in the present work have indicated that the largest 1st- and unmixed 2nd-order sensitivities of the PERP benchmark’s leakage response are with respect to the nuclear parameters related to ²³⁹Pu. Figure 2 presents the comparison of the absolute values of the 1st-order relative sensitivities, for each of the 30 energy groups, of the PERP benchmark’s leakage response with respect to the following groups of imprecisely known parameters related to ²³⁹Pu: (i) the total microscopic cross sections [1]; (ii) the scattering microscopic cross sections [2]; (iii) fission microscopic cross sections [3]; (iv) the average number of neutrons per fission [3]; (v) the source parameters [4]; and (vi) the fission spectrum, as presented in Table 5 in this work. As Figure 2 indicates, the 1st-order relative sensitivities of the PERP’s leakage response with respect to the imprecisely known parameters related to ²³⁹Pu can be ranked in the following order of importance:

(i)

The 1st-order relative sensitivities $S^{\left(1\right)}({\sigma }_{t,1}^g)$ of the leakage response with respect to the total cross sections of ²³⁹Pu have the largest magnitudes for all energy groups;

(ii)

The next largest magnitudes are displayed by the 1st-order relative sensitivities $S^{\left(1\right)}({\nu }_1^g)$ with respect to the average number of neutrons per fission of ²³⁹Pu. The values of these 1st-order sensitivities are about 30% to 50% larger than those corresponding to the fission cross sections, which follow next in the importance ranking.

(iii)

The third largest are the 1st-order relative sensitivities $S^{\left(1\right)}({\sigma }_{f,1}^g)$ of the PERP benchmark’s leakage response with respect to the fission cross sections of ²³⁹Pu.

(iv)

The fourth largest are the 1st-order relative sensitivities $S^{\left(1\right)}({\sigma }_{s,l=0,1}^{g\to g})$ of the leakage response with respect to the 0th-order self-scattering cross sections for ²³⁹Pu. These 1st-order relative sensitivities are approximately one order of magnitude smaller than the corresponding 1st-order relative sensitivities with respect to the fission cross sections of ²³⁹Pu.

(v)

The next-to-last in the importance ranking are the 1st-order relative sensitivities $S^{\left(1\right)}({\chi }_1^g)$ of the leakage response with respect to the fission spectrum parameters;

(vi)

Finally, the 1st-order relative sensitivities $S^{\left(1\right)}(q_1^g)$ of the leakage response with respect to the source parameters of ²³⁹Pu are the smallest of the sensitivities pertaining to ²³⁹Pu.

The largest 2nd-order unmixed sensitivities of the PERP benchmark’s leakage response also involve the imprecisely known parameters pertaining to ²³⁹Pu. Based on the results presented in Parts I−III [1,2,3], Figure 3 illustrates the importance ranking for the absolute values of the unmixed 2nd-order relative sensitivities with respect to the parameters underlying the total, scattering and fission cross sections, and the average number of neutrons per fission of ²³⁹Pu.

The results presented in Figure 3 highlight the following features:

(i)

For every energy group, the 2nd-order unmixed sensitivities with respect to the total cross sections are the largest and significantly larger than any others.

(ii)

The 2nd-order unmixed sensitivities with respect to the parameters underlying the average number of neutrons per fission are the second largest, being approximately 100–200% larger than the 2nd-order unmixed sensitivities with respect to the fission cross sections.

(iii)

The 2nd-order unmixed sensitivities with respect to scattering cross sections are negligibly small comparing to others.

(iv)

Comparing the results shown in Figure 2 and Figure 3 reveals the important conclusion that the 2nd-order unmixed sensitivities of the leakage response with respect to the microscopic total group cross sections of ²³⁹Pu are significantly (over 300–400%) larger than the corresponding 1st-order sensitivities.

The next largest group of 1st- and 2nd-order relative sensitivities of the PERP’s leakage response pertain to parameters related to ²⁴⁰Pu, which are displayed in Figure 4 and Figure 5. Comparing the results displayed in Figure 4 and Figure 5 to the results presented in Figure 2 and Figure 3 indicates that, albeit important, the sensitivities of the PERP’s leakage response to parameters pertaining to ²⁴⁰Pu are at least an order of magnitude smaller than the sensitivities pertaining to ²³⁹Pu.

The PERP benchmark’s leakage response displays remarkably large 1st- and 2nd-order sensitivities with respect to the total microscopic cross section of ¹H in the lowest-energy group. These large sensitivities are illustrated in Figure 6 and Figure 7.

Figure 8, Figure 9, Figure 10 and Figure 11 present comprehensive comparisons of the contributions of the 1st- and, respectively, 2nd-order sensitivities of the PERP’s leakage response to the expected values, variances and skewness of this response, considering that all of the PERP’s model parameters (fission, scattering and fission microscopic cross sections, the average number of neutrons per fission, isotopic number densities, and source parameters) are uncorrelated and have uniform standard deviations of 5% or 10%, respectively. The vertical-axes in Figure 8, Figure 9, Figure 10 and Figure 11 use logarithmic scales.

Thus, Figure 8 illustrates the magnitudes of the 2nd-order contributions, ${\left[E\left(L\right)\right]}_i^{(2,U)},\hspace{0.17em}i=t,N,\nu ,f,s,q$, to the expected value, $E\left(L\right)$, of the PERP’s leakage response. The 2nd-order contributions, ${\left[E\left(L\right)\right]}_i^{(2,U)},$ are computed in [1,2,3,4] using expressions similar to Equation (114) and stem from 2nd-order sensitivities of the parameters underlying the total, scattering and fission cross sections, the average number of neutrons per fission, isotopic number densities, and the source parameters, respectively. Since correlations among parameters are unavailable, these parameters are considered to be uncorrelated and having all uniform standard deviations (SD) of either 5% or 10% respectively. A standard deviation of 5% is typical for the available nuclear data while the standard deviation of 10% is used to illustrate how an increase in the parameter uncertainties would affect the expected value of the leakage response. The nominal value, $L\left({\mathsf{\alpha }}^0\right)$, of the leakage response is also displayed in Figure 8, in order to provide an evident standard for assessing the importance of the 2nd-order contributions ${\left[E\left(L\right)\right]}_i^{(2,U)}$ to the expected value, $E\left(L\right)$, of the PERP’s leakage response. Recall from Equation (113) that if the 2nd-order sensitivities are ignored or are unavailable, the expected value, $E\left(L\right)$, will coincide with the nominal value, $L\left({\mathsf{\alpha }}^0\right)$. The comparison presented in Figure 8 indicates that, assuming a uniform standard deviation of either 5% or 10% for all parameters, the magnitudes of the 2nd-order contributions stemming from the various imprecisely known nuclear data fall into the following ranking order (from small to large):

${\left[E\left(L\right)\right]}_q^{\left(2,U\right)}\ll {\left[E\left(L\right)\right]}_s^{\left(2,U\right)}<{\left[E\left(L\right)\right]}_f^{\left(2,U\right)}<{\left[E\left(L\right)\right]}_{\nu }^{\left(2,U\right)}\ll {\left[E\left(L\right)\right]}_N^{\left(2,U\right)}\ll {\left[E\left(L\right)\right]}_t^{\left(2,U\right)}$.

Figure 9 presents a comparison of contributions to the 1st-order variance, ${\left[\mathrm{var}\left(L\right)\right]}_i^{(1,U,N)}$, computed using expressions similar to Equation (116), of the 1st-order sensitivities of the PERP leakage response to the PERP parameters (considered to be uncorrelated and normally distributed) underlying the total, scattering and fission cross sections, the average number of neutrons per fission, isotopic number densities, and the source parameters, as presented in [1,2,3,4], respectively. The comparison presented in Figure 9 indicates that, assuming a uniform standard deviation of either 5% or 10% for all parameters, the magnitudes of the 2nd-order contributions stemming from the various imprecisely known nuclear data fall into the following ranking order (from small to large):

${\left[\mathrm{var}\left(L\right)\right]}_s^{(1,U,N)}\ll {\left[\mathrm{var}\left(L\right)\right]}_f^{(1,U,N)}<{\left[\mathrm{var}\left(L\right)\right]}_q^{(1,U,N)}<{\left[\mathrm{var}\left(L\right)\right]}_v^{(1,U,N)}\ll {\left[\mathrm{var}\left(L\right)\right]}_N^{(1,U,N)}<{\left[\mathrm{var}\left(L\right)\right]}_t^{(1,U,N)}.$

Figure 10 presents a comparison of contributions to the 2nd-order variance, ${\left[\mathrm{var}\left(L\right)\right]}_i^{(2,U,N)}$, computed using expressions similar to Equation (117), of the 2nd-order sensitivities of the PERP leakage response to the PERP parameters (considered to be uncorrelated and normally distributed) underlying the total, scattering and fission cross sections, the average number of neutrons per fission, isotopic number densities, and the source parameters, as presented in [1,2,3,4], respectively. The comparison presented in Figure 10 indicates that, assuming a uniform standard deviation of either 5% or 10% for all parameters, the magnitudes of the 2nd-order contributions stemming from the various imprecisely known nuclear data fall into the following ranking order (from small to large):

${\left[\mathrm{var}\left(L\right)\right]}_q^{(2,U,N)}\ll {\left[\mathrm{var}\left(L\right)\right]}_s^{(2,U,N)}\ll {\left[\mathrm{var}\left(L\right)\right]}_f^{(2,U,N)}<{\left[\mathrm{var}\left(L\right)\right]}_v^{(2,U,N)}\ll {\left[\mathrm{var}\left(L\right)\right]}_N^{(2,U,N)}\ll {\left[\mathrm{var}\left(L\right)\right]}_t^{(2,U,N)}.$

Figure 11 presents a comparison of contributions to the skewness, ${\left[{\gamma }_1\left(L\right)\right]}_i^{\left(U,N\right)}$, computed using expressions similar to Equation (119), of the PERP leakage response to the PERP parameters (considered to be uncorrelated and normally distributed) underlying the total, scattering and fission cross sections, the average number of neutrons per fission, isotopic number densities, and the source parameters, as presented in [1,2,3,4], respectively. The comparison presented in Figure 11 indicates that, assuming a uniform standard deviation of either 5% or 10% for all parameters, the magnitudes of the 2nd-order contributions stemming from the various imprecisely known nuclear data fall into the following ranking order (from small to large):

${\left[{\gamma }_1\left(L\right)\right]}_s^{\left(U,N\right)}<0<{\left[{\gamma }_1\left(L\right)\right]}_q^{\left(U,N\right)}\ll {\left[{\gamma }_1\left(L\right)\right]}_f^{\left(U,N\right)}<{\left[{\gamma }_1\left(L\right)\right]}_v^{\left(U,N\right)}<{\left[{\gamma }_1\left(L\right)\right]}_t^{\left(U,N\right)}\ll {\left[{\gamma }_1\left(L\right)\right]}_N^{\left(U,N\right)}.$

The results presented in Figure 8, Figure 9 and Figure 10 indicate that the contributions to the expected values and the (1st- and 2nd-order) variances of the PERP leakage stemming from the imprecisely known PERP parameters follow the following order of importance:

(i)

The contributions stemming from the 1st- and 2nd-order sensitivities of the leakage response to the group-averaged uncorrelated total cross sections are the largest, by a significant margin, by comparison to the contributions from any of the other parameters. The contributions from the 2nd-order sensitivities are paramount, causing the following significant effects: (i) the expected value of the leakage response is significantly larger than the corresponding computed value, and (ii) $S{D}^{\left(2\right)}\gg S{D}^{\left(1\right)}$, i.e., the 2nd-order standard deviation $S{D}^{\left(2\right)}$ is significantly larger than the 1st-order standard deviation $S{D}^{\left(1\right)}$.

(ii)

The contributions stemming from the 1st- and 2nd-order sensitivities of the leakage response to the uncorrelated isotopic number densities are second in importance.

(iii)

Ranked third in importance are the contributions stemming from the 1st- and 2nd-order sensitivities of the leakage response to the average number of neutrons per fission.

(iv)

The contributions to the expected values and variances stemming from the group-averaged uncorrelated fission cross section parameters rank overall fourth in importance, even if the relation ${\left[\mathrm{var}\left(L\right)\right]}_f^{(1,U,N)}<{\left[\mathrm{var}\left(L\right)\right]}_q^{(1,U,N)}$ indicates that the contributions stemming from the 1st-order sensitivities of the response to the fission parameters are slightly smaller than the contributions stemming from the 1st-order response sensitivities to the source parameters.

(v)

The contributions stemming from the group-averaged uncorrelated microscopic scattering cross sections and source parameters are smaller by factors of 10³–10⁷ by comparison to the contributions stemming from the 1st- and 2nd-order response sensitivities with respect to the other parameters.

Notably, the results summarized in Figure 11 indicate that the contribution to the skewness stemming from the uncorrelated isotopic number densities is much larger than that stemming from other parameters. It is also remarkable that the skewness stemming solely from the scattering microscopic group cross sections is negative, meaning that if the only imprecisely known PERP parameters were the scattering microscopic group cross sections, then the distribution of the PERP’s leakage response would be skewed towards values lower than the expectation of the response. In contradistinction, the skewness arising from uncertainties in the other parameters are all positive, signifying that uncertainties in all of the other PERP model parameters would cause the distribution of the PERP’s leakage response to be skewed towards values higher than the expectation of the response.

Assuming that all PERP model parameters are uncorrelated and assuming that all of these parameters have uniform relative standard deviations of 5%, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 present results for the following quantities: (i) the expected value $\hspace{0.17em}\left[E\left(L\right)\right]$ of the PERP leakage response, which takes into account the contributions stemming from the corresponding 2nd-order sensitivities; (ii) the standard deviations, $S{D}^{\left(1\right)}$, for the PERP leakage response arising solely from the 1st-order sensitivities (in green); (iii) the standard deviations, $S{D}^{\left(2\right)}$, for the PERP leakage response arising solely from the 2nd-order sensitivities; and (iv) the sum, $S{D}^{\left(1\right)}+S{D}^{\left(2\right)}$, for the PERP leakage response stemming from both the 1st- and 2nd-order sensitivities. In Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, the green-colored plots involve solely 1st-order sensitivities, while the red-colored plots depict the contributions from 2nd-order sensitivities. Consistent with the importance rankings illustrated in Figure 8, Figure 9 and Figure 10, the results presented in Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 indicate that the largest effects of the 2nd-order sensitivities are displayed by the results illustrated in Figure 12 for the total microscopic cross sections, followed, in order of importance, by the contributions involving the isotopic number densities (displayed in Figure 13), and followed the contributions involving the average number of neutrons per fission (displayed in Figure 14). As indicated by the results displayed in Figure 15 and Figure 16, the contributions stemming from the uncorrelated fission microscopic cross sections and, respectively, the scattering microscopic cross sections are negligibly small, as are the contributions stemming from the sensitivities of the leakage response with respect to the source parameters (not shown).

Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21 also present results for: (i) the expected value $\hspace{0.17em}\left[E\left(L\right)\right]$ of the PERP leakage response; (ii) the standard deviations, $S{D}^{\left(1\right)}$, for the PERP leakage response arising solely from the 1st-order sensitivities; (iii) the standard deviations, $S{D}^{\left(2\right)}$, for the PERP leakage response arising solely from the 2nd-order sensitivities; and (iv) the sum, $S{D}^{\left(1\right)}+S{D}^{\left(2\right)}$, for the PERP leakage response stemming from both the 1st- and 2nd-order sensitivities. As in Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, the green-colored plots in Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21 involve solely 1st-order sensitivities, while the red-colored plots in Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21 depict the contributions from 2nd-order sensitivities. In contradistinction with the results displayed in Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, however, the results displayed in Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21 assume that all PERP model parameters are uncorrelated and have uniform relative standard deviations of 10% (rather than 5%). Although standard deviations of 10% are rather large for nuclear data, these larger standard deviations highlight dramatically the effects of the 2nd-order sensitivities in uncertainty analysis.

Consistent with the importance rankings illustrated in Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, the results presented in Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21 indicate that importance ranking of the parameter sensitivities displayed for standard deviations of 5% remains unchanged when the standard deviations are uniformly increased to 10%. Very important, however, are the results displayed in the green-colored plots in Figure 17, which indicate that, if the 2nd-order sensitivities of the leakage response to the total microscopic group cross sections are neglected, the use of only the 1st-order sensitivities yields unphysical results, since the quantity $L\left({\mathsf{\alpha }}^0\right)-S{D}^{\left(1\right)}$ takes on unphysical negative values.

The sequence of works [1,2,3,4,5] and the present work, labeled as Parts I-VI, have applied the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) conceived by Cacuci [6,7] to compute exactly the 21,976 first-order sensitivities and 482,944,576 second-order sensitivities of a polyethylene-reflected plutonium (PERP) benchmark’s leakage response with respect to the benchmark’s imprecisely known parameters. The 2nd-ASAM is the only known methodology, at this time, which enables such large-scale exhaustive computations of the exact expressions of 2nd-order response sensitivities. The results of these pioneering computations have indicated that there are many 2nd-order sensitivities that have values significantly larger than the 1st-order ones. Specifically, 13 first-order sensitivities attain values between 1.0 and 10.0, while 126 second-order relative sensitivities have values greater than 10.0, and 1853 second-order relative sensitivities have values between 1.0 and 10.0. Hence, the number of 2nd-order relative sensitivities having values greater than 1.0 is obviously far larger than the number of 1st-order relative sensitivities that have values greater than 1.0.

Since correlations among the PERP’s model parameters are unavailable, the effects of the 1st- and 2nd-order sensitivities in contributing to the expected value and standard deviation of the PERP’s leakage response have mostly been illustrated by assuming that all model’s parameters are uncorrelated. This assumption eliminates the effects of the mixed 2nd-order sensitivities. On the other hand, the illustrative results presented in Part I [1] for the extreme case of fully correlated total cross sections have shown that neglecting the 2nd-order sensitivities would cause an error as large as 2000% in the expected value of the leakage response, and up to 6000% in the variance of the leakage response for the microscopic group total cross sections. These illustrative examples have highlighted the fact that the importance of the mixed 2nd-order sensitivities increases as the respective cross sections correlations and standard deviations increase. Of course, neither the fully uncorrelated nor the fully correlated illustrative examples presented in this sequence of works [1,2,3,4,5] realistically describe the actual physical situations regarding the parameters describing the total and other microscopic cross section. The fully uncorrelated case underestimates reality while the fully correlated case overestimates it. In reality, cross sections are partially correlated, so reality falls in between the fully uncorrelated and fully correlated cases. Nevertheless, the illustrative examples in [1,2,3,4,5] underscore the need for future experimental research aimed at obtaining values for the correlations that might exist among the various cross sections, which are unavailable at this time. Correlations among the model parameters (e.g., total and fission cross sections, isotopic number densities, and the average number of neutrons per fission) provide contributions to the response distribution moments (i.e., expected values, variance, skewness, etc.) in addition to those stemming from the parameters’ standard deviations.

All in all, this sequence of works [1,2,3,4,5] aimed at drawing attention to the fact that 2nd-order sensitivities are important, and their effects must be assessed for each physical system under consideration. For the PERP benchmark, in particular, it has been shown that the 2nd-order sensitivities are even more important than the 1st-order ones. While the effects of the 2nd-order sensitivities may be less marked for other reactor physics systems, the point is that they are not always negligible, as they have been a priori assumed in hitherto in the published literature. Finally, since this sequence of works [1,2,3,4,5] has revealed that the 2nd-order mixed relative sensitivities of the PERP benchmark’s leakage response to this benchmark’s total cross sections exhibit a very large number of values that are greater than 1.0 (including many having values significantly larger than 10.0), it is of clear interest to compute the magnitude of the corresponding 3rd-order sensitivities of the leakage response with respect to the parameters underlying the PERP’s total cross sections. The computation of these 3rd-order sensitivities is currently under way and will be reported soon.