# Presentation: 675059

## Low Fidelity Covariance Work  at ORNL

**Low Fidelity Covariance Work  at ORNL**

**Mark Williams, Goran Arbanas**
**Mike Dunn, Doro Wiarda  **

**Oak Ridge National Laboratory**

- CSEWG Meeting
- November 2007

## Multigroup covariance matrix is represented by three uncorrelated matrix partitions:
Fast (F)		E  > 0.5 keV
Epithermal (H)	0.5 eV  <  E  < 0.5 keV
Thermal  (T)	E < 0.5 eV

- Multigroup covariance matrix is represented by three uncorrelated matrix partitions:
- Fast (F)		E  > 0.5 keV
- Epithermal (H)	0.5 eV  <  E  < 0.5 keV
- Thermal  (T)	E < 0.5 eV

## Fast Range Partition (being done by BNL and LANL)
only mts 1, 2, 18, 102, 4, 16 considered
nuclear model codes used to propagate assumed model uncertainties into nuclear data uncertainties/correlations

Epithermal and Thermal Partitions (ORNL)
only mts 102, 18, 2 considered
Integral Approximation

**Fast Range Partition**** ****(being done by BNL and LANL)**
- only mts 1, 2, 18, 102, 4, 16 considered
- nuclear model codes used to propagate assumed model uncertainties into nuclear data uncertainties/correlations
**Epithermal and Thermal Partitions**** ****(ORNL)**
- only mts 102, 18, 2 considered
- Integral Approximation

## Thermal cross section at 2200 m/s
Resonance integral (RI) 

**Resonance integral (RI) **

- Intrinsic Integral Parameters

**P****m**** = *****intrinsic***** integral parameter with known variance Var(P****m****)**

**sensitivity coefficient**

## Resonance Integrals

| Nuclide |  ENDF/B-VI | Integral(*) | difference |
| --- | --- | --- | --- |
| Cd-113 | 3.92E+2  | 3.90E+2 ± 10.3% | 0.5% |
| Xe-135 | 7.65E+6  | 7.60E+3 ± 6.6% | 0.7% |
| Sm-149 | 4.02E+4  | 4.01E+4 ± 5.9% | 0.2% |
| Np-237 | 6.60E+2  | 6.40E+2 ± 7.8 % | 3.0% |

*(*) **Atlas of Neutron Resonances,* S. F. Mughabghab

| Nuclide | ENDF/B-VI | Integral(*) | difference |
| --- | --- | --- | --- |
| Cd-113 | 2.06E+4  | 2.06E+4 ± 1.9% | 0% |
| Xe-135 | 2.63E+6  | 2.65E+6 ± 4.1% | -0.7% |
| Sm-149 | 4.02E+4  | 4.01E+4 ± 1.5% | 0.2% |
| Np-237 | 1.81E+2  | 1.76E+2 ± 1.6% | -2.8% |

- Thermal Cross Sections

## Integral Approximation For Covariance

**P R E M I S E**

- 1.  Intrinsic integral quantities computed with differential data are expected to agree within uncertainty of integral measurements

- Rgg’ = correlation matrix

- = standard deviation (diagonal matrix)

## Integral Approximation For Covariance Matrix

_*Thermal Range*_

_*Epithermal Range*_

_*OR*_

- (tri-diagonal)

- ?

##  
Thermal data for 1/v cross section
single degree of freedom => full correlation 

  Thermal data is for non-1/v cross section
ignore uncertainty in shape of data => full correlation 
neglect uncertainty in g-factors 

Most important moderator data is potential elastic which is constant over most of slowing range
assume uniform relative uncertainty with full correlation

Resonance absorption in dilute concentrations depends mainly on magnitude of RI
ignore effects of uncertainty in shape of data
assume uniform relative uncertainty... initially with full correlation

** **
**Thermal data for 1/v cross section**
- single degree of freedom => full correlation** **
**  ****Thermal data is for non-1/v cross section**
- ignore uncertainty in shape of data => full correlation** **
- neglect uncertainty in g-factors
**Most important moderator data is potential elastic which is constant over most of slowing range**
- assume uniform relative uncertainty with full correlation
**Resonance absorption in dilute concentrations depends mainly on *****magnitude of RI***
- ignore effects of uncertainty in shape of data
- assume uniform relative uncertainty... initially with full correlation

## Impact of Using RI to Represent XS

| Critical Benchmark  | Benchmark Dscription | XS Modified | keff |  |
| --- | --- | --- | --- | --- |
|  |  |  | ORIGINAL | FLAT |
| HST013-1  | bare sphere;
 HEU/water solution | 235U: (n,γ), (n,f) | 0.99940 | 0.99932 |
| LST002-1 | H2O-reflected sphere;
 LEU/H2O solution | 235U: (n,γ), (n,f) | 0.99821 | 0.99835 |
| PST002-2 
(PNL-2) | H2O-reflected sphere; 
Pu/water solution] | 239Pu: (n,γ), (n,f) | 1.00563 | 1.00672 |
| PWR pincell  | Hot Full Power; UO2
30 GWD/MT | 235U: (n,γ), (n,f) | 0.99725 | 0.99938 |
| PWR pincell  | Hot Full Power; UO2
30 GWD/MT | 149Sm,143Nd,103Rh,151Sm, 133Cs,155Gd,152Sm:  | 0.99725 | 0.99421 |
| FLATTOP |  [U-reflector, HEU | 35U(n,γ), 235U (n,f) | 1.00051 | 1.00368 |

## ENDF/B-VII versus Lo-Fi Uncertainties

## ENDF/B-VII versus Lo-Fi Thermal Correlation Matrices for 232Th Capture

*(1/v cross section)*

## ENDF/B-VII versus Lo-Fi Thermal Correlation Matrices for 155Gd

*(non-1/v cross section)*

## Hi-Fi versus Lo-Fi Thermal + Epithermal Correlation Matrices for 235U

## Combined BNL and ORNL Lo-Fi Cov

**ORNL used integral approximation to generate COV data for resonance & thermal range of ~300 materials **

**BNL has produced fast COV data for 219 nuclides (FP)**

**ORNL has recently merged the two sets of data **

## SUMMARY

- Integral approximation is simple method to generate low fidelity uncertainty data in thermal and epithermal ranges
- Comparisons with limited high-fidelity data indicate
- approximation tends to underestimate thermal uncertainty
- full correlations in epithermal appear to be overly conservative
- More testing using sensitivity analysis is being done
- Thermal and resonance range covariances for ~300 materials have been generated
- Fast range obtained from BNL for ~200 materials have been combined with ORNL data