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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)

$\mathit{\alpha }$ =

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

$\mathbf{k}(\mathit{\alpha })$ =

I dimensional vector of prior calculated responses obtained with nuclear data $\mathit{\alpha }$, 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)$${\mathrm{F}}_{\mathrm{m}/\mathrm{k}}(\mathrm{i},\mathrm{i})=\frac{{\mathrm{m}}_{\mathrm{i}}}{{\mathrm{k}}_{\mathrm{i}}}$$

${\hat{\mathbf{F}}}_{\mathbf{m}\mathbf{/}\mathbf{k}}$ =

I by I diagonal matrix, where diagonal element

(6.8.57)$${\mathrm{F}}_{\mathrm{m}/\mathrm{k}}(\mathrm{i},\mathrm{i})=\frac{{\mathrm{m}}_{\mathrm{i}}}{{\mathrm{k}}_{\mathrm{i}}}$$

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

${\mathit{\alpha }}^{\prime }$ =

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

m’ =

I dimensional vector of adjusted measured responses produced by GLLS procedure

${\mathbf{k}}^{\prime }({\mathit{\alpha }}^{\prime })$ =

I dimensional vector of adjusted calculated responses obtained with modified nuclear data ${\mathit{\alpha }}^{\prime }$

Note

${\mathbf{k}}^{\prime }({\mathit{\alpha }}^{\prime })={\mathbf{m}}^{\prime }$, due to GLLS adjustment procedure.

$\stackrel{\mathbf{~}}{\mathbf{d}}$ =

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 = $({\mathrm{k}}_{\mathrm{i}}-{\mathrm{m}}_{\mathrm{i}})/{\mathrm{k}}_{\mathrm{i}}$

$\hat{\mathbf{d}}$

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

$[\mathbf{\Delta }\mathit{\alpha }]$ =

M dimensional vector of relative variations in nuclear data = ${\mathbf{A}}^{-1}({\mathit{\alpha }}^{\prime }-\mathit{\alpha })$ where component-i = $\frac{{\alpha }_i^{\prime }-{\alpha }_i}{{\alpha }_i}$

$[\mathbf{\Delta }\mathbf{m}]$ =

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

$[\mathbf{\Delta }\mathbf{m}]$ =

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

$[\mathbf{\Delta }\hat{\mathbf{m}}]$ =

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

$[\mathbf{\Delta }\mathbf{k}]$ =

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

$[\mathbf{\Delta }\mathbf{k}]$ =

I dimensional vector of absolute variations in calculated responses = ${\mathbf{k}}^{\prime }-\mathbf{k}$, where component-i = $\text{k}{{}^{\prime }}_{\text{i}}-{\text{k}}_{\text{i}}$

$[\mathbf{\Delta }\hat{\mathbf{k}}]$ =

I dimensional vector of mixed absolute-relative variations in calculated responses, where component-i = $\text{k}{{}^{\prime }}_{\text{i}}-{\text{k}}_{\text{i}}$ for an absolute formatted response

6.8.7.1.2. Sensitivity Relations

${\tilde{\mathbf{S}}}_{\mathbf{k}\mathit{\alpha }}$ =

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

${\mathbf{S}}_{\mathbf{k}\mathit{\alpha }}$ =

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

${\hat{\mathbf{S}}}_{\mathbf{k}\mathit{\alpha }}$ =

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

(6.8.58)$$\begin{array}{r}\begin{array}{l}[\mathbf{\Delta }\mathbf{k}]={\mathbf{S}}_{\mathbf{k}\mathit{\alpha }}[\mathbf{\Delta }\mathit{\alpha }]\\ [\mathbf{\Delta }\mathbf{k}]={\mathbf{S}}_{\mathbf{k}\mathit{\alpha }}[\mathbf{\Delta }\mathit{\alpha }]\\ [\mathbf{\Delta }\hat{\mathbf{k}}]={\hat{\mathbf{S}}}_{\mathbf{k}\mathit{\alpha }}[\mathbf{\Delta }\mathit{\alpha }]\end{array}\end{array}$$

6.8.7.1.3. Absolute covariances

${\stackrel{\mathbf{~}}{\mathbf{C}}}_{\mathbf{mm}}$ =

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

${\stackrel{\mathbf{~}}{\mathbf{C}}}_{\mathbf{kk}}$ =

I by I covariance matrix for prior calculated responses, where element ${\stackrel{\mathbf{~}}{\mathbf{C}}}_{\mathbf{kk}}$(i,j) = $E\left(\delta k_i\delta k_j\right)$

${\stackrel{\mathbf{~}}{\mathbf{C}}}_{\mathbf{dd}}$ =

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

${\stackrel{\mathbf{~}}{\mathbf{C}}}_{{\mathbf{k}}^{\prime }{\mathbf{k}}^{\prime }}$ =

I by I covariance matrix for adjusted responses, where element ${\stackrel{\mathbf{~}}{\mathbf{C}}}_{{\mathbf{k}}^{\prime }{\mathbf{k}}^{\prime }}$(i,j) = $E\left(\delta k{{}^{\prime }}_i\delta k{{}^{\prime }}_j\right)$

${\mathit{\sigma }}_{\mathbf{m}}$ =

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

${\mathit{\sigma }}_{\mathbf{k}}$ =

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

${\mathit{\sigma }}_{{\mathbf{k}}^{\prime }}$

6.8.7.1.4. Relative covariances

$C_{\mathbf{mm}}$ =

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

$C_{\mathit{\alpha }\mathit{\alpha }}$ =

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

$C_{\mathbf{kk}}$ =

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

$C_{\mathbf{dd}}$

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

${\mathit{\sigma }}_{\mathbf{m}}$ =

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

${\mathit{\sigma }}_{\mathbf{k}}$ =

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

${\mathit{\sigma }}_{\mathit{\alpha }}$ =

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

6.8.7.1.5. Mixed absolute-relative covariances

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

(6.8.59)$$\begin{array}{r}\begin{array}{rl}{\hat{\mathrm{C}}}_{\mathrm{kk}}(\mathrm{i},\mathrm{j})& ={\mathrm{C}}_{\mathrm{kk}}(\mathrm{i},\mathrm{j})\\ {\hat{\mathrm{C}}}_{\mathrm{mm}}(\mathrm{i},\mathrm{j})& ={\mathrm{C}}_{\mathrm{mm}}(\mathrm{i},\mathrm{j})\\ {\hat{\mathrm{C}}}_{\mathrm{dd}}(\mathrm{i},\mathrm{j})& ={\mathrm{C}}_{\mathrm{dd}}(\mathrm{i},\mathrm{j})\\ {\hat{\mathrm{C}}}_{{\mathrm{k}}^{\prime }{\mathrm{k}}^{\prime }}(\mathrm{i},\mathrm{j})& ={\mathrm{C}}_{{\mathrm{k}}^{\prime }{\mathrm{k}}^{\prime }}(\mathrm{i},\mathrm{j})\end{array}\end{array}$$

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

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

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

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

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)$$\begin{array}{r}{\hat{\sigma }}_{\mathrm{k}}(\mathrm{i},\mathrm{i})=\{\begin{array}{ll}{\sigma }_{\mathrm{k}}(\mathrm{i},\mathrm{i})& \text{ absolute-formatted }\\ {\sigma }_{\mathrm{k}}(\mathrm{i},\mathrm{i})& \text{ relative-formatted }\end{array}\end{array}$$

(6.8.63)$$\begin{array}{r}{\hat{\sigma }}_{\mathrm{m}}(\mathrm{i},\mathrm{i})=\{\begin{array}{ll}{\sigma }_{\mathrm{m}}(\mathrm{i},\mathrm{i})& \text{ absolute-formatted }\\ {\sigma }_{\mathrm{m}}(\mathrm{i},\mathrm{i})& \text{ relative-formatted }\end{array}\end{array}$$

(6.8.64)$$\begin{array}{r}{\hat{\sigma }}_{\mathrm{d}}(\mathrm{i},\mathrm{i})=\{\begin{array}{ll}{\sigma }_{\mathrm{d}}(\mathrm{i},\mathrm{i})& \text{ absolute-formatted }\\ {\sigma }_{\mathrm{d}}(\mathrm{i},\mathrm{i})& \text{ relative-formatted }\end{array}\end{array}$$

(6.8.65)$$\begin{array}{r}{\hat{\sigma }}_{{\mathrm{k}}^{\prime }}(\mathrm{i},\mathrm{i})=\{\begin{array}{ll}{\sigma }_{{\mathrm{k}}^{\prime }}(\mathrm{i},\mathrm{i})& \text{ absolute-formatted }\\ {\sigma }_{{\mathrm{k}}^{\prime }}(\mathrm{i},\mathrm{i})& \text{ relative-formatted }\end{array}\end{array}$$

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 ${\mathrm{R}}_{\mathrm{mm}}(\mathrm{i},\mathrm{j})$ $\frac{{\mathrm{C}}_{\mathrm{mm}}(\mathrm{i},\mathrm{j})}{{\sigma }_{\mathrm{m}}(\mathrm{i},\mathrm{i}){\sigma }_{\mathrm{m}}(\mathrm{j},\mathrm{j})}=\frac{{\mathrm{C}}_{\mathrm{mm}}(\mathrm{i},\mathrm{j})}{{\sigma }_{\mathrm{m}}(\mathrm{i},\mathrm{i}){\sigma }_{\mathrm{m}}(\mathrm{j},\mathrm{j})}=\frac{{\hat{\mathrm{C}}}_{\mathrm{mm}}(\mathrm{i},\mathrm{j})}{{\hat{\sigma }}_{\mathrm{m}}(\mathrm{i},\mathrm{i}){\hat{\sigma }}_{\mathrm{m}}(\mathrm{j},\mathrm{j})}$

${\mathbf{R}}_{\mathit{\alpha }\mathit{\alpha }}$ =

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 ${\mathrm{R}}_{{\mathrm{k}}^{\prime }{\mathrm{k}}^{\prime }}(\mathrm{i},\mathrm{j})$ = $\frac{C_{{\mathrm{k}}^{\prime }{\mathrm{k}}^{\prime }}(\mathrm{i},\mathrm{j})}{{\sigma }_{{\mathrm{k}}^{\prime }}(\mathrm{i},\mathrm{i}){\sigma }_{{\mathrm{k}}^{\prime }}(\mathrm{j},\mathrm{j})}=\frac{{\mathrm{C}}_{{\mathrm{k}}^{\prime }{\mathrm{k}}^{\prime }}(\mathrm{i},\mathrm{j})}{{\sigma }_{{\mathrm{k}}^{\prime }}(\mathrm{i},\mathrm{i}){\sigma }_{{\mathrm{k}}^{\prime }}(\mathrm{j},\mathrm{j})}=\frac{{\hat{\mathrm{C}}}_{{\mathrm{k}}^{\prime }{\mathrm{k}}^{\prime }}(\mathrm{i},\mathrm{j})}{{\hat{\sigma }}_{{\mathrm{k}}^{\prime }}(\mathrm{i},\mathrm{i}){\hat{\sigma }}_{{\mathrm{k}}^{\prime }}(\mathrm{j},\mathrm{j})}$