The sensitivity coefficient for k _eff with respect to multi-group microscopic cross section σ x , g i can be expressed as: S k e f f , σ x , g i = σ x , g i k e f f d k e f f d σ x , g i (1) x is reaction type identifier, i is nuclide identifier and g is energy group identifier. Considering that the effective resonance cross section of reaction y nuclide j in group h can be perturbated by σ x , g i , using the chain rule and Eq. 1 can be then written as (B.T. Rearten and Jessee, 2016): S k e f f , σ x , g i t o t = σ x , g i k e f f d k e f f d σ x , g i = σ x , g i k e f f ∂ k e f f ∂ σ x , g i + ∑ j σ y , h j k e f f ∂ k e f f ∂ σ y , h j × σ x , g i σ y , h j ∂ σ y , h j ∂ σ x , g i         = S k e f f , σ x , g i exp + ∑ j S k e f f , σ y , h j exp S σ y , h j , σ x , g i i m p (2) It can be found that the total sensitivity coefficient for k _eff with respect to multi-group cross section σ x , g i consists of two parts: S σ y , h j , σ x , g i i m p is the implicit sensitivity coefficient for effective resonance cross section σ y , h j with respect to multi-group cross section σ x , g i ; S k e f f , σ x , g i exp and S k e f f , σ y , h j exp are both explicit sensitivity coefficients which represents the impact of cross sections on the responses directly through neutron transport equation. The methods for quantifying the implicit and explicit sensitivity are given below.

It needs to be emphasized that according to the division pattern for resonance nuclides and resonance energy group, total sensitivity analysis can be divided into three conditions:

1) For multi-group cross sections of non-resonance nuclides in resonance energy group, both implicit and explicit effect need to be taken into consideration;

The classical perturbation theory can be used to quantify the sensitivity coefficients of k _eff to the effective resonance cross sections of different reaction type of different nuclides. In nuclear reactor physics system, the forward and adjoint neutron transport equations can be written as the operator forms, ( A − λ B ) Ψ = 0 (3a) ( A * − λ * B * ) Ψ * = 0 (3b) A represents neutron leakage, absorption and scattering term, B represents the fission source term, Ψ is the forward neutron angular flux. A * and B * are the adjoint form of operators A and B . Ψ * represents the adjoint neutron angular flux. λ and λ * are the eigenvalue of forward and adjoint neutron transport equations, respectively. According to the mathematical derivation, the explicit sensitivity coefficient of k _eff due to the perturbation of effective resonance cross sections σ can be eventually expressed as (J. Ma et al., 2020), S k e f f , σ = − σ 〈 Ψ ∗ ( ∂ A σ − 1 k e f f ∂ B σ ) Ψ 〉 / 〈 Ψ ∗ 1 k e f f B Ψ 〉 (4)

Based on Eq. 4, the sensitivity coefficient of k _eff with respect to the effective resonance cross sections can be quantified, and only once forward transport calculation and adjoint calculation is required. Equation 4 indicates that the denominator is the function of forward flux, adjoint flux, eigenvalue and the fission source operator, whereas numerator would vary with targeted reaction type. The denominator and the derivation term in numerator could be obtained based on the discrete form of neutron transport equation according to our previous research (J. Ma et al., 2020), and the exact form of explicit sensitivity coefficients won’t be given in this paper.

As introduced in Method of Quantifying the Total Sensitivity Coefficient, the key issue in the study of implicit sensitivity is to study the effect of multi-group microscopic cross section on the effective resonance cross section. Generally, the effective resonance cross section in group g can be written as: σ x , g = ∫ g σ x ( E ) ϕ ( E ) d E / ∫ g ϕ ( E ) d E (5) ϕ ( E ) is the neutron flux in group g; σ x ( E ) is fine-group cross section in group g。According to the definition of sensitivity coefficient, the relative sensitivity of σ x , g with respect to multi-group microscopic cross section α g is: S σ x , g , α g = d σ x , g / σ x , g d α g / α g = α g { ∫ g ϕ ( E ) ∂ σ x ( E ) ∂ α g d E ∫ g ϕ ( E ) σ x ( E ) d E + ∫ g σ x ( E ) ∂ ϕ ( E ) ∂ α g d E ∫ g ϕ ( E ) σ x ( E ) d E − ∫ g ∂ ϕ ( E ) ∂ α g d E ∫ g ϕ ( E ) d E } = α g ∫ g ϕ ( E ) ∂ σ x ( E ) ∂ α g d E ∫ g ϕ ( E ) σ x ( E ) d E ︸ S d i r + α g ∫ g ( σ x ( E ) ∫ g ϕ ( E ) σ x ( E ) d E − 1 ∫ g ϕ ( E ) d E ) ∂ ϕ ( E ) ∂ α g d E ︸ S i n d i r (6)

It can be found that Eq. 6 can be divided into two parts: the first part is the direct part, it represent the change of effective resonance cross section σ x ( E ) caused directly by the perturbation of multi group cross section α g ; the second part is the indirect part, it represent multi group cross section α g firstly influence the neutron flux ϕ ( E ) , and further influence the effective resonance cross section σ x ( E ) .

The neutron flux ϕ ( E ) in Eq. 5 is the solution of neutron slowing-down equation. The operator form of slowing-down equation can be expressed as: B ϕ ( E ) = Q ( E ) (7) B is the disappearing term, Q is the effective source term. Take the derivative on each term of Eq. 7 with multi group cross section α : B ∂ ϕ ( E ) ∂ α = ∂ Q ( E ) ∂ α − ∂ B ∂ α ϕ ( E ) (8)

Equation 8 describes the basic relationship between the perturbation of microscopic cross-section α and the weighting function ϕ ( E ) . Theoretically, if the variation of the operators Q and B are obtained, the derivative term ∂ ϕ ( E ) / ∂ α can be solved directly based on Eq. 8, and the relative sensitivity coefficients of effective resonance cross section σ x , g with respect to microscopic cross section α can be calculated using Eq. 6. In fact, the derivative terms ∂ B / ∂ α and ∂ Q / ∂ α can be calculated directly utilizing the direct perturbation method. However, the perturbation calculations need to be achieved for different nuclides, energy groups, and reaction types, which will introduce an unacceptable calculated amount and computational complexity in the study of implicit sensitivity analysis.

Fortunately, the indirect part of Eq. 6 can be determined for multiple perturbations by combining the generalized adjoint equation and the slowing equation instead of explicitly calculating the derivative term ∂ ϕ ( E ) / ∂ α . Based on the generalized perturbation theory, introducing a generalized adjoint function Γ x , g ∗ ( E ) for reaction type x and energy group g, which represents the value of weighted function ϕ ( E ) to the effective resonance section σ x , g rather than the neutron value. Then the generalized adjoint form of the slowing down equation can be written as: B * Γ x , g ∗ ( E ) = σ x ( E ) ∫ g ϕ ( E ) σ x ( E ) d E − 1 ∫ g ϕ ( E ) d E (9) B * is the adjoint form of operator B .

Multiply both sides of Eq. 8 by the generalized adjoint function Γ x , g ∗ ( E ) , and integrating over group g: ∫ g ( B ∂ ϕ ( E ) ∂ α ) Γ x , g ∗ ( E ) d E = ∫ g ( ∂ Q ( E ) ∂ α − ∂ B ∂ α ϕ ( E ) ) Γ x , g ∗ ( E ) d E (10)

According to the characteristics of the adjoint operator, the left term of Eq. 10 can be written as: ∫ g ( B ∂ ϕ ( E ) ∂ α ) Γ x , g ∗ ( E ) d E = ∫ g ( ∂ ϕ ( E ) ∂ α ) B ∗ Γ x , g ∗ ( E ) d E (11)

Substituting Eqs. 9–11 into Eq. 6, the relative sensitivity coefficient of effective resonance cross section with respect to a certain type microscopic cross section can be calculated by: S σ x , g , α g = α g ∫ g ∂ σ x ( E ) ∂ α g ϕ ( E ) d E ∫ g σ x ( E ) ϕ ( E ) d E ︸ S d i r + α g ∫ g Γ x , g ∗ ( E ) ( ∂ Q ( E ) ∂ α g − ∂ B ∂ α g ϕ ( E ) ) d E ︸ S i n d i r (12)

Subgroup method is using to perform resonance self-shielding calculation in HNET. Before performing the implicit sensitivity analysis, it is necessary to briefly introduce the subgroup resonance calculation method. Differ from the traditional resonance calculation method, which subdivides the energy group depending on the value of neutron energy, subgroups are defining according to the cross section itself which drastic changes. S o a few subgroups can contain the resonance energy interval. The neutron flux varies weakly due to the smooth value of cross section in one subgroup, which leads to a high efficiency resonance computation compared with the traditional resonance self-shielding calculation method, and subgroup method currently became one of the most widely used resonance calculation method.

According to the subgroup method, the cross section in the resonance energy group is divided into several intervals within one cross section range, and each interval is a so-called subgroup. Each subgroup corresponds to several discrete energy segments, the energy segment set is expressed as: Δ E g , i ∈ { E | σ g , i < σ ≤ σ g , i + 1 } (13)

Subgroup cross section and subgroup probability, which are called subgroup parameters, are used to describes the properties of subgroup, and can be expressed as: σ x , g , i = ∫ Δ E g , i σ x ( E ) ϕ ( E ) d E ∫ Δ E g , i ϕ ( E ) d E (13a)   p g , i = Δ E g , i Δ E g (13b)

After obtaining the subgroup parameters, the subgroup flux density can be obtained by solving the subgroup transport equation, then the subgroup flux is used as weight function to obtain the effective resonance self-shielding cross section. For energy group g, the transport equation of subgroup i is: Ω ⋅ ∇ ϕ g , i + Σ t , g , i ϕ g , i = Q g , i (13a) Q g , i = p g , i Σ p (13b) Q g , i is source term in subgroup i, Σ p is potential cross section, Σ t , g , i is the total cross section in subgroup i. The operator form of slowing-down equation in subgroup i can be expressed as: B g , i ϕ g , i = Q g . i (14)

The flux density of subgroup i can be obtained by solving Eq. 14, then the effective resonance self-shielding cross section can be calculated by: σ x , g = ∑ i = 1 I σ x , g , i ϕ g , i / ∑ i = 1 I ϕ g , i (15)

Based on the generalized perturbation theory, the generalized adjoint subgroup transport equation needs to be established firstly in order to obtain the relative sensitivity coefficient of the effective resonance self-shielding cross section with respect to the multi-group cross section, the subgroup generalized adjoint transport equation can be written as: - Ω ∇ Γ g , i ∗ + Σ t , g , i Γ g , i ∗ = Q g , i ∗ (16)

The Operator Form Is

B g , i ∗ Γ g , i ∗ = Q g . i ∗ (17) B g ∗ is subgroup adjoint transport term, Γ g ∗ is the generalized subgroup adjoint neutron flux, Q g ∗ is the generalized adjoint source, which is defined by: Q g , i ∗ = σ x , g , i ∑ i = 1 N ∫ V ∫ Ω ϕ g , i σ x , g , i d V d Ω − 1 ∑ i = 1 N ∫ V ∫ Ω ϕ g , i d V d Ω (18)

Once obtaining the specific form of the generalized adjoint source term, it then can be used as the external source term of the generalized adjoint equation solver in HNET in order to obtain the generalized adjoint function Γ x , g ∗ .

Finally, considering the exact form of σ x ( E ) and Γ x , g ∗ when subgroup method is used to apply the subgroup method and combining Eq. 18, the relative sensitivity coefficient of the effective resonance self-shielding cross section with respect to the multi-group microscopic cross section can be calculated by: S σ x , g j , α y , g ′ k =   α y , g ′ k ∑ i = 1 I ( ∫ Ω ∂ σ x , g , i j ∂ α y , g ′ k ϕ g , i d Ω ∫ Ω σ x , g , i j ϕ g , i d Ω ) + α y , g ′ k ∑ i = 1 I ∑ Z V Z ∫ Ω Γ x , g , i ∗ ( ∂ Q g , i ∂ α y , g ′ k − ∂ L g , i ∂ α y , g ′ k ϕ g , i ) d Ω (19)

After obtaining the sensitivity vectors, the relative uncertainty can be then calculated using “Sandwich Formula”. In the following part, the derivation of Sandwich Formula will be introduced briefly.

The nuclear reactor physics system k _eff can be written as the function of a series of multi-group microscopic cross sections, which is shown in Eq. 20. k e f f = R ( σ 1 , σ 2 , … … σ n ) (20) σ represents the multi-group microscopic cross section for a nuclide reaction type, and k ¯ , σ ¯ represent the expected value. Using the first order linearity approximation, the Taylor expansion form of Eq. 20 can be rewritten as, k e f f = k ¯ e f f + δ k e f f = R ( σ ¯ 1 , σ ¯ 2 , ⋯ σ ¯ n ) + ∑ i n ∂ k e f f ∂ σ i δ σ i (21)

Presumptively, the input parameters in reactor system satisfy the p ( σ 1 , σ 2 , ⋯ , σ n ) , which is a joint probability density function, the variance of k _eff can be calculated using the following form, v a r ( k e f f ) = ∫ ( ∑ i = 1 n ( ∂ k e f f ∂ σ i ) δ σ i ) 2 p ( σ 1 , σ 2 , ⋯ , σ n ) d σ 1 d σ 2 ⋯ d σ n = ∑ i = 1 n ( ∂ k e f f ∂ σ i ) 2 D σ i + 2 ∑ i ≠ j = 1 n ∂ k e f f ∂ σ i ∂ k e f f ∂ σ j Σ σ i σ j (22) k Σ is the covariance of two parameters, D is the variance of a specific parameter. Then the relative variance of k _eff, i.e. the square of the k _eff relative uncertainty, due to multi-group microscopic cross sections can be written as v a r ( k e f f ) k e f f 2 = ∑ i = 1 n ( σ i k e f f ∂ k e f f ∂ σ i ) 2 D σ i σ i 2 + 2 ∑ i ≠ j = 1 n ( σ i k e f f ∂ k e f f ∂ σ i ) ( σ j k e f f ∂ k e f f ∂ σ j ) Σ σ i σ j σ i σ j = ∑ i = 1 n ( S k e f f , σ i ) 2 D σ i r e l a t i v e + 2 ∑ i ≠ j = 1 n S k e f f , σ i S k e f f , σ j Σ σ i σ j r e l a t i v e (23)

Considering the definition of sensitivity coefficient, the matrix form of relative uncertainty of k _eff can be rewritten as, v a r ( k e f f ) k e f f 2 = S k e f f , σ Σ r e l a t i v e ( S k e f f , σ ) T (24)

Equation 24 is the so-called “Sandwich Rule”. Once obtaining the sensitivity vectors for k _eff with respect to multi-group microscopic cross sections and the relative covariance matrix, Eq. 24 can be used to quantify the relative uncertainty of nuclear reactor physics system k _eff.

In this section, the total sensitivity and uncertainty analysis scheme with the calculation flow is established, and the details will be discussed. Figure 1 shows the implementation flow of the total sensitivity and uncertainty analysis. The fundamental evaluated nuclear data library is WIMS 69 g library, which is the basic input for the resonance and transport calculations. And the ZZ-SCALE6.0/COV-44G library, which contains the uncertainty and correlation information for multi-group cross sections, is the basic uncertainty source. High-fidelity Neutron Transport program (HNET), which is a self-developed deterministic 3D high-fidelity neutron transport code, is applied to solve the subgroup transport equations, forward neutron transport equations, adjoint neutron transport equations, and the generalized adjoint equation. The effective resonance multi-group cross sections are calculated by using subgroup resonance calculation method. Then a total sensitivity and uncertainty analysis module is developed for quantifying the sensitivity and uncertainty of k _eff propagated from the multi-group microscopic nuclear cross sections in HNET. The computational steps of generalized perturbation theory-based total sensitivity and uncertainty (SU) analysis can be summarized as follows:

1) The relative covariance matrix in 69 g energy group structure is generated from the well-developed 44 g energy group relative covariance library by using home-developed covariance matrix generation code T-COCOO.

Sensitivity coefficient calculated using central difference direct perturbation method is considered relatively precise if the linear relation is strong and the perturbation is feasible. In this way, this method is always used to verify the sensitivity coefficient calculated by other methods. For sensitivity coefficient of k _eff with respect to certain multi-group microscopic cross section using central difference direct perturbation method, the k _eff of the system is computed for three times: firstly, with the unperturbed cross sections α and obtains unperturbed k e f f ; then with an adequate increased cross section α + and obtains k e f f α + ; with the same value decreased input cross section α − and obtains k e f f α − . Finally, DP method-based sensitivity coefficient of k _eff with respected to cross section α can be calculated by S k e f f , α = ( k e f f α + − k e f f α − ) / k e f f ( α + − α − ) / α (25)

In this paper, DP method is used to verify the sensitivity coefficient calculated using proposed generalized perturbation theory. Three types of sensitivity coefficient, including implicit, explicit and total sensitivity coefficient are verified, and the calculation methods can be summarized as follows:

1) Total sensitivity coefficients: Complete calculation including resonance self-shielding calculation and neutron transport calculation need to be performed for three times, with unperturbed, positive perturbed and negative perturbed multi-group cross sections respectively.

A two-dimensional fuel pin cell problem was chosen for the purpose of verifying the sensitivity coefficient calculated by using generalized perturbation theory proposed in this paper and perform total sensitivity and uncertainty analysis. More detail descriptions of this problem can be found in (A.T. Godfrey, 2013). The main calculation parameters of this problem is summarized in Table 1. The geometry information and computing mesh are illustrated in Figure 2. Meanwhile, the WIMS 69 energy group nuclear data library is used to perform subgroup resonance calculation, adjoint and transport calculations.

The sensitivity vectors are calculated using the WIMS 69-group structure nuclear data library, which is different from the 44 g group structure relative covariance library, so the problem-related 69 g relative covariance library needs to be firstly constructed from the well-evaluated 44 g group structure using a former proposed covariance matrix transforming method (D. Wang, et al., 2016). The comparison of relative covariance matrix of two typical nuclide neutron reaction cross section, including ²³⁵U 5) and ²³⁸U (n, γ) in 44 g group structure and 69 g group structure are shown in Figures 3, 4 respectively.

The 2D pin cell problem is firstly used to verify the implicit sensitivity coefficients calculated based on the proposed generalized perturbation theory method. According to the linear relationship test, 2% relative perturbation factor is in the region with strongest linear relationship, so a 2% relative perturbation of multigroup cross sections is used in the DP method for sensitivity verification. The comparisons of implicit sensitivity coefficients, which is divided into resonance nuclides and non-resonance nuclides, calculated by using the DP and GPT methods for some typical nuclides are illustrated in Figures 5, 6, respectively.

Figures 5, 6 suggest that the implicit sensitivity coefficients for both resonance and non-resonance nuclides calculated by GPT and DP methods agree well for selected reaction types in all resonance energy groups. Most relative error in these energy groups is less than 1%, but in group 22, the relative error of implicit sensitivity coefficients of absorption cross section of ²³⁵U with respect to radiative capture cross section of ²³⁵U is 1.89%, which is a little large, but it is still acceptable. These comparisons indicate that acceptable accuracy of implicit sensitivity analysis is achievable by using generalized perturbation theory proposed in this paper.

It is worth noted that the ability of calculation for explicit sensitivity coefficient is verified in our precious work (J. Ma et al., 2020) and verification will not be performed in this paper. For the verification of total sensitivity, the integral sensitivity coefficients (integrating with volume and energy) of selected cross sections of representative isotopes and reaction types in resonance energy groups are summarized in Table 2 for comparison. The comparison results also demonstrate that accurate total sensitivity coefficient can be obtained, and the total sensitivity and uncertainty can be then performed.

Before performing the total sensitivity analysis, it is worth investigate that the relative importance of implicit sensitivity compared with the explicit part. Figure 7 illustrates the implicit sensitivity, explicit sensitivity and total sensitivity of k _eff with respect to elastic scattering cross section of ¹H. According to Figure 7, in most energy group, the explicit sensitivity coefficients group are positive, while the implicit part are negative, thus the total sensitivity is less than explicit part in these energy groups. It suggests that if the implicit sensitivity is neglected, the sensitivity will be overestimated. It leads to a fact that the implicit sensitivity needs to be considered in detail when performing total SU analysis.

Three kinds of sensitivity coefficients, including total, explicit and implicit part of k _eff with respect to some representative reaction type cross section of both resonance and non-resonance nuclides are illustrated in Table 3. It also can be found in Table.3 that the implicit effect has a nonnegligible impact for the sensitivity analysis for LWR, especially for the sensitivity coefficient for resonance nuclides in resonance energy groups.

The uncertainty contribution of some important nuclide reaction cross section with and without considering the implicit part is presented in Table 4. It can be found in Table 4 that although the implicit part is essential in sensitivity analysis, it has nonsignificant influences on uncertainty results. The first reason is that compared with the explicit part, the absolute value of implicit part is relatively small; the second reason is that uncertainty is quantified by using Sandwich Formula, the nuclide reaction cross section may have significant influence on eigenvalue, yet it has small uncertainty itself. Although the total uncertainty results barely effected by implicit sensitivity, considering the implicit influence ensure the reasonable propagation for uncertainty of input parameters, the implicit sensitivity needs to be fully investigated when performing total sensitivity and uncertainty analysis.

The total uncertainty contributions of some important reactions to k _eff, taken both implicit and explicit influence into consideration, are shown in Table 5. The most three significant contributors to the total uncertainty of k _eff are σ _γ of ²³⁸U, ν of ²³⁵U and σ _γ of ²³⁵U respectively. This conclusion is similar to the uncertainty contribution results without considering the implicit effect. The total uncertainty contribution due to these representative nuclides reaction types to k _eff is 0.517%, compared with the results of 0.522% when implicit part is ignored, it also suggests that the implicit effect has little influence on the uncertainty quantification results.