This work extends to third-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a poly ethylene- reflected plutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The PERP benchmark comprises 21,976 imprecisely known (uncertain) model parameters. Previous works have used the adjoint sensitivity analysis methodology to compute exactly and efficiently all of the 21,976 first-order and (21,976) 2 second-order sensitivities of the PERP benchmark’s leakage response to all of the benchmark’s uncertain parameters, showing that the largest and most consequential 1 st- and 2 nd-order response sensitivities are with respect to the total microscopic cross sections. These results have motivated extending the previous adjoint-based derivations to third-order, leading to the derivation, in this work, of the exact mathematical expressions of the (180) 3 third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections. The formulas derived in this work are valid not only for the PERP benchmark but can also be used for computing the 3 rd-order sensitivities of the leakage response of any nuclear system involving fissionable material and internal or external neutron sources. Subsequent works will use the adjoint-based mathematical expressions obtained in this work to compute exactly and efficiently the numerical values of these (180) 3 third-order sensitivities (which turned out to be very large and consequential) and use them for a third-order uncertainty analysis of the PERP benchmark’s leakage response.
Until recently, only the first-order sensitivities (i.e., functional derivatives) of a computational model’s responses (i.e., quantities of interest) to the respective model’s imprecisely known (i.e., uncertain) parameters have been taken into account when assessing the uncertainties induced in the respective responses by the parameter uncertainties. The second- and higher-order sensitivities could not be computed, except for very simple models comprising a handful of parameters, so these sensitivities were ignored. The Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) recently conceived by Cacuci [
All of the 21,976 first-order sensitivities and (21,976)2 second-order sensitivities of the PERP leakage response with respect to the benchmark’s parameters were computed, ranked, and analyzed in [
This work is organized as follows: Section 2 describes the methodology for computing the leakage response of the PERP benchmark. Section 3 presents the derivation of the exact analytical expressions of the third-order mixed sensitivities of the PERP leakage response to total cross sections. Section 4 concludes this work.
This section presents the derivation of the exact analytical expressions of the 3rd-order sensitivities of the polyethylene-reflected plutonium (acronym: PERP) metal sphere OECD/NEA benchmark [
B g ( α ) φ g ( r , Ω ) = Q g ( r ) , g = 1 , ⋯ , G , (1)
φ g ( r d , Ω ) = 0 , Ω ⋅ n < 0 , g = 1 , ⋯ , G , (2)
where r d is the radius of the PERP sphere, and where
B g ( α ) φ g ( r , Ω ) ≜ Ω ⋅ ∇ φ g ( r , Ω ) + Σ t g ( α ; r ) φ g ( r , Ω ) − ∑ g ′ = 1 G ∫ 4 π d Ω ′ φ g ′ ( r , Ω ′ ) [ Σ s g ′ → g ( α ; r , Ω ′ → Ω ) + χ g ( α ; r ) ( ν Σ ) f g ′ ( α ; r ) ] , (3)
Q g ( r ) ≜ ∑ k = 1 N f λ k N k , 1 F k S F ν k S F ( 2 π a k 3 b k e − a k b k 4 ) ∫ E g + 1 E g d E e − E / a k sinh b k E . (4)
The Boltzmann-operator B g ( α ) in Equation (3) contains implicitly a factor 1 / 4 π in its scattering and fission terms, to conform to the convention used by PARTISN [
The mathematical expression of the PERP benchmark’s leakage response, denoted as L ( α ) , is provided below:
L ( α ) ≜ ∫ S b d S ∑ g = 1 G ∫ Ω ⋅ n > 0 d Ω Ω ⋅ n φ g ( r , Ω ) . (5)
The vector α , which appears in the expressions of the Boltzmann-operator B g ( α ) and the leakage response L ( α ) , in Equation (3) and Equation (5), respectively, represents the “vector of imprecisely known model parameters”, and is defined as follows:
α ≜ [ α 1 , ⋯ , α J α ] † ≜ [ σ t ; σ s ; σ f ; ν ; p ; q ; N ] † , J α = J σ t + J σ s + J σ f + J ν + J p + J q + J n . (6)
The components of the vector α are described in the Appendix. Since only the effects of the uncertainties in the total macroscopic cross sections will be considered in this work, only the components of the vector t for the parameters of macroscopic total cross sections will be explicitly used; they are reproduced below from the Appendix:
t ≜ [ t 1 , ⋯ , t J t ] † ≜ [ t 1 , ⋯ , t J σ t ; n 1 , ⋯ , n J n ] † ≜ [ σ t ; N ] † , J t = J σ t + J n , (7)
where
σ t ≜ [ t 1 , ⋯ , t J σ t ] † ≜ [ σ t , i = 1 1 , σ t , i = 1 2 , ⋯ , σ t , i = 1 G , ⋯ , σ t , i g , ⋯ , σ t , i = I 1 , ⋯ , σ t , i = I G ] † , i = 1 , ⋯ , I = 6 ; g = 1 , ⋯ , G = 30 ; J σ t = I × G , (8)
N ≜ [ n 1 , ⋯ , n J n ] † ≜ [ N 1 , 1 , N 2 , 1 , N 3 , 1 , N 4 , 1 , N 5 , 2 , N 6 , 2 ] † , J n = 6. (9)
In Equations (7) through (9), the dagger denotes “transposition,” σ t , i g denotes the microscopic total cross section for isotope i and energy group g, N i , m denotes the respective isotopic number density for isotope i and material m, J σ t denotes the total number of microscopic total cross sections, and J n denotes the total number of isotopic number densities in the model. Thus, the vector t comprises a total of J t = J σ t + J n = 30 × 6 + 6 = 186 uncertain model parameters. However, only the components σ t ≜ [ t 1 , ⋯ , t J σ t ] † ≜ [ σ t , i = 1 1 , σ t , i = 1 2 , ⋯ , σ t , i = 1 G , ⋯ , σ t , i g , ⋯ , σ t , i = I 1 , ⋯ , σ t , i = I G ] † , i = 1 , ⋯ , I = 6 , g = 1 , ⋯ , G = 30 , J σ t = I × G = 180 , will be considered in this work. Thus, the numbers of sensitivities of the PERP leakage response with respect to the total microscopic cross sections are as follows: 1) 180 first-order sensitivities; 2) 32,400 second-order sensitivities, of which 16,290 are distinct; and 3) 5,832,000 third-order sensitivities, of which 988,260 are distinct. The 180 first-order and 16,290 second-order sensitivities were obtained and analyzed in [
This Section will present the derivation of the 3rd-order mixed sensitivities ∂ 3 L ( α ) / ∂ t j ∂ t k ∂ t l , j , k , l = 1 , ⋯ , J σ t of the PERP leakage response with respect to the group-averaged microscopic total cross sections. These 3rd-order sensitivities will be derived by using two alternative procedures, which will be presented in sections 3.1 and 3.2, respectively. In section 3.1, the sensitivities ∂ 3 L ( α ) / ∂ t j ∂ t k ∂ t l , j , k , l = 1 , ⋯ , J σ t will be obtained as a particular case of the general expressions derived in [
The general expression of the 3rd-order sensitivities of a reaction-rate type response to the nuclear data characterizing a physical system modeled by the multigroup neutron transport equation has been derived in the Appendix of [
Σ d g ( α ) ≜ Ω ⋅ n δ ( r − r d ) , (10)
α i 1 → t j ≡ σ t , i j g j ; i 1 → j = 1 , ⋯ , J σ t ; α i 2 → t k ≡ σ t , i k g k ; i 2 → k = 1 , ⋯ , J σ t ; α i 3 → t l ≡ σ t , i l g l ; i 3 → l = 1 , ⋯ , J σ t . (11)
It has been shown in [
∂ Σ t g ( t ) ∂ t j ≜ ∂ Σ t g ( t ) ∂ σ t , i j g j = ∂ ( ∑ m = 1 M ∑ i = 1 I N i , m σ t , i g ) ∂ σ t , i j g j = δ g j g N i j , m j , (12)
∂ 2 Σ t g ( t ) ∂ t j ∂ t k ≜ ∂ 2 Σ t g ( t ) ∂ σ t , i j g j ∂ σ t , i k g k = ∂ ( δ g j g N i j , m j ) ∂ σ t , i k g k = 0 , (13)
where the subscripts i j , g j and m j denote the isotope, the energy group and material associated with the parameter σ t , i j g j , respectively; the subscripts i k , g k and m k denote the isotope, the energy group and material associated with the parameter σ t , i k g k , respectively; and δ g j g denotes the Kronecker-delta functional (e.g., δ g j g = 1 if g j = g ; δ g j g = 0 if g j ≠ g ).
It has been shown in the Appendix of [
∂ 3 R ( α , φ ) ∂ α i 3 ∂ α i 2 ∂ α i 1 = 〈 φ g ( r , Ω ) , ∂ 3 Σ d g ( α ) ∂ α i 3 ∂ α i 2 ∂ α i 1 〉 ( 1 ) + 〈 ψ ( 1 ) , g ( r , Ω ) , ∂ 3 Q ( 1 ) , g ( α ) ∂ α i 3 ∂ α i 2 ∂ α i 1 〉 ( 1 ) + 〈 ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) , ∂ 2 Q ( 1 ) , g ( α ) ∂ α i 3 ∂ α i 2 〉 ( 1 ) + 〈 ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) , ∂ 2 Q ( 2 ) , g ( α ) ∂ α i 3 ∂ α i 2 〉 ( 1 ) + 〈 ψ 1 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , ∂ Q ( 1 ) , g ( α ) ∂ α i 3 〉 ( 1 ) + 〈 ψ 2 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , ∂ Q ( 2 ) , g ( α ) ∂ α i 3 〉 ( 1 ) + 〈 ψ 3 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , ∂ Q 1 ( 3 ) , g ( α ) ∂ α i 3 〉 ( 1 ) + 〈 ψ 4 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , ∂ Q 2 ( 3 ) , g ( α ) ∂ α i 3 〉 ( 1 ) , for i 1 = 1 , ⋯ , J α ; i 2 = 1 , ⋯ , i 1 ; i 3 = 1 , ⋯ , i 2 , (14)
where the inner product 〈 u ( r , Ω ) , w ( r , Ω ) 〉 ( 1 ) between two vector-values functions, each having G components of the form u ( r , Ω ) ≜ [ u 1 ( r , Ω ) , ⋯ , u g ( r , Ω ) , ⋯ , u G ( r , Ω ) ] † and w ( r , Ω ) ≜ [ w 1 ( r , Ω ) , ⋯ , w g ( r , Ω ) , ⋯ , w G ( r , Ω ) ] † , is defined as follows:
〈 u g ( r , Ω ) , w g ( r , Ω ) 〉 ( 1 ) ≜ ∑ g = 1 G ∫ d V ∫ 4 π d Ω u g ( r , Ω ) w g ( r , Ω ) . (15)
In Equation (14), the 3rd-level adjoint functions ψ j ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , j = 1 , 2 , 3 , 4 ; g = 1 , ⋯ , G are the solutions of the following multigroup form of the “third-level adjoint sensitivity system” (3rd-LASS) presented in the Appendix of [
A g ( α ) ψ 4 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) = ∇ ψ 2 ( 2 ) ( i 1 ) R ( 2 ) , g ( α ) , (16)
ψ 4 ( 3 ) , g ( i 1 , i 2 ; r d , Ω ) = 0 , Ω ⋅ n > 0 , i 1 = 1 , ⋯ , J α ; i 2 = 1 , ⋯ , i 1 ; (17)
A g ( α ) ψ 1 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) = ∇ φ R ( 2 ) , g ( α ) − [ F 41 ( 3 ) , g ] * ψ 4 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , (18)
ψ 1 ( 3 ) , g ( i 1 , i 2 ; r d , Ω , E ) = 0 , Ω ⋅ n > 0 , i 1 = 1 , ⋯ , J α ; i 2 = 1 , ⋯ , i 1 ; (19)
B g ( α ) ψ 3 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) = ∇ ψ 1 ( 2 ) ( i 1 ) R ( 2 ) , g ( α ) , (20)
ψ 3 ( 3 ) , g ( i 1 , i 2 ; r d , Ω ) = 0 , Ω ⋅ n < 0 , i 1 = 1 , ⋯ , J α ; i 2 = 1 , ⋯ , i 1 ; (21)
B g ( α ) ψ 2 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) = ∇ ψ ( 1 ) R ( 2 ) , g ( α ) − [ F 32 ( 3 ) , g ] * ψ 3 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , (22)
ψ 2 ( 3 ) , g ( i 1 , i 2 ; r d , Ω ) = 0 , Ω ⋅ n < 0 , i 1 = 1 , ⋯ , J α ; i 2 = 1 , ⋯ , i 1 , (23)
where
A g ( α ) ψ 1 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) ≜ − Ω ⋅ ∇ ψ 1 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) + Σ t g ( t ) ψ 1 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) − ∑ g ′ = 1 G ∫ 4 π d Ω ′ ψ 1 ( 3 ) , g ′ ( i 1 , i 2 ; r , Ω ) [ Σ s g → g ′ ( s ; Ω → Ω ′ ) + χ g ′ ( p ) ν Σ f g ( f ) ] , (24)
and where the various quantities serving as “sources” on the right-sides of the 3rd-LASS are defined as follows:
∂ Q ( 1 ) , g ( α ; φ ) ∂ α i 2 ≜ ∂ Q g ( α ; r , Ω ) ∂ α i 2 − ∂ [ B g ( α ) φ g ( r , Ω ) ] ∂ α i 2 , (25)
∂ 2 Q ( 1 ) , g ( α ; φ ) ∂ α i 3 ∂ α i 2 ≜ ∂ 2 Q g ( α ; r , Ω ) ∂ α i 3 ∂ α i 2 − ∂ 2 [ B g ( α ) φ g ( r , Ω ) ] ∂ α i 3 ∂ α i 2 , (26)
∂ Q ( 2 ) , g ( α ; ψ ( 1 ) ) ∂ α i 2 = ∂ Σ d g ( α ; r , Ω ) ∂ α i 2 − ∂ [ A g ( α ) ψ ( 1 ) , g ( r , Ω ) ] ∂ α i 2 , (27)
∇ ψ 2 ( 2 ) ( i 1 ) R ( 2 ) , g ( α ) ≜ ∂ Q ( 2 ) , g ( α ) ∂ α i 2 , (28)
[ F 41 ( 3 ) , g ] * ψ 4 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) ≜ ψ 4 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) ∂ Σ t g ( t ) ∂ α i 1 − ∑ g ′ = 1 G ∫ 4 π d Ω ′ ψ 4 ( 3 ) , g ′ ( i 1 , i 2 ; r , Ω ′ ) × { ∂ Σ s g → g ′ ( s ; Ω → Ω ′ ) ∂ α i 1 + ∂ [ ν Σ f g ( f ) ] ∂ α i 1 χ g ′ + ν Σ f g ( f ) ∂ χ g ′ ∂ α i 1 } , (29)
∇ ψ 1 ( 2 ) ( i 1 ) R ( 2 ) , g ( α ) ≜ ∂ Q ( 1 ) , g ( α , φ ) ∂ α i 2 , (30)
∇ φ R ( 2 ) , g ( α ) ≜ ∂ 2 Σ d g ( α ; r , Ω ) ∂ α i 1 ∂ α i 2 − ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) ∂ Σ t g ( t ) ∂ α i 2 + ∑ g ′ = 1 G ∫ 4 π d Ω ′ ψ 1 ( 2 ) , g ′ ( i 1 ; r , Ω ′ )
× { ∂ Σ s g → g ′ ( s ; Ω → Ω ′ ) ∂ α i 2 + ν Σ f g ( f ) ∂ χ g ′ ∂ α i 2 + ∂ [ ν Σ f g ( f ) ] ∂ α i 2 χ g ′ } − ψ ( 1 ) , g ( r , Ω ) ∂ 2 Σ t g ( t ) ∂ α i 1 ∂ α i 2 + ∑ g ′ = 1 G ∫ 4 π d Ω ′ ψ ( 1 ) , g ′ ( r , Ω ′ ) × { ∂ 2 Σ s g → g ′ ( s ; Ω → Ω ′ ) ∂ α i 1 ∂ α i 2 + χ g ′ ∂ 2 [ ν Σ f g ( f ) ] ∂ α i 1 ∂ α i 2 + ∂ [ ν Σ f g ( f ) ] ∂ α i 1 ∂ χ g ′ ∂ α i 2 + ∂ χ g ′ ∂ α i 1 ∂ [ ν Σ f g ( f ) ] ∂ α i 2 + ν Σ f g ( f ) ∂ 2 χ g ′ ∂ α i 1 ∂ α i 2 } , (31)
∇ ψ ( 1 ) R ( 2 ) , g ( α ) ≜ − φ g ( r , Ω ) ∂ 2 Σ t g ( t ) ∂ α i 2 ∂ α i 1 + ∑ g ′ = 1 G ∫ 4 π d Ω ′ φ g ′ ( r , Ω ′ ) { ∂ 2 Σ s g ′ → g ( s ; Ω ′ → Ω ) ∂ α i 2 ∂ α i 1 + ∂ χ g ∂ α i 2 ∂ [ ( ν Σ f ) g ′ ] ∂ α i 1 + χ g ∂ 2 [ ( ν Σ f ) g ′ ] ∂ α i 2 ∂ α i 1 + ∂ [ ( ν Σ f ) g ′ ] ∂ α i 2 ∂ χ g ∂ α i 1 + ( ν Σ f ) g ′ ∂ 2 χ g ∂ α i 2 ∂ α i 1 } − ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) ∂ Σ t g ( t ) ∂ α i 2 + ∑ g ′ = 1 G ∫ 4 π d Ω ′ ψ 2 ( 2 ) , g ′ ( i 1 ; r , Ω ′ ) × { ∂ Σ s g ′ → g ( s ; Ω ′ → Ω ) ∂ α i 2 + ∂ [ ( ν Σ f ) g ′ ] ∂ α i 2 χ g + ( ν Σ f ) g ′ ∂ χ g ∂ α i 2 } , (32)
[ F 32 ( 3 ) , g ] * ψ 3 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) ≜ ψ 3 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) ∂ Σ t g ( t ) ∂ α i 1 − ∑ g ′ = 1 G ∫ 4 π d Ω ′ ψ 3 ( 3 ) , g ′ ( i 1 , i 2 ; r , Ω ) × { ∂ Σ s g ′ → g ( s ; Ω ′ → Ω ) ∂ α i 1 + ∂ [ ( ν Σ f ) g ′ ] ∂ α i 1 χ g + ( ν Σ f ) g ′ ∂ χ g ∂ α i 1 } . (33)
The particular forms taken on by the general expressions given Equations (25) through (33) when considering solely the group-averaged microscopic total cross sections are obtained specializing these expressions to the particular cases α i 1 → t j , α i 2 → t k , α i 3 → t l . Thus, the expression in Equation (25) becomes:
{ ∂ Q ( 1 ) , g ( α ; φ ) ∂ α i 2 } α i 2 → t k ≜ ∂ Q g ( α ; r , Ω ) ∂ t k − ∂ ∂ t k { Ω ⋅ ∇ φ g ( r , Ω ) + Σ t g ( t ) φ g ( r , Ω ) − ∑ g ′ = 1 G ∫ 4 π d Ω ′ φ g ′ ( r , Ω ′ ) [ Σ s g ′ → g ( s ; Ω ′ → Ω ) + χ g ( ν Σ f ) g ′ ( f ) ] } = − φ g ( r , Ω ) ∂ Σ t g ( t ) ∂ σ t , i k g k = − φ g ( r , Ω ) δ g k g N i k , m k . (34)
In view of the result obtained in Equation (34), it follows that the expression in Equation (26) vanishes, i.e.,
∂ 2 Q ( 1 ) , g ( α ; φ ) ∂ α i 3 ∂ α i 2 = 0 , for α i 2 → t k , α i 3 → t l . (35)
The expression in Equation (27) becomes:
{ ∂ Q ( 2 ) , g ( α ; ψ ( 1 ) , g ) ∂ α i 2 } α i 2 → t k ≜ ∂ Σ d g ( α ; r , Ω ) ∂ t k − ∂ ∂ t k { − Ω ⋅ ∇ ψ ( 1 ) , g ( r , Ω ) + Σ t g ( t ) ψ ( 1 ) , g ( r , Ω ) + ∑ g ′ = 1 G ∫ 4 π d Ω ′ ψ ( 1 ) , g ′ ( r , Ω ′ ) [ Σ s g → g ′ ( s ; Ω → Ω ′ ) + ν Σ f g ( f ) χ g ′ ] } = − ψ ( 1 ) , g ( r , Ω ) ∂ Σ t g ( t ) ∂ σ t , i k g k = − ψ ( 1 ) , g ( r , Ω ) δ g k g N i k , m k . (36)
In view of Equation (36), the expression in Equation (28) becomes:
{ ∇ ψ 2 ( 2 ) ( i 1 ) R ( 2 ) , g ( α ) } α i 2 → t k = ∂ Q ( 2 ) , g ( α ; ψ ( 1 ) , g ) ∂ t k = − ψ ( 1 ) , g ( r , Ω ) δ g k g N i k , m k . (37)
The expression in Equation (31) becomes:
{ ∇ φ R ( 2 ) , g ( α ) } α i 2 → t k ≜ − ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) ∂ Σ t g ( t ) ∂ σ t , i k g k = − ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) δ g k g N i k , m k . (38)
The expression in Equation (29) becomes:
{ [ F 41 ( 3 ) , g ] * ψ 4 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) } α i 1 → t j = ψ 4 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) ∂ Σ t g ( t ) ∂ t j − ∑ g ′ = 1 G ∫ 4 π d Ω ′ ψ 4 ( 3 ) , g ′ ( i 1 , i 2 ; r , Ω ′ ) × { ∂ Σ s g → g ′ ( s ; Ω → Ω ′ ) ∂ t j + ∂ [ ν Σ f g ( f ) ] ∂ t j χ g ′ + ν Σ f g ( f ) ∂ χ g ′ ∂ t j } = ψ 4 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) ∂ Σ t g ( t ) ∂ σ t , i j g j = ψ 4 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) δ g j g N i j , m j . (39)
In view of Equation (34), the expression in Equation (30) becomes:
{ ∇ ψ 1 ( 2 ) ( i 1 ) R ( 2 ) , g ( α ) } α i 2 → t k = ∂ Q ( 1 ) , g ( α ; φ ) ∂ t k = − φ g ( r , Ω ) ∂ Σ t g ( t ) ∂ σ t , i k g k = − φ g ( r , Ω ) δ g k g N i k , m k . (40)
The expression in Equation (32) becomes:
{ ∇ ψ ( 1 ) R ( 2 ) , g ( α ) } α i 1 → t j α i 2 → t k = − φ g ( r , Ω ) ∂ 2 Σ t g ( t ) ∂ t k ∂ t j + ∑ g ′ = 1 G ∫ 4 π d Ω ′ φ g ′ ( r , Ω ′ ) × { ∂ 2 Σ s g ′ → g ( s ; Ω ′ → Ω ) ∂ t k ∂ t j + ∂ χ g ∂ t k ∂ [ ( ν Σ f ) g ′ ] ∂ t j + χ g ∂ 2 [ ( ν Σ f ) g ′ ] ∂ t k ∂ t j + ∂ [ ( ν Σ f ) g ′ ] ∂ t k ∂ χ g ∂ t j + ( ν Σ f ) g ′ ∂ 2 χ g ∂ t k ∂ t j } − ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) ∂ Σ t g ( t ) ∂ t k + ∑ g ′ = 1 G ∫ 4 π d Ω ′ ψ 2 ( 2 ) , g ′ ( i 1 ; r , Ω ′ )
× { ∂ Σ s g ′ → g ( s ; Ω ′ → Ω ) ∂ t k + ∂ [ ( ν Σ f ) g ′ ] ∂ t k χ g + ( ν Σ f ) g ′ ∂ χ g ∂ t k } = − ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) ∂ Σ t g ( t ) ∂ t k = − ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) ∂ Σ t g ( t ) ∂ σ t , i k g k = − ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) δ g k g N i k , m k . (41)
The expression in Equation (33) becomes:
{ [ F 32 ( 3 ) , g ] * ψ 3 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) } α i 1 → t j α i 2 → t k = ψ 3 ( 3 ) , g ( j , k ; r , Ω ) ∂ Σ t g ( t ) ∂ t j − ∑ g ′ = 1 G ∫ 4 π d Ω ′ ψ 3 ( 3 ) , g ′ ( j , k ; r , Ω ) × { ∂ Σ s g ′ → g ( s ; Ω ′ → Ω ) ∂ t j + ∂ [ ( ν Σ f ) g ′ ] ∂ t j χ g + ( ν Σ f ) g ′ ∂ χ g ∂ t j } = ψ 3 ( 3 ) , g ( j , k ; r , Ω ) ∂ Σ t g ( t ) ∂ σ t , i j g j = ψ 3 ( 3 ) , g ( j , k ; r , Ω ) δ g j g N i j , m j . (42)
Collecting the results obtained in Equations (34) through (42), inserting them into Equations (16) through (23) and recalling that i 1 → j , i 2 → k yields the following particular form for the 3rd-LASS, specialized for computing the 3rd-level adjoint functions needed for computing the 3rd-order sensitivities of the PERP leakage response to the group-averaged total microscopic cross sections:
A g ( α ) ψ 4 ( 3 ) , g ( j , k ; r , Ω ) = − ψ ( 1 ) , g ( r , Ω ) δ g k g N i k , m k , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , (43)
ψ 4 ( 3 ) , g ( j , k ; r d , Ω ) = 0 , Ω ⋅ n > 0 , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , (44)
A g ( α ) ψ 1 ( 3 ) , g ( j , k ; r , Ω ) = − [ ψ 1 ( 2 ) , g ( j ; r , Ω ) δ g k g N i k , m k + ψ 4 ( 3 ) , g ( j , k ; r , Ω ) δ g j g N i j , m j ] , (45)
ψ 1 ( 3 ) , g ( j , k ; r d , Ω , E ) = 0 , Ω ⋅ n > 0 , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , (46)
B g ( α ) ψ 3 ( 3 ) , g ( j , k ; r , Ω ) = − φ g ( r , Ω ) δ g k g N i k , m k , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , (47)
ψ 3 ( 3 ) , g ( j , k ; r d , Ω ) = 0 , Ω ⋅ n < 0 , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , (48)
B g ( α ) ψ 2 ( 3 ) , g ( j , k ; r , Ω ) = − [ ψ 2 ( 2 ) , g ( j ; r , Ω ) δ g k g N i k , m k + ψ 3 ( 3 ) , g ( j , k ; r , Ω ) δ g j g N i j , m j ] , (49)
ψ 2 ( 3 ) , g ( j , k ; r d , Ω ) = 0 , Ω ⋅ n < 0 , j = 1 ⋯ , J σ t , k = 1 , ⋯ , j . (50)
The 2nd-level adjoint functions ψ 1 ( 2 ) , g ( j ; r , Ω ) and ψ 2 ( 2 ) , g ( j ; r , Ω ) are the solutions of the following 2nd-LASS [
A g ( α ) ψ 1 ( 2 ) , g ( j ; r , Ω ) = − δ g j g N i j , m j ψ ( 1 ) , g ( r , Ω ) , j = 1 , ⋯ , J σ t ; g = 1 , ⋯ , G , (51)
ψ 1 , j ( 2 ) , g ( r d , Ω ) = 0 , Ω ⋅ n > 0 ; j = 1 , ⋯ , J σ t ; g = 1 , ⋯ , G , (52)
B g ( α ) ψ 2 ( 2 ) , g ( j ; r , Ω ) = − δ g j g N i j , m j φ g ( r , Ω ) , j = 1 , ⋯ , J σ t ; g = 1 , . ⋯ , G , (53)
ψ 2 , j ( 2 ) , g ( r d , Ω ) = 0 , Ω ⋅ n < 0 ; j = 1 , ⋯ , J σ t ; g = 1 , ⋯ , G , (54)
while the 1st-level adjoint function ψ ( 1 ) , g ( r , Ω ) is the solution of the following 1st-LASS [
A g ( α ) ψ ( 1 ) , g ( r , Ω ) = Ω ⋅ n δ ( r − r d ) , g = 1 , ⋯ , G , (55)
ψ ( 1 ) , g ( r d , Ω ) = 0 , Ω ⋅ n > 0 , g = 1 , ⋯ , G . (56)
Next, each of the quantities on the right-side of Equation (14) will be specialized to obtain the specific expression for computing the 3rd-order sensitivities of the leakage response in the PERP benchmark to the 180 total cross sections σ t , i g , i = 1 , ⋯ , I = 6 ; g = 1 , ⋯ , G = 30 defined in Equation (8). Thus, in view of Equation (10), it follows that the first term on the right-side of Equation (14) vanishes, since
∂ 3 Σ d g ( α ) ∂ α i 3 ∂ α i 2 ∂ α i 1 = ∂ 3 Σ d g ( α ) ∂ t j ∂ t k ∂ t l = 0. (57)
In view of Equation (35), the second term on the right-side of Equation (14) vanishes, i.e.,
〈 ψ ( 1 ) , g ( r , Ω ) , ∂ 3 Q ( 1 ) , g ( α ) ∂ α i 3 ∂ α i 2 ∂ α i 1 〉 ( 1 ) = 0. (58)
In view of Equation (35), the third term on the right-side of Equation (14) vanishes, i.e.,
〈 ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) , ∂ 2 Q ( 1 ) , g ( α ) ∂ α i 3 ∂ α i 2 〉 ( 1 ) = 0. (59)
In view of Equation (36), the fourth term on the right-side of Equation (14) vanishes, i.e.,
〈 ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) , ∂ 2 Q ( 2 ) , g ( α ) ∂ α i 3 ∂ α i 2 〉 ( 1 ) = 0. (60)
In view of Equation (34), the fifth term on the right-side of Equation (14) takes on the following particular form:
〈 ψ 1 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , ∂ Q ( 1 ) , g ( α ) ∂ α i 3 〉 ( 1 ) = − 〈 ψ 1 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , φ g ( r , Ω ) δ g l g N i l , m l 〉 ( 1 ) . (61)
In view of Equation (36), the sixth term on the right-side of Equation (14) takes on the following particular form:
〈 ψ 2 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , ∂ Q ( 2 ) , g ( α ) ∂ α i 3 〉 ( 1 ) = − 〈 ψ 2 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , ψ ( 1 ) , g ( r , Ω ) δ g l g N i l , m l 〉 ( 1 ) , (62)
The seventh term on the right-side of Equation (14) takes on the following particular form:
〈 ψ 3 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , ∂ Q 1 ( 3 ) , g ( α ) ∂ α i 3 〉 ( 1 ) = − 〈 ψ 3 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) δ g l g N i l , m l 〉 ( 1 ) , (63)
since
∂ Q 1 ( 3 ) , g [ α ; r , Ω ; ψ 1 ( 2 ) , g ( i 1 ) ] ∂ α i 3 ≜ − ∂ [ A g ( α ) ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) ] ∂ α i 3 + ∂ [ ∇ φ R ( 1 ) ( α ; ψ ( 1 ) , g ) ] ∂ α i 3 = − ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) ∂ Σ t g ( t ) ∂ σ t , i l g l = − ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) δ g l g N i l , m l . (64)
The result obtained in Equation (64) stems from the fact that repeating the derivation shown in Equation (36) yields the results ∂ [ A ( α ) ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) ] / ∂ α i 3 = ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) [ ∂ Σ t g ( t ) / ∂ σ t , i l g l ] , and from the fact that the second term on the right side of Equation (64) vanishes, as shown below:
{ ∂ [ ∇ φ R ( 1 ) ( α ; ψ ( 1 ) , g ) ] ∂ α i 3 } α i 1 → t j α i 3 → t l ≜ ∂ 2 Σ d g ( α ; r ) ∂ t l ∂ t j − ψ ( 1 ) , g ( r , Ω ) ∂ 2 Σ t g ( t ) ∂ t l ∂ t j + ∑ g ′ = 1 G ∫ 4 π d Ω ′ ψ ( 1 ) , g ′ ( r , Ω ′ )
× { ∂ 2 Σ s g → g ′ ( s ; Ω ′ → Ω ) ∂ t l ∂ t j + χ g ′ ∂ 2 [ ( ν Σ f ) g ] ∂ t l ∂ t j + ∂ [ ( ν Σ f ) g ] ∂ t j ∂ χ g ′ ∂ t l + ∂ [ ( ν Σ f ) g ] ∂ t l ∂ χ g ′ ∂ t j + ( ν Σ f ) g ∂ 2 χ g ′ ∂ t l ∂ t j } = 0. (65)
The eighth term on the right-side of Equation (14) takes on the following particular form:
〈 ψ 4 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , ∂ Q 2 ( 3 ) , g ( α ) ∂ α i 3 〉 ( 1 ) = − 〈 ψ 4 ( 3 ) , g ( i 1 , i 2 ; r , Ω ) , ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) δ g l g N i l , m l 〉 ( 1 ) (66)
since
∂ Q 2 ( 3 ) , g [ α ; r , Ω ; ψ 2 ( 2 ) , g ( i 1 ) ] ∂ α i 3 ≜ − ∂ [ B g ( α ) ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) ] ∂ α i 3 + ∂ [ ∇ ψ ( 1 ) R ( 1 ) ( α ; φ ) ] ∂ α i 3 = − ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) ∂ Σ t g ( t ) ∂ σ t , i l g l = − ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) δ g l g N i l , m l , (67)
as can be shown by repeating the derivations used in Equation (34) to obtain the result ∂ [ B g ( α ) ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) ] / ∂ α i 3 = ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) [ ∂ Σ t g ( α ; r ) / ∂ σ t , i l g l ] , and by noting that
{ ∂ [ ∇ ψ ( 1 ) R ( 1 ) ( α ; φ ) ] ∂ α i 3 } α i 1 → t j α i 3 → t l = ∂ 2 Q g ( α ; r , Ω ) ∂ t l ∂ t j − φ g ( r , Ω ) ∂ 2 Σ t g ( t ) ∂ t l ∂ t j + ∑ g ′ = 1 G ∫ 4 π d Ω ′ φ g ′ ( r , Ω ′ ) × { ∂ 2 Σ s g ′ → g ( s ; Ω ′ → Ω ) ∂ t l ∂ t j + ∂ χ g ∂ t l ∂ [ ( ν Σ f ) g ′ ] ∂ t j + χ g ∂ 2 [ ( ν Σ f ) g ′ ] ∂ t l ∂ t j + ∂ [ ( ν Σ f ) g ′ ] ∂ t l ∂ χ g ∂ t j + ( ν Σ f ) g ′ ∂ 2 χ g ∂ t l ∂ t j } = 0. (68)
Collecting the results obtained in Equations (57)-(63), and (66), and replacing them in Equation (14) yields the following expression:
∂ 3 L ( α ) ∂ t j ∂ t k ∂ t l = − { 〈 ψ 1 ( 3 ) , g ( j , k ; r , Ω ) , φ g ( r , Ω ) 〉 ( 1 ) + 〈 ψ 2 ( 3 ) , g ( j , k ; r , Ω ) , ψ ( 1 ) , g ( r , Ω ) 〉 ( 1 ) + 〈 ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , ψ 1 ( 2 ) , g ( j ; r , Ω ) 〉 ( 1 ) + 〈 ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , ψ 2 ( 2 ) , g ( j ; r , Ω ) 〉 ( 1 ) } δ g l g N i l , m l , for j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , l = 1 , ⋯ , k . (69)
In Ref. [
∂ 2 L ∂ t j ∂ t k = ∂ 2 L ∂ σ t , i j g j ∂ σ t , i k g k = − N i k , m k ∫ V d V ∫ 4 π d Ω [ ψ 2 , j ( 2 ) , g k ( r , Ω ) ψ ( 1 ) , g k ( r , Ω ) + ψ 1 , j ( 2 ) , g k ( r , Ω ) φ g k ( r , Ω ) ] , for j = 1 , ⋯ , J σ t ; k = 1 , ⋯ , J σ t , g = 1 , ⋯ , G , (70)
where the first-level adjoint functions ψ ( 1 ) , g ( r , Ω ) , g = 1 , ⋯ , G are the solutions of the 1st-Level Adjoint Sensitivity System (1st-LASS) show in Equations (55) and (56), while the 2nd-level adjoint functions ψ 1 , j ( 2 ) , g ( r , Ω ) and ψ 2 , j ( 2 ) , g ( r , Ω ) are the solutions of the 2nd-LASS presented in Equations (51) through (54).
The total G-differential of Equation (70) provides the 3rd-order sensitivities of the leakage response involving the 2nd-order derivatives to the total cross sections. Since this work is limited to computing the sensitivities of the PERP leakage response solely with respect to the group-averaged microscopic total cross sections, it follows that the sensitivities ∂ 3 L ( α ) / ∂ t j ∂ t k ∂ t l , j , k , l = 1 , ⋯ , J σ t are obtained by taking the following G-differential of Equation (70), limited to variations just in the group-averaged total microscopic cross sections, namely:
δ ( ∂ 2 L ∂ t j ∂ t k ) = − N i k , m k ∫ V d V ∫ 4 π d Ω [ δ ψ 2 , j ( 2 ) , g k ( r , Ω ) ψ ( 1 ) , g k ( r , Ω ) + ψ 2 , j ( 2 ) , g k ( r , Ω ) δ ψ ( 1 ) , g k ( r , Ω ) + δ ψ 1 , j ( 2 ) , g k ( r , Ω ) φ g k ( r , Ω ) + ψ 1 , j ( 2 ) , g k ( r , Ω ) δ φ g k ( r , Ω ) ] = ∑ l = 1 J σ t ∂ 3 L ( α ) ∂ t j ∂ t k ∂ t l δ t l , for j , k = 1 , ⋯ , J σ t , (71)
where the functions δ ψ ( 1 ) , g k ( r , Ω ) and δ φ g k ( r , Ω ) are the solutions of the 1st-LFSS and, respectively, 2nd-LFSS, which are reproduced below from Ref. [
B g ( α ) δ φ g ( r , Ω ) = Q ( 1 ) , g ( α , φ ; δ α ) , g = 1 , ⋯ , G , (72)
δ φ g ( r d , Ω ) = 0 , Ω ⋅ n < 0 , g = 1 , ⋯ , G , (73)
A g ( α ) δ ψ ( 1 ) , g ( r , Ω ) = Q ( 2 ) , g ( α , ψ ( 1 ) ; δ α ) , g = 1 , ⋯ , G , (74)
δ ψ ( 1 ) , g ( r d , Ω ) = 0 , Ω ⋅ n > 0 , (75)
while the functions δ ψ 1 , j ( 2 ) , g k and δ ψ 2 , j ( 2 ) , g k are the solutions of the G-differentiated 2nd-LASS defined by Equations (51) through (54), namely:
A g ( α ) [ δ ψ 1 , j ( 2 ) , g ( r , Ω ) ] + δ g j g N i j , m j [ δ ψ ( 1 ) , g ( r , Ω ) ] = Q 1 ( 3 ) , g [ α , ψ 1 ( 2 ) ( i 1 ) ; δ α ] , (76)
δ ψ 1 , j ( 2 ) , g ( r d , Ω ) = 0 , Ω ⋅ n > 0 , j = 1 , ⋯ , J σ t , g = 1 , ⋯ , G , (77)
B g ( α ) [ δ ψ 2 , j ( 2 ) , g ( r , Ω ) ] + δ g j g N i j , m j [ δ φ g ( r , Ω ) ] = Q 2 ( 3 ) [ α , ψ 2 ( 2 ) ( i 1 ) ; δ α ] , (78)
δ ψ 2 , j ( 2 ) , g ( r d , Ω ) = 0 , Ω ⋅ n < 0 , j = 1 , ⋯ , J σ t , g = 1 , ⋯ , G . (79)
The source Q ( 1 ) , g ( α , φ ; δ α ) on the right side in Equation (72) is defined as follows:
Q ( 1 ) , g ( α , φ ; δ α ) ≜ ∑ i 3 = 1 J σ t ∂ Q ( 1 ) , g ( α , φ ) ∂ α i 3 δ α i 3 , (80)
where, in view of Equation (34),
{ ∂ Q ( 1 ) , g ( α , φ ) ∂ α i 3 } α i 3 = t l ≜ − φ g ( r , Ω ) ∂ Σ t g ( t ) ∂ σ t , i l g l = − φ g ( r , Ω ) δ g l g N i l , m l . (81)
The source Q ( 2 ) , g ( α , ψ ( 1 ) ; δ α ) on the right side in Equation (74) is defined as follows:
Q ( 2 ) , g ( α , ψ ( 1 ) ; δ α ) ≜ ∑ i 3 = 1 J σ t ∂ Q ( 2 ) , g ( α , ψ ( 1 ) ) ∂ α i 3 δ α i 3 , (82)
where, in view of Equation (36),
{ ∂ Q ( 2 ) , g ( α , ψ ( 1 ) ) ∂ α i 3 } α i 3 = t l ≜ − ψ ( 1 ) , g ( r , Ω ) ∂ Σ t g ( t ) ∂ σ t , i l g l = − ψ ( 1 ) , g ( r , Ω ) δ g l g N i l , m l . (83)
The source Q 1 ( 3 ) , g [ α , ψ 1 ( 2 ) ( i 1 ) ; δ α ] on the right side in Equation (76) is defined as follows:
Q 1 ( 3 ) , g [ α , ψ 1 ( 2 ) ( i 1 ) ; δ α ] ≜ ∑ i 3 = 1 J σ t ∂ Q 1 ( 3 ) , g [ α , ψ 1 ( 2 ) ( i 1 ) ] ∂ α i 3 δ α i 3 , (84)
where, in view of Equation (64),
{ ∂ Q 1 ( 3 ) , g [ α , ψ 1 ( 2 ) ( i 1 ) ] ∂ α i 3 } α i 3 = t l ≜ − ∂ [ A g ( α ) ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) ] ∂ t l − ∂ [ δ g j g N i j , m j ψ ( 1 ) , g ( r , Ω ) ] ∂ t l = − ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) ∂ Σ t g ( t ) ∂ σ t , i l g l = − ψ 1 ( 2 ) , g ( i 1 ; r , Ω ) δ g l g N i l , m l . (85)
The source Q 2 ( 3 ) [ α , ψ 2 ( 2 ) ( i 1 ) ; δ α ] on the right side in Equation (78) is defined as follows:
Q 2 ( 3 ) [ α , ψ 2 ( 2 ) ( i 1 ) ; δ α ] ≜ ∑ i 3 = 1 J σ t ∂ Q 2 ( 3 ) [ α , ψ 2 ( 2 ) ( i 1 ) ] ∂ α i 3 δ α i 3 , (86)
where, in view of Equation (67),
{ ∂ Q 2 ( 3 ) [ α , ψ 2 ( 2 ) ( i 1 ) ] ∂ α i 3 } α i 3 = t l ≜ − ∂ [ B g ( α ) ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) ] ∂ t l − ∂ [ δ g j g N i j , m j φ g ( r , Ω ) ] ∂ t l = − ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) ∂ Σ t g ( α ; r ) ∂ σ t , i l g l = − ψ 2 ( 2 ) , g ( i 1 ; r , Ω ) δ g l g N i l , m l . (87)
The 3rd-Level Forward Sensitivity System (3rd-LFSS) comprises Equations (72) through (79). The 3rd-Level Adjoint Sensitivity System (3rd-LASS) that corresponds to the 3rd-LFSS is derived by following the general procedure described in Ref. [
A. Use the definition of the inner product provided in Equation (15) to perform the following operations:
1) Form the inner product of Equation (72) with a yet undefined vector-valued function ψ 1 ( 3 ) , g ( j , k ; r , Ω ) ≜ [ ψ 1 ( 3 ) , g = 1 ( j , k ; r , Ω ) , ⋯ , ψ 1 ( 3 ) , g ( j , k ; r , Ω ) , ⋯ , ψ 1 ( 3 ) , g = G ( j , k ; r , Ω ) ] † having G square-integrable components ψ 1 ( 3 ) , g ( j , k ; r , Ω ) , to obtain the following relation:
〈 ψ 1 ( 3 ) , g ( j , k ; r , Ω ) , B g ( α ) δ φ g ( r , Ω ) 〉 ( 1 ) = 〈 ψ 1 ( 3 ) , g ( j , k ; r , Ω ) , ∑ l = 1 J σ t [ − δ g l g N i l , m l φ g ( r , Ω ) ] δ t l 〉 ( 1 ) . (88)
2) Form the inner product of Equation (74) with a yet undefined vector-valued function ψ 2 ( 3 ) , g ( j , k ; r , Ω ) ≜ [ ψ 2 ( 3 ) , g = 1 ( j , k ; r , Ω ) , ⋯ , ψ 2 ( 3 ) , g ( j , k ; r , Ω ) , ⋯ , ψ 2 ( 3 ) , g = G ( j , k ; r , Ω ) ] † having G square-integrable components ψ 2 ( 3 ) , g ( j , k ; r , Ω ) to obtain the following relation:
〈 ψ 2 ( 3 ) , g ( j , k ; r , Ω ) , A g ( α ) δ ψ ( 1 ) , g ( r , Ω ) 〉 ( 1 ) = 〈 ψ 2 ( 3 ) , g ( j , k ; r , Ω ) , ∑ l = 1 J σ t [ − δ g l g N i l , m l ψ ( 1 ) , g ( r , Ω ) ] δ t l 〉 ( 1 ) . (89)
3) Form the inner product of Equation (76) with a yet undefined vector-valued function ψ 3 ( 3 ) , g ( j , k ; r , Ω ) ≜ [ ψ 3 ( 3 ) , g = 1 ( j , k ; r , Ω ) , ⋯ , ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , ⋯ , ψ 3 ( 3 ) , g = G ( j , k ; r , Ω ) ] † having G square-integrable components ψ 3 ( 3 ) , g ( j , k ; r , Ω ) to obtain the following relation:
〈 ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , A g ( α ) [ δ ψ 1 , j ( 2 ) , g ( r , Ω ) ] 〉 ( 1 ) + δ g j g N i j , m j 〈 ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , δ ψ ( 1 ) , g ( r , Ω ) 〉 ( 1 ) = 〈 ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , ∑ l = 1 J σ t [ − δ g l g N i l , m l ψ 1 , j ( 2 ) , g ( r , Ω ) ] δ t l 〉 ( 1 ) . (90)
4) Form the inner product of Equation (78) with a yet undefined vector-valued function ψ 4 ( 3 ) , g ( j , k ; r , Ω ) ≜ [ ψ 4 ( 3 ) , g = 1 ( j , k ; r , Ω ) , ⋯ , ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , ⋯ , ψ 4 ( 3 ) , g = G ( j , k ; r , Ω ) ] † having G square-integrable components square integrable function ψ 4 ( 3 ) , g ( j , k ; r , Ω ) to obtain the following relation:
〈 ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , B g ( α ) [ δ ψ 2 , j ( 2 ) , g ( r , Ω ) ] 〉 ( 1 ) + δ g j g N i j , m j 〈 ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , δ φ g ( r , Ω ) 〉 ( 1 ) = 〈 ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , ∑ l = 1 J σ t [ − δ g l g N i l , m l ψ 2 , j ( 2 ) , g ( r , Ω ) ] δ t l 〉 ( 1 ) . (91)
5) Add Equations (88) through (91) to obtain the following relation:
〈 ψ 1 ( 3 ) , g ( j , k ; r , Ω ) , B g ( α ) δ φ g ( r , Ω ) 〉 ( 1 ) + 〈 ψ 2 ( 3 ) , g ( j , k ; r , Ω ) , A g ( α ) δ ψ ( 1 ) , g ( r , Ω ) 〉 ( 1 ) + 〈 ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , A g ( α ) [ δ ψ 1 , j ( 2 ) , g ( r , Ω ) ] 〉 ( 1 ) + δ g j g N i j , m j 〈 ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , δ ψ ( 1 ) , g ( r , Ω ) 〉 ( 1 ) + 〈 ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , B g ( α ) [ δ ψ 2 , j ( 2 ) , g ( r , Ω ) ] 〉 ( 1 )
+ δ g j g N i j , m j 〈 ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , δ φ g ( r , Ω ) 〉 ( 1 ) = ∑ l = 1 J σ t [ − δ g l g N i l , m l 〈 ψ 1 ( 3 ) , g ( j , k ; r , Ω ) , φ g ( r , Ω ) 〉 ( 1 ) − δ g l g N i l , m l 〈 ψ 2 ( 3 ) , g ( j , k ; r , Ω ) , ψ ( 1 ) , g ( r , Ω ) 〉 ( 1 ) − δ g l g N i l , m l 〈 ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , ψ 1 , j ( 2 ) , g ( r , Ω ) 〉 ( 1 ) − δ g l g N i l , m l 〈 ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , ψ 2 , j ( 2 ) , g ( r , Ω ) 〉 ( 1 ) ] δ t l . (92)
B. Use the relations between the forward and adjoint multigroup transport operators to recast the left side of Equation (92) into the following form:
〈 ψ 1 ( 3 ) , g ( j , k ; r , Ω ) , B g ( α ) δ φ g ( r , Ω ) 〉 ( 1 ) + 〈 ψ 2 ( 3 ) , g ( j , k ; r , Ω ) , A g ( α ) δ ψ ( 1 ) , g ( r , Ω ) 〉 ( 1 )
+ 〈 ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , A g ( α ) δ ψ 1 , j ( 2 ) , g ( r , Ω ) 〉 ( 1 ) + δ g j g N i j , m j 〈 ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , δ ψ ( 1 ) , g ( r , Ω ) 〉 ( 1 )
+ 〈 ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , B g ( α ) δ ψ 2 , j ( 2 ) , g ( r , Ω ) 〉 ( 1 ) + δ g j g N i j , m j 〈 ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , δ φ g ( r , Ω ) 〉 ( 1 ) = 〈 δ φ g ( r , Ω ) , A g ( α ) ψ 1 ( 3 ) , g ( j , k ; r , Ω ) + δ g j g N i j , m j ψ 4 ( 3 ) , g ( j , k ; r , Ω ) 〉 ( 1 ) + 〈 δ ψ ( 1 ) , g ( r , Ω ) , B g ( α ) ψ 2 ( 3 ) , g ( j , k ; r , Ω ) + δ g j g N i j , m j ψ 3 ( 3 ) , g ( j , k ; r , Ω ) 〉 ( 1 )
+ 〈 δ ψ 1 , j ( 2 ) , g ( r , Ω ) , B g ( α ) ψ 3 ( 3 ) , g ( j , k ; r , Ω ) 〉 ( 1 ) + 〈 δ ψ 2 , j ( 2 ) , g ( r , Ω ) , A g ( α ) ψ 4 ( 3 ) , g ( j , k ; r , Ω ) 〉 ( 1 ) + P ( 3 ) [ δ φ g , δ ψ ( 1 ) , g ; δ ψ 1 ( 2 ) , g ( i 1 ) , δ ψ 2 ( 2 ) , g ( i 1 ) ; ψ 1 ( 3 ) , g ( i 1 , i 2 ) , ψ 2 ( 3 ) , g ( i 1 , i 2 ) , ψ 3 ( 3 ) , g ( i 1 , i 2 ) , ψ 4 ( 3 ) , g ( i 1 , i 2 ) ] , (93)
where P ( 3 ) denotes the corresponding bilinear concomitant on the domain’s boundary in the phase-space of independent variables.
C. Use the boundary conditions provided in Equations (73), (75), (77) and (79), and impose on the functions ψ m ( 3 ) , g ( j , k ; r , Ω ) , m = 1 , 2 , 3 , 4 , the boundary conditions provided in Equations (95), (97), (99) and (101), below, in order to cause the bilinear concomitant P ( 3 ) in Equation (93) to vanish.
D. Identify the right side of Equation (93) with the right side of Equations (71) to obtain the following equations for the 3rd-LASS, where
j = 1 , ⋯ , J σ t ; k = 1 , ⋯ , j ; g = 1 , ⋯ , G :
A g ( α ) ψ 1 ( 3 ) , g ( j , k ; r , Ω ) + δ g j g N i j , m j ψ 4 ( 3 ) , g ( j , k ; r , Ω ) = − δ g k g N i k , m k ψ 1 , j ( 2 ) , g ( r , Ω ) , (94)
ψ 1 ( 3 ) , g ( j , k ; r d , Ω ) = 0 , Ω ⋅ n > 0 , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , g = 1 , ⋯ , G , (95)
B g ( α ) ψ 2 ( 3 ) , g ( j , k ; r , Ω ) + δ g j g N i j , m j ψ 3 ( 3 ) , g ( j , k ; r , Ω ) = − δ g k g N i k , m k ψ 2 , j ( 2 ) , g ( r , Ω ) , (96)
ψ 2 ( 3 ) , g ( j , k ; r d , Ω ) = 0 , Ω ⋅ n < 0 , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , g = 1 , ⋯ , G , (97)
B g ( α ) ψ 3 ( 3 ) , g ( j , k ; r , Ω ) = − δ g k g N i k , m k φ g ( r , Ω ) , (98)
ψ 3 ( 3 ) , g ( j , k ; r d , Ω ) = 0 , Ω ⋅ n < 0 , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , g = 1 , ⋯ , G , (99)
A g ( α ) ψ 4 ( 3 ) , g ( j , k ; r , Ω ) = − δ g k g N i k , m k ψ ( 1 ) , g ( r , Ω ) , (100)
ψ 4 ( 3 ) , g ( j , k ; r d , Ω ) = 0 , Ω ⋅ n > 0 , j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , g = 1 , ⋯ , G . (101)
E. It follows from the 3rd-LASS defined in Equations (94) through (101) that the right-sides of Equations (92) and (71) represent the same functional, which implies that
∂ 3 L ( α ) ∂ t j ∂ t k ∂ t l = − δ g j g N i j , m j { 〈 ψ 1 ( 3 ) , g ( j , k ; r , Ω ) , φ g ( r , Ω ) 〉 ( 1 ) + 〈 ψ 2 ( 3 ) , g ( j , k ; r , Ω ) , ψ ( 1 ) , g ( r , Ω ) 〉 ( 1 ) + 〈 ψ 3 ( 3 ) , g ( j , k ; r , Ω ) , ψ 1 , j ( 2 ) , g ( r , Ω ) 〉 ( 1 ) + 〈 ψ 4 ( 3 ) , g ( j , k ; r , Ω ) , ψ 2 , j ( 2 ) , g ( r , Ω ) 〉 ( 1 ) } , for j = 1 , ⋯ , J σ t , k = 1 , ⋯ , j , l = 1 , ⋯ , k . (102)
As expected, the 3rd-LASS obtained in Equations (94) through (101) is identical to the 3rd-LASS obtained in Equations (43) through (50), and the expression obtained for the 3rd-order sensitivities ∂ 3 L ( α ) / ∂ t j ∂ t k ∂ t l , j , k , l = 1 , ⋯ , J σ t obtained in Equation (102) is identical to the expression obtained in Equation (69).
This work has presented the derivation of the exact mathematical expressions of the (180)3 third-order sensitivities of the PERP leakage response with respect to the total microscopic cross sections. By using the solution of the 3rd-Level Adjoint Sensitivity System, the expressions of these 3rd-order sensitivities can be computed selectively and most efficiently. The formulas derived in this work are valid not only for the PERP benchmark but can also be used for computing the 3rd-order sensitivities of the leakage response of any nuclear system involving fissionable material and internal or external neutron sources. Subsequent works will use the adjoint-based mathematical expressions obtained in this work to compute exactly and efficiently the numerical values of these (180)3 third-order sensitivities (which turned out to be very large and consequential) and use them for a third-order uncertainty analysis of the PERP benchmark’s leakage response.
The authors declare no conflicts of interest regarding the publication of this paper.
Cacuci, D.G. and Fang, R.X. (2020) Third-Order Adjoint Sensitivity Analysis of an OECD/NEA Reactor Physics Benchmark: I. Mathematical Framework. American Journal of Computational Mathematics, 10, 503-528. https://doi.org/10.4236/ajcm.2020.104029
The dimensions and material composition of the polyethylene-reflected plutonium (PERP) metal sphere considered in this work are presented in
The scattering cross section Σ s g ′ → g ( r , Ω ′ → Ω ) → Σ s g ′ → g ( s ; r , Ω ′ → Ω ) will be characterized by the vector of parameters s , which is defined as follows:
s ≜ [ s 1 , ⋯ , s J s ] † ≜ [ s 1 , ⋯ , s J σ s ; n 1 , ⋯ , n J n ] † ≜ [ σ s ; N ] † , J s ≜ J σ s + J n , (103)
| Materials | Isotopes | Weight Fraction | Density (g/cm3) | Zones |
|---|---|---|---|---|
| Material 1 (plutonium metal) | Isotope 1 (239Pu) | 9.3804 × 10−1 | 19.6 | Homogeneous sphere of radius r 1 = 3.794 cm , designated as “material 1” and assigned to zone 1 |
| 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 | Homogeneous spherical shell of inner radius r 1 = 3.794 cm and outer radius r 2 = 7.604 cm , designated as “material 2” and assigned to zone 2 |
| Isotope 6 (1H) | 1.4370 × 10−1 |
| g | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| E g | 1.50 × 101 | 1.35 × 101 | 1.20 × 101 | 1.00 × 101 | 7.79 × 100 | 6.07 × 100 |
| E g − 1 | 1.70 × 101 | 1.50 × 101 | 1.35 × 101 | 1.20 × 101 | 1.00 × 101 | 7.79 × 100 |
| g | 7 | 8 | 9 | 10 | 11 | 12 |
| E g | 3.68 × 100 | 2.87 × 100 | 2.23 × 100 | 1.74 × 100 | 1.35 × 100 | 8.23 × 10−1 |
| E g − 1 | 6.07 × 100 | 3.68 × 100 | 2.87 × 100 | 2.23 × 100 | 1.74 × 100 | 1.35 × 100 |
| g | 13 | 14 | 15 | 16 | 17 | 18 |
| E g | 5.00 × 10−1 | 3.03 × 10−1 | 1.84 × 10−1 | 6.76 × 10−2 | 2.48 × 10−2 | 9.12 × 10−3 |
| E g − 1 | 8.23 × 10−1 | 5.00 × 10−1 | 3.03 × 10−1 | 1.84 × 10−1 | 6.76 × 10−2 | 2.48 × 10−2 |
| g | 19 | 20 | 21 | 22 | 23 | 24 |
| E g | 3.35 × 10−3 | 1.24 × 10−3 | 4.54 × 10−4 | 1.67 × 10−4 | 6.14 × 10−5 | 2.26 × 10−5 |
| E g − 1 | 9.12 × 10−3 | 3.35 × 10−3 | 1.24 × 10−3 | 4.54 × 10−4 | 1.67 × 10−4 | 6.14 × 10−5 |
| g | 25 | 26 | 27 | 28 | 29 | 30 |
| E g | 8.32 × 10−6 | 3.06 × 10−6 | 1.13 × 10−6 | 4.14 × 10−7 | 1.52 × 10−7 | 1.39 × 10−10 |
| E g − 1 | 2.26 × 10−5 | 8.32 × 10−6 | 3.06 × 10−6 | 1.13 × 10−6 | 4.14 × 10−7 | 1.52 × 10−7 |
where
σ s ≜ [ s 1 , ⋯ , s J σ s ] † ≜ [ σ s , l = 0 , i = 1 g ′ = 1 → g = 1 , σ s , l = 0 , i = 1 g ′ = 2 → g = 1 , ⋯ , σ s , l = 0 , i = 1 g ′ = G → g = 1 , σ s , l = 0 , i = 1 g ′ = 1 → g = 2 , σ s , l = 0 , i = 1 g ′ = 2 → g = 2 , ⋯ , σ s , l , i g ′ → g , ⋯ , σ s , I S C T , i = I G → G ] † , l = 0 , ⋯ , I S C T ; i = 1 , ⋯ , I ; g , g ′ = 1 , ⋯ , G ; J σ s = ( G × G ) × I × ( I S C T + 1 ) . (104)
The quantity ( ν Σ f ) g → ( ν Σ f ) g ( f ; r ) in the fission integral ∫ 4 π ( ν Σ ) f g ′ ( r ) φ g ′ ( r , Ω ′ ) d Ω ′ depends on the vector of parameters f , which is defined as follows:
f ≜ [ f 1 , ⋯ , f J σ f ; f J σ f + 1 , ⋯ , f J σ f + J ν ; f J σ f + J ν + 1 , ⋯ , f J f ] † ≜ [ σ f ; ν ; N ] † , J f = J σ f + J ν + J n , (105)
with
σ f ≜ [ σ f , i = 1 1 , σ f , i = 1 2 , ⋯ , σ f , i = 1 G , ⋯ , σ f , i g , ⋯ , σ f , i = N f 1 , ⋯ , σ f , i = N f G ] † ≜ [ f 1 , ⋯ , f J σ f ] † , i = 1 , ⋯ , N f ; g = 1 , ⋯ , G ; J σ f = G × N f , (106)
ν ≜ [ ν i = 1 1 , ν i = 1 2 , ⋯ , ν i = 1 G , ⋯ , ν i g , ⋯ , ν i = N f 1 , ⋯ , ν i = N f G ] † ≜ [ f J σ f + 1 , ⋯ , f J σ f + J ν ] † , i = 1 , ⋯ , N f ; g = 1 , ⋯ , G ; J ν = G × N f , (107)
where σ f , i g denotes the microscopic fission cross section for isotope i and energy group g, ν i g denotes the average number of neutrons per fission for isotope i and energy group g, and N f denotes the total number of fissionable isotopes.
The fission spectrum is considered to depend on the vector of parameters p , defined as follows:
p ≜ [ p 1 , ⋯ , p J p ] † ≜ [ χ i = 1 g = 1 , χ i = 1 g = 2 , ⋯ , χ i = 1 G , ⋯ , χ i g , ⋯ , χ N f G ] † , for i = 1 , ⋯ , N f ; g = 1 , ⋯ , G ; J p = G × N f . (108)
The quantities χ g ( r ) further depend on the parameters χ i g , N i , m , f i g , ( ν σ f ) i g , but these latter dependences can be taken into account by applying the chain rule on the 1st-order sensitivities ∂ L / ∂ χ g , once these sensitivities will have been obtained.
The source Q g ( r ) → Q g ( q ; N ; r ) depends on the vector of model parameters q , defined as follows:
q ≜ [ q 1 , ⋯ , q J q ] † ≜ [ λ 1 , λ 2 ; F 1 S F , F 2 S F ; a 1 , a 2 ; b 1 , b 2 ; ν 1 S F , ν 2 S F ] † , J q = 10. (109)
In summary, the model parameters characterizing the PERP benchmark can all be considered to be the components of the following “vector of model parameters:”
α ≜ [ α 1 , ⋯ , α J α ] † ≜ [ σ t ; σ s ; σ f ; ν ; p ; q ; N ] † , J α = J σ t + J σ s + J σ f + J ν + J p + J q + J n . (110)
Thus, the total number of imprecisely known model parameters for the PERP benchmark is:
J α = ( I × G ) + ( G × G ) × I × ( I S C T + 1 ) + 2 ( G × N f ) + G × N f + 10 + J n = 21976 .
a k , b k : parameters used in Watt’s fission spectra approximation for isotope k
B: forward Boltzmann operator
E g : boundary of energy group g
F k S F : fraction of isotope k decays that are spontaneous fission events
f j , f m 2 : parameters in vector σ f indexed by j and m 2
G: total number of energy groups
I: total number of isotopes
J n : total number of parameters in vector N
J p : total number of parameters in vector p
J q : total number of parameters in vector q
J σ f : total number of parameters in vector σ f
J σ s : total number of parameters in vector σ s
J σ t : total number of parameters in vector σ t
J t : total number of parameters in vector t
J ν : total number of parameters in vector ν
l : variable for the order of Legendre-expansion of the microscopic scattering cross sections, l = 1 , ⋯ , I S C T
L ( α ) : total neutron leakage from the PERP sphere
M: total number of materials
N f : total number of fissionable isotopes
N i , m : atom number density for isotope i and material m
P l ( Ω ′ ⋅ Ω ) : Legendre and associated Legendre polynomials appreciate for the geometry
P l ( μ ) : spherical harmonics appreciate for the geometry
Q g ( r ) : source term in group g
r: spatial variable
r d : external radius of the PERP benchmark
S b : outer surface of the PERP sphere
s f , i g : standard deviation associated with the model parameter σ f , i g
s ν , i g : standard deviation associated with the model parameter ν i g
s j , s m 2 : parameters in vector σ s indexed by j and m 2
t j , t m 2 : parameters in vector σ t indexed by j and m 2
α : vector of imprecisely known model parameters, α ≜ [ σ t ; σ s ; σ f ; ν ; p ; q ; N ] †
α 0 : nominal values of the parameters in the vector α
t : vector of imprecisely known total parameters, t ≜ [ σ t ; N ] †
s : vector of imprecisely known scatter parameters, s ≜ [ σ s ; N ] †
f : vector of imprecisely known fission parameters, f ≜ [ σ f ; ν ; N ] †
σ t : vector of imprecisely known total cross sections
σ s : vector of imprecisely known scattering cross sections
σ f : vector of imprecisely known fission cross sections
ν : vector of imprecisely known parameters underlying the average number of neutrons per fission
N : vector of imprecisely known atom number densities
p : vector of imprecisely known fission spectrum parameters
q : vector of imprecisely known source parameters
S ( 1 ) : vector of first-order relative sensitivities of the leakage response
S ( 2 ) : matrix of first-order relative sensitivities of the leakage response
λ k : decay constant for isotope k
ν k S F : the spontaneous emission of an average neutrons of an isotope k
σ f , i g : microscopic fission cross section in group g of isotope i
σ s , l , i g ′ → g : the l t h order Legendre-expanded microscopic scattering cross section from energy group g ′ into energy group g for isotope i
σ t , i g : microscopic total cross section in group g of isotope i
Σ t g ( t ; r ) : macroscopic total cross section for energy group g
Σ f g ( f ; r ) : macroscopic fission cross section for energy group g
Σ s g ′ → g ( s ; r , Ω ′ → Ω ) : macroscopic scattering transfer cross section from energy group g ′ into energy group g
φ g ( r , Ω ) : forward angular flux in group g at point r in direction Ω
φ 0 g ( r ) : zeroth order of forward flux moment in group g at point r
φ l g ( r ) : l t h ( l = 1 , ⋯ , I S C T ) order forward flux moment in group g at point r, φ l g ( r ) ≜ ∫ 4 π d Ω P l ( μ ) φ g ( r , Ω ) , l = 1 , ⋯ , I S C T
χ g ( r ) : material fission spectrum in energy group g
ψ ( 1 ) , g ( r , Ω ) : adjoint angular flux in group g at point r in direction Ω
Ω , Ω ′ : directional variable
f: fission
g , g ′ : energy group variable g , g ′ = 1 , ⋯ , G
g j , g m 2 : energy group associated with parameter indexed by j (e.g., f j , t j and s j ) or m 2 (e.g., f m 2 , t m 2 and s m 2 )
i: index variable for isotopes, i = 1 , ⋯ , I
i j , i m 2 : isotope associated with the parameter indexed by j (e.g., f j , t j and s j ) or m 2 (e.g., f m 2 , t m 2 and s m 2 )
j: index variable for parameters
k: index variable for isotopes, k = 1 , ⋯ , I
l j , l m 2 : order of Legendre expansion associated with the microscopic scattering cross section parameters indexed by j (e.g., s j ) or m 2 (e.g., s m 2 )
ν : number of neutrons produced per fission
m: index variable for materials, m = 1 , ⋯ , M
m 2 : index variable for parameters
m j , m m 2 : material associated with parameter indexed by j (e.g., f j , t j and s j ) or m 2 (e.g., f m 2 , t m 2 and s m 2 )
t: total
s: scatter
1st-LASS: 1st-Level adjoint sensitivity system
2nd-ASAM: second-order adjoint sensitivity analysis methodology
2nd-CASAM: comprehensive second-order adjoint sensitivity analysis methodology
2nd-LASS: 2nd-Level adjoint sensitivity system
3rd-LASS: 3rd-Level adjoint sensitivity system
Nu: the average number of neutrons per fission
ISCT: order of the finite expansion in Legendre polynomial
PERP: polyethylene-reflected plutonium
sigf: fission cross sections
sigs: scattering cross sections
sigt: total cross sections