Covariance Generation using CONRAD and SAMMY Computer Codes L. Leal a, C. De Saint Jean b, H. Derrien a, G. Noguere b, B. Habert b, and J. M. Ruggieri b a Oak Ridge National Laboratory b CEA, DEN, Cadarache 2 nd International Workshop On Nuclear Data Evaluation for Reactor applications Organized by CEA and NEA 29 th September 2 nd October, 2009 CEA Cadarache Château, France
OBJECTIVES Crosscheck methodologies for covariance generation in the resonance region Nuclear data evaluation codes CONRAD and SAMMY Resonance parameters covariance generation methodologies: Retroactive/Bayes (CONRAD and SAMMY) Retroactive/Analytical or Monte Carlo Marginalization (CONRAD) 2
Computer Code SAMMY Used for analysis of neutron, charged-particle crosssection data. Uses Bayes method (generalized least squares) to find parameter values. Uses R-matrix theory, Reich-Moore approximation (default) or multi- or single-level Breit-Wigner theory. Generates covariance and sensitivity parameters for resonance region calculations. 3
Computer Code CONRAD Analysis of microscopic, semi-integral and integral measurements Bayesian parameters estimations (GLS), Uncertainty propagation and evaluation with Analytical or Monte Carlo method Nuclear Reaction Models in CONRAD Resolved resonance region: R-matrix derived formalism (Reich-Moore, MLBW, etc) Unresolved resonance range: average R-matrix Statistical models: Hauser-Feshbach, Moldauer, GOE, etc) Continuum: ECIS wrapper 4
Covariance Data Evaluation in the Resolved Resonance Region with CONRAD and SAMMY Can be obtained as: 1. During the resonance parameter evaluation; 2. Retroactively when no experimental data and resonance parameters covariance exist. The methodologies available are: a) Slightly modification in the application of the Bayes Equations (CONRAD/SAMMY); b) Analytical/Monte Carlo Marginalization (CONRAD); 5
Retroactive covariance scheme Retrieve best set of available resonance parameters that presumably describe the experimental data; Use the parameters to simulate experimental cross sections for total, scattering, fission, and capture cross sections; Assign a global experimental uncertainty to the experimental data based on the uncertainty in the thermal capture cross section and the capture resonance integral, data normalization, data background, etc; Run CORAD/SAMMY code with the option to generate resonance-covariance retroactively; Convert the resonance-covariance results into the ENDF File 32 format. 6
Mathematical Details Bayes Equations (generalized least-squares) P = P + M Y M = (M -1 + W ) -1 Y = G t V -1 ( D T ) W = G t V -1 G Notation: (primes indicate updated values) P = parameters M = covariance matrix for parameters D = experimental data T = theoretical calculation G = partial derivatives (sensitivity matrix) V = covariance matrix for experimental data 7
Slightly Modification on the Bayes Equation Bayes Equations in as slightly different form P = P + M Y with Y = G t V -1 ( D T ) i i i i i where Y = Σ i Y i for data set i with W = (G t V -1 G ) i i i i where W = Σ i W i for data set i M = (M -1 + W ) -1 Treat individual data sets separately, calculating Y i and W i values of resonance parameters Add Y i s and W i s to obtain Y and W Solve Bayes equations once to fit all data sets using known 8
Mathematical Details Check whether output parameter values = input values Question: Is it true that P P? Answer: Probably, because Y = G t V -1 ( D T ) 0 because D was chosen T Assume M is appropriate for P Write M in ENDF format 9
CONRAD Marginalization Technique Thorough description of the marginalization technique in CONRAD just presented by Cyrille De Saint Jean Fast range Covariance Estimation with CONRAD 10
Application Reich-Moore resonance parameters for 48 Ti Energy region from 10-5 ev to 300 kev Only s-wave resonance parameters and capture cross sections considered Experimental capture cross section was generated with a global statistical uncertainty of 6 percent Prior resonance parameter uncertainties were assumed to be 100 percent Systematic uncertainty of 4 percent was assumed in the normalization 11
Application 48 Ti s-wave Capture Cross Section 12
Application 13 s-wave resonances in the energy region 10-5 ev to 300 kev and two resonances above 300 kev(15 resonances) 3 varied parameters: E r Γ γ (resonance energy) (gamma width) Γ n (neutron width) Total of 45 varied parameters: 15 resonances 3 parameters 13
Results: SAMMY (retroactive) 14
SAMMY (retroactive) 15
CONRAD (retroactive) 16
CONRAD (retroactive) 17
CONRAD (Analytical Marginalization) 18
CONRAD (Analytical Marginalization) 19
CONRAD (Monte Carlo Marginalization) 20
CONRAD (Monte Carlo Marginalization) 21
Average Capture Cross Sections Uncertainties Calculated with CONRAD and SAMMY Average capture cross-section uncertainties calculated using 44-energy groups Multigroup uncertainty calculations provide a good indication whether SAMMY/CONRAD off-diagonal terms of the RPC are in agreement 4% uncertainty in the cross section is attained when 4 % change in the normalization is introduced 22
Average Capture Cross Sections Uncertainties Calculated with CONRAD and SAMMY 23
Average Capture Cross Sections Uncertainties Calculated with CONRAD and SAMMY 24
Concluding Remarks The methodologies for generating RPC with the computer codes SAMMY and CONRAD have been investigated S-wave resonance parameters for 48 Ti in the energy region 10-5 ev to 300 kev were used for testing RPC covariance was done using the retroactive/bayes approach (SAMMY/CONRAD) and using retroactive/monte Carlo and analytical marginalization (CONRAD) Results of SAMMY and CONRAD methodologies are in excellent agreement 25