Covariance Matrix Evaluation for Independent Fission Yields

Transcription

1 Covariance Matrix Evaluation for Independent Fission Yields N. Terranova 1, O. Serot 2, P. rchier 2, C. De Saint Jean 2, M. Sumini 1 1 Dipartimento di Ingegneria Industriale (DIN), Università di Bologna, Italy 2 CE, DEN, Cadarache, F-1318 Saint Paul les Durance, France Meeting May 12, 214, NE, Paris

2 Scope of the work Main goal How To produce reliable covariance information on fission yields evaluated in JEFF Development of fission yields models based on free parameters able to reproduce JEFF Bayesian method to generate variance/covariance matrices. t this stage only independent fission yields (IFY) have been considered. IFY are related to secondary fission fragments, after prompt neutron emission. 235 U(n th, f) and 239 Pu(n th, f) have been used as test fission reactions Computational Environment CONRD (COde for Nuclear Reaction nalysis and Data assimilation), developed at CE (Cadarache)

3 Table of contents 1 Post-Neutron Mass Fission Yields Pre-Neutron Mass Fission Yields Saw-Tooth 2 Covariance Generation Bayesian pproach 3 Preliminary Results djusted parameters Preliminary Correlation Matrices 239 Pu 4 Conclusions and Outlook

4 Table of contents 1 Post-Neutron Mass Fission Yields Pre-Neutron Mass Fission Yields Saw-Tooth 2 Covariance Generation Bayesian pproach 3 Preliminary Results djusted parameters Preliminary Correlation Matrices 239 Pu 4 Conclusions and Outlook

5 Post-neutron mass fission yield Y POST (14) = Y PRE (14) p 14 ()+Y PRE (141) p 141 (1)+Y PRE (142) p 142 (2)+ (1) Y POST () = Y PRE ( + ν i ) p +νi (ν i ) (2) ν i = Y POST (): Yield for a secondary fission fragment of mass number. Y PRE ( + ν i ): Yield for a primary fission fragment of mass + ν i who can contribute to the post-neutron yield Y POST () emitting ν i prompt neutrons. p +νi (ν i ): Probability to emit ν i prompt neutrons for a primary fission fragment of mass + ν i.

6 Pre-Neutron Mass Fission Yields Pre-neutron mass fission yields Y POST () = ν i = Y PRE ( + ν i ) p }{{} +νi (ν i ) (3) pre-neutron Y POST (): Yield for a secondary fission fragment of mass number. Y PRE ( + ν i ): Yield for a primary fission fragment of mass + ν i who can contribute to the post-neutron yield Y POST () emitting ν i prompt neutrons. p +νi (ν i ): Probability to emit ν i prompt neutrons for a primary fission fragment of mass + ν i.

7 Pre-Neutron Mass Fission Yields The Brosa model ccording to the Brosa model, the pre-neutron distribution for actinides can be described by 5 Gaussians, which correspond to 3 fission modes: Super Long Symmetric pre-scission shape (one Gaussian for the symmetric region). Standard I symmetric pre-scission shape (two Gaussians located symmetrically for light and heavy fragments ). Standard II symmetric pre-scission shape (two Gaussians located symmetrically for light and heavy fragments). Y PRE () = c P cy c(), c = SL, SI, SII (4) Y c() = 1 [ 2πσ 2 c exp( ( PRE c) 2 2σ 2 c ) + exp( ( PRE f + c) 2 ] ) 2σc 2 (5) U. Brosa et al., Nuclear Scission, Physics Reports, 197, 4 (199).

8 Pre-Neutron Mass Fission Yields Pre-neutron Parameters Each fission mode has three parameters: weight: P c. FY e-5 1e-6 Multi Gaussian Fission Yield Model Representation U235T Multi-Gaussian deviation: D c c = f 2 + Dc. width: σ c 1e-7 1e-8 1e We do not have 9 free parameters but 7: SL is centered in f, 2 where f is the compound nucleus mass. c Pc = 1 Y PRE () = 2. FY e-5 1e-6 1e-7 1e-8 1e-9 Fission modes Standard I Standard II Super Long

9 Saw-Tooth Prompt neutron emission probability Y POST () = ν i = Y PRE ( + ν i ) p +νi (ν i ) }{{} prompt n prob. (6) Y POST (): Yield for a secondary fission fragment of mass number. Y PRE ( + ν i ): Yield for a primary fission fragment of mass + ν i who contributes to the post-neutron yield Y POST () emitting ν i prompt neutrons. p +νi (ν i ): Probability to emit ν i prompt neutrons for a primary fission fragment of mass + ν i.

10 Saw-Tooth Prompt neutron emission probability (con t) We suppose that the probability of the prompt neutron emission for each mass follows a Gaussian law with a constant width. p (ν) = ν+.5 ν.5 N G ν() (ν )dν (7) G ν() (ν): Gauss distribution p(nu) Prompt neutron probabilities for U235T p(nu) at =9 p(nu) at =11 p(nu) at =12 p(nu) at =14 N: Normalization factor N G ν() (ν)dν = nu

11 Saw-Tooth Prompt neutron emission probability (con t) Free Parameters p PRE (ν i ) = N [ ( 2 νi +.5 ν ) ( νi.5 ν )] erf erf 2σ 2σ ν( PRE ): central value of the re-normalized Gaussians N G ν(pre ). σ: width of the same function, supposed independent from the mass of the primary fragment.

12 Saw-Tooth Prompt neutron emission probability (con t) Saw-Tooth Experimental information about the first momentum of p PRE (ν i ) are available saw-tooth curve; ν is then related to the average number of prompt neutrons emitted by primary fragments given by the Saw-tooth; ν i ν i p PRE (ν i ) =< ν > ST ; This equation is iteratively solved in the model. <nu> SWTOOTH CURVE 3 Wahl evaluation Vorobyev Nishio 1995 Boldeman

13 Table of contents 1 Post-Neutron Mass Fission Yields Pre-Neutron Mass Fission Yields Saw-Tooth 2 Covariance Generation Bayesian pproach 3 Preliminary Results djusted parameters Preliminary Correlation Matrices 239 Pu 4 Conclusions and Outlook

14 What we want to adjust Y PRE : Fission modes parameters, D c, σ c, P c p PRE (ν i ): Prompt-neutron parameters, ν( PRE ), σ Evaluated data JEFF U235T Total prompt neutron emission probability U235T 8 7 JEFF Boldeman FY% P(nu) nu ν tot = f /2 Y PRE () < ν > ST ν L = Y PRE () < ν > ST ν H = Y PRE () < ν > ST f /2

15 Bayesian pproach Generalized Least Square Method It is based on the Bayes theorem: p( x y, U) = p( y x, U) p( x U) d x p( x U) p( y x, U) (8) From the Maximum Entropy Theorem we have [ posterior = p( x y, U) exp 1 2 ( x x prior) T 1 prior Mx ( x x prior ) + ( f ( x) y) T M 1 y ( f ( x) y) ] (9) x: parameter vector; y: experimental values; f ( x): theoretical model function.

16 Bayesian pproach Modeling environment: CONRD Using Newton-Raphson to minimize the cost function [ x (n) = x (n 1) M (n 1) x G (n 1)T M 1 y M (n 1) x = ( f ( x (n 1) 1 prior ) y) + Mx ( x (n 1) x prior ) 1 prior [M x + G (n 1)T M 1 y G (n 1)] 1 ] (1) (11) CONRD COde for Nuclear Reaction nalysis and Data assimilation. Developed at CE (Cadarache). The mass fission yield model has been implemented in CONRD.

17 Table of contents 1 Post-Neutron Mass Fission Yields Pre-Neutron Mass Fission Yields Saw-Tooth 2 Covariance Generation Bayesian pproach 3 Preliminary Results djusted parameters Preliminary Correlation Matrices 239 Pu 4 Conclusions and Outlook

18 djusted parameters PRIORS 3 Saw-Tooth Curve for U235T Vorobyev 21 Sawtooth Curve Parameter Value Error Deviation Standard1 (amu) Deviation Standard2 (amu) StdDev Standard StdDev Standard StdDev Super Long Weight Standard Weight Super Long S. Zeynalov, V. Furman, F.-J. Hambsch, Investigation of mass-tke distributions of fission fragments from <nu> the 235 U(n,f)-reaction in resonances, Proc. of the 13th ISINN Workshop, May 25, Dubna..S. Vorobyev et al., Investigation of the prompt neutron emission mechanism in low energy fission of 235,233 U(n th, f) and 252 Cf(sf). EPJ Web Conf., 8, October 21.

19 djusted parameters Parameter Prior Posterior Deviation Standard1 (amu) Deviation Standard2 (amu) StdDev Standard StdDev Standard StdDev Super Long Weight Standard Weight Super Long JEFF Fission Yields djustment Post-Neutron Fission Yields U235T FY% 4 P(nu) Total prompt neutron emission probability U235T Boldeman 1985 P(nu) adjustment Parameter Best Estimate σ prompt-n emission prob nu

20 djusted parameters djusted Saw-Tooth Comparison between experimental and adjusted sawtooth Vorobyev 21 djusted ST 2 <nu>

21 Preliminary Correlation Matrices Correlation matrix for fission yields U235T fter adjustment, statistical errors only Only fission mode parameters propagated ll parameters propagated

22 Preliminary Correlation Matrices Correlation matrices have been obtained after adjustment, only statistical uncertainties have been considered. Propagating saw-tooth parameters covariances reduces correlations given by the fission modes. Without a physical model, the saw-tooth can give correlations only locally. The pre-neutron modes give strong correlations between light and heavy fragments. Standard1 and Standard2 are in competition and give anti-correlations.

23 Preliminary Correlation Matrices Relative Errors JEFF Relative Error The uncertainties in JEFF for mass fission yields are over-estimated. They are quadratic sums of the errors on isotopic fission yields. In the adjustment procedure we modified the JEFF error to get a better representation on the peaks (99% of the total yield). Rel. Err Relative error for evaluated JEFF vs posterior FY JEFF Post-analysis (only f.m.) Post-analysis (all) Marginalization is needed to get more reliable relative errors on the final values.

24 239 Pu Parameter Prior Posterior Deviation Standard1 (amu) Deviation Standard2 (amu) StdDev Standard StdDev Standard StdDev Super Long Weight Standard Weight Super Long JEFF Fission Yields djustment Post-Neutron Fission Yields Pu239T FY% 4 P(nu) Total prompt neutron emission probability Pu239T Gwin 1984 P(nu) adjustment Parameter Best Estimate σ prompt-n emission prob nu

25 239 Pu Correlation matrix for fission yields Pu239T fter adjustment, statistical errors only Only fission mode parameters propagated ll parameters propagated

26 239 Pu djusted Saw-Tooth Pu239T Saw-Tooth for Pu239T Comparison between evaluated and adjusted sawtooth Pu239T Wahl Evaluation palin 1965 Nishio 1995 Batenkov 24 Tsuchiya Wahl Evaluation djusted Saw-Tooth <nu> 3 <nu>

27 Table of contents 1 Post-Neutron Mass Fission Yields Pre-Neutron Mass Fission Yields Saw-Tooth 2 Covariance Generation Bayesian pproach 3 Preliminary Results djusted parameters Preliminary Correlation Matrices 239 Pu 4 Conclusions and Outlook

28 Outlook Conclusions We have developed a method able to represent faithfully JEFF evaluations for mass fission yields. The adjustment of the parameters for the pre-neutron fission modes and the saw-tooth curve has given acceptable results. Preliminary correlation information have been produced for mass fission yields, considering only statistical uncertainties. Systematic errors must be properly considered in this exercise, using marginalization techniques. To get more reliable correlation information and variance-covariance matrices, a model for the saw-tooth curve should be found. The same exercise can be done for isotopic and isomeric fission yields adding features in CONRD. Other fissioning systems will be processed in the future.

29 THNK YOU FOR YOUR TTENTION