Update on sensitivity coefficient precise computations

1. Update on sensitivity coefficient precise computations E. Ivanov WPEC/Sg. 39 Meeting December 4, 2015 11, Rue Pierre et Marie Curie, France

2. Layout of contribution in report • Introduction • Remarks on theory of Sk computations • What was done in 2014 • Brief description of the benchmarks • Computed sensitivities • Updates in 2015 • Nature of negative n,2n contribution • Metrics for data presentations • Conclusions Updates are in the way how to illustrate the results I.Kodeli updated deterministically computed profiles

3. Theory of Sk precise computations Probability to cause fission in r by neutron born in r’ The only assumption is the equivalence k+ and k Note: Importance function Ψ(r,E,Ω) is defined as change of asymptotic power (after all transients relaxation) if inject one neutron in the point of phase space with coordinates (r,E,Ω) 3

4. References on the theoretical analysis • Alexandr A. Blyskavka, “On Monte Carlo Estimators of Keff Derivatives and Perturbations,” Preprint IPPE-920, Obninsk, 1979 (in Russian). • A. A. Blyskavka, K. F.Raskach, A. M.Tsiboulia, “Algorithm for calculating keff sensitivities to group cross sections using Monte Carlo method and features of its implementation in the MMKKENO code”, Proceedings of Int. Conf. on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C-2005), Avignon, France, September 12-15, 2005. • Kirill F. Raskach, “An Improvement of the Monte Carlo Generalized Differential Operator Method by Taking into Account First- and Second-Order Perturbations of Fission Source”, Nuclear Science and Engineering, Volume 162, Number 2, June 2009, Pages 158-166 • Brian C. Kiedrowski, Forrest B. Brown, Paul P. H. Wilson “Adjoint-Weighted Tallies for k-Eigenvalue Calculations with Continuous-Energy Monte Carlo”, Nuclear Science and Engineering, Volume 168, Number 3, July 2011, Pages 226-241 • Brian C. Kiedrowski, “Adjoint Weighting for Continuous-Energy Monte Carlo Radiation Transport”, doctoral dissertation, University of Wisconsin (2009). • Christopher M. Perfetti, “Advanced Monte Carlo Methods for Eigenvalue Sensitivity Coefficient Calculations”, doctoral dissertation, University of Michigan (2012)

5. Contributors • James Dyrda AWE • Evgeny Ivanov, Maria Brovchenko IRSN • Ian Hill NEA • Mathieu Hursin, Sandro Pelloni PSI • Brian Kiedrowski LANL • Ivan Kodeli IJS • Christopher Perfetti, Bradley Rearden ORNL

6. Selected ICSBEP Benchmarks • Popsy (Flattop-Pu) is a plutonium (94 wt% 239Pu) sphere surrounded by a thick reflector of natural uranium. • PU-MET-FAST-006. Spherical model. • Topsy (Flattop 25) is a highly enriched (93 wt% ) uranium sphere surrounded by a thick reflector of natural uranium. • HEU-MET-FAST-028. Spherical model. • ZPR 9/34 loading 303 is a highly enriched uranium/ iron benchmark, reflected by steel. • HEU-MET-INTER-001. RZ model. • ZPR 6/10 loading 24 is the core with heterogeneous plutonium metal fuel with carbon/stainless steel dilutions, and a steel reflector. • PU-MET-INTER-002. RZ model.

7. Profiles Computations • 238-gr. sensitivities converted into 33 gr. by IRSN BERING code (E. Ivanov) • 33-gr. sensitivities converted to SCALE/sdf format by IRSN scripts (E. Ivanov) • MCNP6 output converted into SCALE/sdf format by NEA script (I. Hill) • Sensitivity profiles are presented using SCALE/Javapeno that reads *.sdf

8. PMF-006 (Flattop-Pu or Popsy) • Plutonium (94 wt239Pu) sphere surrounded by a thick reflector of natural uranium. Sensitive to scattering on heavy metals, and threshold reactions. • kbench=1.0000±0.0030

9. HMF-028 (Flattop-25 or Topsy) • Highly enriched (93 wt%) uranium sphere surrounded by a thick reflector of natural uranium. • kbench=1.0000±0.0030

10. HMI-001 (ZPR 9/34 loading 303) • Highly enriched uranium/ iron benchmark, moderated, reflected by steel. RZ model • kbench=0.9966±0.0026 • 56Fe-tot sensitivity in core = 0.0785, in reflector = 0.0166(0.015)

11. PMI-002 (ZPR 6/10 loading 24) • Core with heterogeneous plutonium metal fuel with carbon/stainless steel dilutions, and a steel reflector. • kbench=0.9862±0.0005 (kZPR6/10=1.0009±0.0007)

12. Example of negative Sn,2n MONK with JEFF U-238 n,2n reaction MONK with ENDF U-238 n,2n reaction SERPENT U-238 n,2n reaction SCALE 6.2β U-238 n,2n reaction Zero level

13. PMF-006 importance function The negative n,2n contribution is possible if importance of emitted neutrons more than factor of two lower than importance of neutron caused the reaction Group-wise (238 gr) spectra of importance function have been computed using TSUNAMI-1D

14. PMF-006 importance function The negative n,2n contribution is possible if importance of emitted neutrons more than factor of two lower than importance of neutron caused the reaction Two-times depression of importance exists => negative sensitivity to n,2n reaction appears because sum of secondary particles importance is lower than importance of the initial neutron

15. Metrics for comparison • Groups limits in 33 group structure “Fast” energy region from 1st to 6th groups “Intermediate” energy region from 7th to 30th groups “Thermal” energy region from 31st to 33rd groups • Uncertainty UTOTAL=<ST×COV×S> due to nuclide-reaction UFAST=<SFASTT×COV×SFAST> in fast energy region UINTER=<SINTERT×COV×SINTER> in intermediate energy region

16. PMF total uncertainties

17. PMF fast energy region

18. PMF intermediate energy region

19. HMF-028 total uncertainties

20. HMF-028 fast energy region

21. HMF-028 intermediate energy region

22. Overview of representativityfactors • Representativity factor (CEA, France) • Derived and proposed for experimental data selection in the 70s using GLLSM formalism in assumptions of non-correlated integral experiments • Ck factors (ORNL, USA) • Representativity factor included in SCALE code system • Complex similarity criteria : vector of similarity coefficients • Vector of the weights representing entire set of the benchmarks The similarity criteria derived using perturbation theory and data covariance can be further used to validate any codes

23. Example of representativity evolution PMI002 Sk collapsed in 238 groups (MCNP6) and in 33 groups (SERPENT) RF_INTR – ultimate contribution to adjust 239Pu fission cross section in intermediate energy region δR~ω1×(C/E)1 +ω2×(C/E)2 +…+ωN×(C/E)N

24. Example of representativity evolution PMF006 Sk computed using MCNP6 (by LANL) and SCALE 6.2β (by PSI) Indicator is ultimate contribution to adjust 239Pu fission cross section in fast energy region Difference is invisible the background of target resolution of adjustment

25. Summary and Conclusions • Nowadays precise methodologies to compute sensitivity coefficients are available worldwide • Accuracy of precise computations were tested against 4 benchmarks using 8 codes/methods (6 MC codes and 3 deterministic SN codes) presented by 7 organizations • Theoretical analysis: Recommended references to be updated • Additional updates: • observed negative sensitivity to n,2n reaction has been addressed • Two metrics are proposed to the results presentation: • collapse of sensitivity coefficients inside given energy regions, and • consideration of the impact on representativity factors for given applications

26. Thank you for your attention

27. Popsy: Integrated Sensitivities (1/3) 27

28. Topsy: Integrated Sensitivities (1/3) 28

29. ZPR-9/34: Integrates Sensitivities (1/3)

30. ZPR 6/10: Integrated Sensitivities (1/3)