6.8.7.1. TSURFER Appendix A: Sensitivity/Uncertainty Notation

In the following expressions, the notation E[X] represents the expected value of random variable X, which is equal to the integral of X weighted by its probability density function over the range of allowable values.

6.8.7.1.1. Basic variables

I =

number of integral response (experiment and applications) used in GLLS analysis

M =

number of nuclear data parameters used in transport calculations (i.e., number of unique nuclide-reaction pairs multiplied by the number of energy groups)

\(\boldsymbol{\alpha}\) =

M dimensional vector of prior nuclear data parameters, where component-i = \(\alpha\)_i

A =

M by M diagonal matrix of prior nuclear data parameters, where diagonal element A(i,i) = \(\alpha\)_i

m =

I dimensional vector of prior measured responses, where component-i = m_i

M =

I by I diagonal matrix of prior measured responses, where diagonal element M(i,i) = m_i

\(\mathbf{k}(\boldsymbol{\alpha})\) =

I dimensional vector of prior calculated responses obtained with nuclear data \(\boldsymbol{\alpha}\), where component I = k_i

K =

I by I diagonal matrix of prior calculated responses, where diagonal element K(i,i) = k_i

\(\mathbf{F}_{\mathbf{m} / \mathbf{k}}\) =

I by I diagonal matrix of “E/C” values =

(6.8.55)\[\begin{array}{l} \mathbf{M} \mathbf{K}^{-1}=\mathbf{K}^{-\mathbf{1}} \mathbf{M} \end{array}\]

where diagonal element

(6.8.56)\[\mathrm{F}_{\mathrm{m} / \mathrm{k}}(\mathrm{i}, \mathrm{i})=\frac{\mathrm{m}_{\mathrm{i}}}{\mathrm{k}_{\mathrm{i}}}\]

\({{\mathbf{\hat{F}}}_{\mathbf{m/k}}}\) =

I by I diagonal matrix, where diagonal element

(6.8.57)\[\mathrm{F}_{\mathrm{m} / \mathrm{k}}\left(\mathrm{i}, \mathrm{i}\right)=\frac{\mathrm{m}_{\mathrm{i}}}{\mathrm{k}_{\mathrm{i}}}\]

for a relative-formatted response and \(\mathrm{F}_{\mathrm{m} / \mathrm{k}}(\mathrm{i}, \mathrm{i})=1\)

\(\boldsymbol{\alpha}'\) =

M dimensional vector of adjusted nuclear data parameters produced by GLLS procedure

m’ =

I dimensional vector of adjusted measured responses produced by GLLS procedure

\(\mathbf{k}'(\boldsymbol{\alpha}')\) =

I dimensional vector of adjusted calculated responses obtained with modified nuclear data \(\boldsymbol{\alpha}'\)

Note

\(\mathbf{k}'(\boldsymbol{\alpha}') = \mathbf{m}'\), due to GLLS adjustment procedure.

\(\mathbf{\tilde{d}}\,\) =

original absolute discrepancy vector = \(\mathbf{k}-\mathbf{m}\) , where component-i= \({{k}_{i}}-{{m}_{i}}\)

d =

original relative discrepancy vector = \(\mathbf{K}^{-1}(\mathbf{k}-\mathbf{m})\) , where component-i = \(\left(\mathrm{k}_{\mathrm{i}}-\mathrm{m}_{\mathrm{i}}\right) / \mathrm{k}_{\mathrm{i}}\)

\(\mathbf{\hat{d}}\)

original mixed absolute-relative discrepancy vector, where component-i = \(\left(\mathrm{k}_{\mathrm{i}}-\mathrm{m}_{\mathrm{i}}\right) / \mathrm{k}_{\mathrm{i}}\) for a relative-formatted response and \(\left(\mathrm{k}_{i}-\mathrm{m}_{\mathrm{i}}\right)\) for an absolute-formatted response

\([\boldsymbol{\Delta} \boldsymbol{\alpha}]\) =

M dimensional vector of relative variations in nuclear data = \(\mathbf{A}^{-1}\left(\boldsymbol{\alpha}^{\prime}-\boldsymbol{\alpha}\right)\) where component-i = \(\frac{\alpha_{i}^{\prime}-\alpha_{i}}{\alpha_{i}}\)

\([\mathbf{\Delta m}]\) =

I dimensional vector of relative variations in measured responses = \(\mathbf{M}^{-1}\left(\mathbf{m}^{\prime}-\mathbf{m}\right)\) where component-i = \(\frac{\text{m}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{m}}_{\text{i}}}}{{{\text{m}}_{\text{i}}}}\to \frac{\text{k}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{m}}_{\text{i}}}}{{{\text{m}}_{\text{i}}}}\)

\([\mathbf{\Delta m}]\) =

I dimensional vector of absolute variations in measured responses = \(\mathbf{m}^{\prime}-\mathbf{m}\) where component-i \(\text{m}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{m}}_{\text{i}}}\to \text{k}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{m}}_{\text{i}}}\)

\([\mathbf{\Delta} \hat{\mathbf{m}}]\) =

I dimensional vector of mixed absolute-relative variations in measured responses, where component-i = \(\frac{\text{m}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{m}}_{\text{i}}}}{{{\text{m}}_{\text{i}}}}\) for a relative-formatted response and \(\text{m}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{m}}_{\text{i}}}\) for an absolute-formatted response

\([\boldsymbol{\Delta} \mathbf{k}]\) =

I dimensional vector of relative variations in calculated responses = \(\mathbf{K}^{-1}\left(\mathbf{k}^{\prime}-\mathbf{k}\right)\) where component-i = \(\frac{\text{k}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{k}}_{\text{i}}}}{{{\text{k}}_{\text{i}}}}\)

\([\boldsymbol{\Delta} \mathbf{k}]\) =

I dimensional vector of absolute variations in calculated responses = \(\mathbf{k'}-\mathbf{k}\), where component-i = \(\text{k}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{k}}_{\text{i}}}\)

\([\boldsymbol{\Delta} \hat{\mathbf{k}}]\) =

I dimensional vector of mixed absolute-relative variations in calculated responses, where component-i = \(\text{k}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{k}}_{\text{i}}}\) for an absolute formatted response

6.8.7.1.2. Sensitivity Relations

\(\widetilde{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}\) =

I by M absolute sensitivity matrix; where element \(\widetilde{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}(\mathrm{i}, \mathrm{n})=\alpha_{\mathrm{n}} \frac{\partial \mathrm{k}_{\mathrm{i}}}{\partial \alpha_{\mathrm{n}}}\)

\(\mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}\) =

I by M relative sensitivity matrix = \(\mathbf{K}^{-1} \mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}\), where element \(\mathbf{S}_{k \alpha}(i, n)=\frac{\alpha_{n}}{k_{i}} \frac{\partial k_{i}}{\partial \alpha_{n}}\).

\(\hat{\mathbf{S}}_{\mathbf{k}\boldsymbol{\alpha}}\) =

I by M mixed absolute-relative sensitivity matrix, where element \(\hat{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}(\mathrm{i}, \mathrm{n})=\frac{\alpha_{\mathrm{n}}}{\mathrm{k}_{\mathrm{i}}} \frac{\partial \mathrm{k}_{\mathrm{i}}}{\partial \alpha_{\mathrm{n}}}\) if response-i is relative-formatted and \(\hat{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}(\mathrm{i}, \mathrm{n})=\alpha_{\mathrm{n}} \frac{\partial \mathbf{k}_{\mathrm{i}}}{\partial \alpha_{\mathrm{n}}}\) if response-i is absolute-formatted

(6.8.58)\[\begin{split}\begin{array}{l} {[\boldsymbol{\Delta} \mathbf{k}]=\quad \mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}[\mathbf{\Delta} \boldsymbol{\alpha}]} \\ {{[\boldsymbol{\Delta} \mathbf{k}}]=\mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}[\mathbf{\Delta} \boldsymbol{\alpha}]} \\ {[\boldsymbol{\Delta} \hat{\mathbf{k}}]=\hat{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}[\mathbf{\Delta} \boldsymbol{\alpha}]} \end{array}\end{split}\]

6.8.7.1.3. Absolute covariances

\({{\mathbf{\tilde{C}}}_{\mathbf{mm}}}\) =

I by I covariance matrix for prior measured experiment responses where element \({{\mathbf{\tilde{C}}}_{\mathbf{mm}}}\)(i,j) = \(E\left( \delta {{m}_{i}}\,\delta {{m}_{j}} \right)\)

\({{\mathbf{\tilde{C}}}_{\mathbf{kk}}}\) =

I by I covariance matrix for prior calculated responses, where element \({{\mathbf{\tilde{C}}}_{\mathbf{kk}}}\)(i,j) = \(E\left( \delta {{k}_{i}}\,\delta {{k}_{j}} \right)\)

\({{\mathbf{\tilde{C}}}_{\mathbf{dd}}}\) =

I by I covariance matrix for the discrepancies (k-m), where element \({{\mathbf{\tilde{C}}}_{\mathbf{dd}}}\)(i,j) = \(E\left( \delta {{d}_{i}}\,\delta {{d}_{j}} \right)\) = \(\mathrm{E}\left(\delta\left(\mathrm{k}_{\mathrm{i}}-\mathrm{m}_{\mathrm{i}}\right) \delta\left(\mathrm{k}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}}\right)\right)\)

\({{\mathbf{\tilde{C}}}_{\mathbf{k'k'}}}\) =

I by I covariance matrix for adjusted responses, where element \({{\mathbf{\tilde{C}}}_{\mathbf{k'k'}}}\)(i,j) = \(E\left( \delta k{{'}_{i}}\,\delta k{{'}_{j}} \right)\)

\(\boldsymbol{\sigma}_{\mathbf{m}}\) =

I by I diagonal matrix containing standard deviations in prior measured responses, where diagonal element \(\widetilde{\sigma}_{\mathrm{m}}\left(\mathrm{i} \mathrm{i}\right)=\sqrt{\widetilde{\mathrm{C}}_{\mathrm{mm}}(\mathrm{i}, \mathrm{i})}\)

\(\boldsymbol{\sigma}_{\mathbf{k}}\) =

I by I diagonal matrix containing standard deviations in prior calculated responses, where diagonal element \(\widetilde{\sigma}_{\mathrm{k}}(\mathrm{i}, \mathrm{i})=\sqrt{\widetilde{\mathrm{C}}_{\mathrm{kk}}(\mathrm{i}, \mathrm{i})}\)

\(\boldsymbol{\sigma}_{\mathbf{k}^{\prime}}\)

6.8.7.1.4. Relative covariances

\({{C}_{\mathbf{mm}}}\) =

I by I relative covariance matrix for prior measured responses, = \(\mathbf{M}^{-1}\left[\tilde{\mathbf{C}}_{\mathbf{m m}}\right] \mathbf{M}^{-1}\) \(C_{m m}(i, j)=\frac{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\mathrm{m}_{\mathrm{i}} \mathrm{m}_{\mathrm{j}}}\)

\({{C}_{\boldsymbol{\alpha \alpha}}}\) =

M by M relative covariance matrix for prior nuclear data, where element \(\widetilde{C}_{\alpha \alpha}(i, j)\) = \(\frac{\mathrm{E}\left(\delta \alpha_{\mathrm{i}} \delta \alpha_{\mathrm{j}}\right)}{\alpha_{\mathrm{i}} \alpha_{\mathrm{j}}}\)

\({{C}_{\mathbf{kk}}}\) =

I by I relative covariance matrix for prior calculated responses = \(\mathbf{K}^{-1}\left[\mathbf{C}_{\mathrm{kk}}\right] \mathbf{K}^{-1}\) where element \(C_{k k}(i, j)=\frac{\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})}{\mathrm{k}_{\mathrm{i}} \mathrm{k}_{\mathrm{j}}}\)

\({{C}_{\mathbf{dd}}}\)

I by I relative covariance matrix for response discrepancies; = \(\mathbf{K}^{-1}\left[\mathbf{C}_{\mathbf{dd}}\right] \mathbf{K}^{-1}\), where element \(C_{d d}(i, j)=\frac{\mathrm{C}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j})}{\mathrm{k}_{\mathrm{i}} \mathrm{k}_{\mathrm{j}}}\)

\(\boldsymbol{\sigma}_{\mathbf{m}}\) =

I by I diagonal matrix containing relative standard deviations in measured responses, where diagonal element \(\sigma_{\mathrm{m}}\left(\mathrm{i}, \mathrm{i}\right)=\sqrt{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{i})}\)

\(\boldsymbol{\sigma}_{\mathbf{k}}\) =

I by I diagonal matrix containing relative standard deviations in calculated responses, where diagonal element \(\sigma_{k^{\prime}}\left(\mathrm{i}, \mathrm{i}\right)=\sqrt{\mathrm{C}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i})}\)

\(\boldsymbol{\sigma}_{\boldsymbol{\alpha}}\) =

M by M diagonal matrix containing standard deviations in nuclear data, where diagonal element \(\boldsymbol{\sigma}_{\boldsymbol{\alpha}}(\mathrm{i}, \mathfrak{i})=\sqrt{C_{\alpha \alpha}(i, i)}\)

6.8.7.1.5. Mixed absolute-relative covariances

If response-i and response-j are both absolute formatted, then

(6.8.59)\[\begin{split}\begin{aligned} \hat{\mathrm{C}}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j}) &=\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j}) \\ \hat{\mathrm{C}}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j}) &=\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j}) \\ \hat{\mathrm{C}}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j}) &=\mathrm{C}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j}) \\ \hat{\mathrm{C}}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j}) &=\mathrm{C}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j}) \end{aligned}\end{split}\]

Likewise, if both response-i and response-j are relative-formatted, then

(6.8.60)\[\begin{split}\begin{array}{l} \hat{\mathrm{C}}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})=\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})}{\mathrm{k}_{\mathrm{i}} \mathrm{k}_{\mathrm{j}}} \\ \hat{\mathrm{C}}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})=\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\mathrm{m}_{\mathrm{i}} \mathrm{m}_{\mathrm{j}}} \\ \hat{\mathrm{C}}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j})=\mathrm{C}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j})}{\mathrm{d}_{\mathrm{i}} \mathrm{d}_{\mathrm{j}}} \\ \hat{\mathrm{C}}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j})=\mathrm{C}_{\mathrm{kk}^{\prime}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{kk}^{\prime}}(\mathrm{i}, \mathrm{j})}{\mathrm{k}_{\mathrm{i}}^{\prime} \mathrm{k}_{\mathrm{j}}^{\prime}} \end{array}\end{split}\]

If response-i is absolute-formatted and response-j is relative-formatted, then

(6.8.61)\[\begin{split}\begin{array}{l} \hat{\mathrm{C}}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})}{\mathrm{k}_{\mathrm{j}}} \\ \hat{\mathrm{C}}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\mathrm{m}_{\mathrm{j}}} \\ \hat{\mathrm{C}}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j})}{\mathrm{d}_{\mathrm{j}}} \hat{\mathrm{C}}_{\mathrm{k}^{\prime}\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j})}{\mathrm{k}_{\mathrm{j}}^{\prime}} \end{array}\end{split}\]

Similar expressions can be derived if response-i is relative-formatted, and response-j is absolute-formatted. The I by I diagonal matrices of standard deviation values are the following:

(6.8.62)\[\begin{split}\hat{\sigma}_{\mathrm{k}}(\mathrm{i}, \mathrm{i})=\left\{\begin{array}{ll} \sigma_{\mathrm{k}}(\mathrm{i}, \mathrm{i}) & \text { absolute-formatted } \\ \sigma_{\mathrm{k}}(\mathrm{i}, \mathrm{i}) & \text { relative-formatted } \end{array}\right.\end{split}\]

(6.8.63)\[\begin{split}\hat{\sigma}_{\mathrm{m}}(\mathrm{i}, \mathrm{i})=\left\{\begin{array}{ll} \sigma_{\mathrm{m}}(\mathrm{i}, \mathrm{i}) & \text { absolute-formatted } \\ \sigma_{\mathrm{m}}(\mathrm{i}, \mathrm{i}) & \text { relative-formatted } \end{array}\right.\end{split}\]

(6.8.64)\[\begin{split}\hat{\sigma}_{\mathrm{d}}(\mathrm{i}, \mathrm{i})=\left\{\begin{array}{ll} \sigma_{\mathrm{d}}(\mathrm{i}, \mathrm{i}) & \text { absolute-formatted } \\ \sigma_{\mathrm{d}}(\mathrm{i}, \mathrm{i}) & \text { relative-formatted } \end{array}\right.\end{split}\]

(6.8.65)\[\begin{split}\hat{\sigma}_{\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i})=\left\{\begin{array}{ll} \sigma_{\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i}) & \text { absolute-formatted } \\ \sigma_{\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i}) & \text { relative-formatted } \end{array}\right.\end{split}\]

6.8.7.1.6. Correlation matrices

\(\mathbf{R}_{\mathbf{kk}}\) =

I by I correlation matrix for prior calculated responses, where element \(\mathrm{R}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})\) = \(\frac{\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{k}}(\mathrm{i}, \mathrm{i}) \sigma_{\mathrm{k}}(\mathrm{j}, \mathrm{j})}=\frac{\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{k}}(\mathrm{i}, \mathrm{i}) \sigma_{\mathrm{k}}(\mathrm{j}, \mathrm{j})}=\frac{\hat{\mathrm{C}}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})}{\hat{\sigma}_{\mathrm{k}}(\mathrm{i}, \mathrm{i}) \hat{\sigma}_{\mathrm{k}}(\mathrm{j}, \mathrm{j})}\)

\(\mathbf{R}_{\mathbf{mm}}\) =

I by I correlation matrix for prior measured responses, where element \(\mathrm{R}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})\) \(\frac{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{m}}(\mathrm{i}, \mathrm{i}) \sigma_{\mathrm{m}}(\mathrm{j}, \mathrm{j})}=\frac{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{m}}(\mathrm{i}, \mathrm{i}) \sigma_{\mathrm{m}}(\mathrm{j}, \mathrm{j})}=\frac{\hat{\mathrm{C}}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\hat{\sigma}_{\mathrm{m}}(\mathrm{i}, \mathrm{i}) \hat{\sigma}_{\mathrm{m}}(\mathrm{j}, \mathrm{j})}\)

\(\mathbf{R}_{\boldsymbol{\alpha} \boldsymbol{\alpha}}\) =

M by M correlation matrix for prior nuclear data, where element \(\mathrm{R}_{\alpha \alpha}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\alpha \alpha}(\mathrm{i}, \mathrm{j})}{\sigma_{\alpha}(\mathrm{i}, \mathrm{i}) \sigma_{\alpha}(\mathrm{j}, \mathrm{j})}\)

\(\mathbf{R}_{\mathbf{k}^{\prime} \mathbf{k}^{\prime}}\) =

I by I correlation matrix for adjusted responses, where element \(\mathrm{R}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}} \quad(\mathrm{i}, \quad \mathrm{j})\) = \(\frac{C_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i}) \sigma_{\mathrm{k}^{\prime}}(\mathrm{j}, \mathrm{j})}=\frac{\mathrm{C}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i}) \sigma_{\mathrm{k}^{\prime}}(\mathrm{j}, \mathrm{j})}=\frac{\hat{\mathrm{C}}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j})}{\hat{\sigma}_{\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i}) \hat{\sigma}_{\mathrm{k}^{\prime}}(\mathrm{j}, \mathrm{j})}\)