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CCFE is the fusion research arm of the United Kingdom Atomic Energy AuthorityInventory Uncertainty Quantification using TENDL Covariance Data in Fispact-II

James W. Eastwood, J. Guy Morgan and Jean-Christophe C. Sublet*

Culham Electromagnetics Ltd, *United Kingdom Atomic Energy AuthorityCulham Science CentreAbingdon, OX14 3DB United Kingdom

Simulation in space, energy and timeBoltzmann equationtransporttime independentenergy and spatial simulationprimary responseBateman equationinventory time dependentsecondary response

Application Program Interface: interfaces to connect Boltzmann and Bateman solvers for non-linear t- and T-dependent transportMCNP6USEASY-IIEUAPIMultifaceted interfaceMaterial evolutionTENDLEU2TRIPOLIEUSERPENTEUCarouselsNuclear DataEASY-II frameworkFISPACT-II is a modern engineering prediction tool for activation-transmutation, depletion, inventories at the heart of the EASY-II system that relies on the n-TAL collaboration to provide the nuclear data libraries.

FISPACT-II was designed to be a functional replacement for the code FISPACT-2007 but now includes enhanced capabilities.

a, g, d, p, n-Transport Activation Library: TENDL-2013 from the TENDL collaboration but also ENDF/B, JENDL, JEFF

All nuclear data processing is handled by NJOY (LANL), PREPRO (LLNL), and CALENDF (CEA-CCFE)3

Theory: H. Bateman, Cambridge 1910Set of stiff Ordinary Differential Equations to be solved

Here i and i are respectively the total decay constant and cross-section for reactions on nuclide i

ij is the cross-section for reactions on nuclide j producing nuclide i, and for fission it is given by the product of the fission cross-section and the fission yield fractions, as for radionuclide production yield

ij is the constant for the decay of nuclide j to nuclide i4Depletion: by decay or reactionSource: by decay or reactionSi: fission and/or radionuclide production4Rate equations: numerical aspectsLSODES, Livermore Solver for Ordinary Differential Equations with general sparse Jacobian matrices

Backward Differentiation Formula (BDF) methods (Gears method) in stiff cases to advance the inventoryAdams methods (predictor-corrector) in non-stiff casemakes error estimates and automatically adjusts its internal time-stepsYale sparse matrix solver efficiently exploits the sparsityability to handle time-dependent matrixno need for equilibrium approximationhandles short (1ns) time interval and high fluxes

LSODES wrapped in portable Fortran 95 codedynamic memory allocationminor changes to Livermore code to ensure portability

5EASY-II TENDLs libraries 200 MeVn-tendl-2013 (2012), multi temperature, 709 group library; 2630 (2434) targets full set of covariances probability tables in the RRR and URR xs, dpa, kerma, gas, radionuclide productionJENDL-4.0u, ENDF/B-VII.1, JEFF-3.2, 709 groups libraries; circa 400 targets each

g-tendl-2013, 162 groups xs library, 2626 targetsp-tendl-2013, 162 groups xs library, 2626 targetsd-tendl-2013, 162 groups xs library, 2626 targetsa-tendl-2013, 162 groups xs library, 2626 targets

6EASY-II other libraries Decay-2013, 3873 isotopes (23 decay modes; 7 single and 16 multi-particle ones)Ingestion and inhalation, clearance and transport indices libraries, 3873 isotopesJEFF-3.1, UKFY4.2 fission yields

EAF-2010 decay data: 2233 isotopesEAF-2010 ingestion and inhalation, clearance and transport indices libraries, 2233 isotopesEAF-2003 libraries; 293K, 774 targets (20 MeV)EAF-2007 libraries; 293K, 816 targets (55 MeV) EAF-2010 libraries; 293K, 816 targets (55 MeV)EAFs uncertainty files

7EASY-II &TENDL & ENDF/B, JENDL or JEFF8

Processing steps: burnup, transmutationMulti-particles groupwise, multi-temperature libraries with NJOY12-021, PREPRO-2013, probability tables in the RRR & URR with CALENDF-2010

For the inventory code FISPACT-II

From a, g, p, d, n-TENDL-2013 (2012)FISPACT-II parses directly the TENDLs and other libraries covariance information

Transport and activation application libraries now stem from unique, truly general purpose files9FISPACT-II irradiation scenarios Single irradiation pulse followed by cooling Multiple irradiation pulses changing flux amplitude cooling Multi-stepchanging flux amplitude and spectrumchanging cross-section (e.g., temperature dependence)cooling Pathways and sensitivity for all cases

10What FISPACT-II does Extracts and reduces nuclear and radiological dataSolves rate equations for time evolution of inventory Computes and outputs derived radiological quantitiesIdentifies and quantifies key reactions and decay processes: dominant nuclidespathways and uncertaintyMonte-Carlo sensitivity and uncertainty reduced model calculations Uncertainty calculation input cross-section and decay uncertaintiesoutput uncertainties for all radiological quantities

11Extract and reduce library data Condense run extracts from decay files: decay constant decay constant uncertainty Collapse constructs flux spectrum weighted averages:

12Data used in codecollapsed cross-section collapsed uncertainty Library inputcross-section vs energy covariances vs energyflux spectrum vs energy

Covariance collapsereactions X and Yenergy bins i and j [1,N] with N = 709uses Cov (Xi,Yj) for X = Y only in Monte-Carlocollapse Cov (Xi,Xj) to get uncertainty for

Collapse Cov(Xi,Yj) to get Cov( , ) for X YCov data in ENDF file 33 & 40, NI type LB=1, 5, 6Cov data in wider energy bins k [1, M], M ~ 4013

1 TENDL, 3 EAF

Covariance, variance

The projection operator Sik maps cross-section energy bins to covariance energy binsThe ENDF style covariance data forms, different LBs are read directly without the need of pre-processing 14Covariance, variance

Using Sik, the formula to construct estimates of the covariance matrix are as follows:The LB=1 case is the one that was applied to thecomputation of for the EAFs libraries15Xi and Yi terms or missing on the LB=815Uncertainty in FISPACT-II Given{ , }select irradiation scenariosolve for radiological quantities

Use {X, } to estimate uncertaintiesmethod 1: pathways to dominant nuclidesmethod 2: Monte-Carlo sensitivitymethod 3: reduced model Monte-Carlo sensitivity

16Uncertainty in FISPACT-II Pathways are used to identify the dominant contributors to the activation products for the specific irradiation scenario under consideration. This makes the calculation of uncertainties more practicable for all methods (random-walk approximation and Monte-Carlo). The standard uncertainty output uses a random-walk approximation to estimate error bounds. This estimate is much quicker than Monte-Carlo, but is likely to give larger bounds since it ignores many possible correlations.

17Uncertainty from pathwaysgiven initial inventory and irradiation scenariosort dominant nuclides at end of irradiation phase topxx (=20) controls number8 categories - activity, heat production, dose, etc. construct pathways from initial to dominant nuclides path_floor (=0.005) and loop_floor (=0.01)iterate on single-visit breadth-first search tree compute inventory contributions of pathways construct error estimate

1818Pathwayskeep pathways providing > path_floor of target inventorykeep loop providing > loop_floor of pathway inventory

19Error estimate

20Nt (atoms) and qt (radiological quantity) from rate equationtp, Ntp, Nt from pathwaysRr and Re pulse averaged reaction ratesreactions uncorrelated, fission correlated Given a set of target nuclide St, the uncertainty on the radiological quantity Q . is given by delta Q

Nt is the number of atoms of target t, So is a subset of SNtp is the number of atoms of target t formed along path pThe split is due to correlated fission pathways, other pathways are not

Se is the set of edges on pathway p, Sr set of reactions on edge eRr pulse averaged reaction rate of reaction rRe is the total pulse averaged reaction rate on edge e

20Using LB = 6 data The TENDL library contains MF=33, LB=6 data for different reactions X1, X2, ... for a given parent, i.e., p(n, X1)d1, p(n, X2)d2, . . . . These covariance data cov(X1,X2) for X1, X2 are stored as fractional values fX1X2 and are tabulated in the same energy bins as used respectively for the LB=5 covariance data fX1X1, fX2X2 for reactions X1, X2If the COVARIANCE keyword is used, FISPACT-II reads these data for all energy bins k and l and corrects for any instances where

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Using LB = 6 data Then the code uses the corrected data to compute collapsed covariance cov(X1,X2). Covariances are mapped to MF=10 by assuming that all isomeric daughters of a given pair of reactions with rates X1, X2 have the same collapsed correlation function, corr(X1,X2). Tables of all reactions which have covariance data and their collapsed covariances and correlations are printed by the collapse run. Inspection of these data will show those cases where the assumption of zero correlation between reactions of a given parent is not good. The effect of non-negligible correlations on uncertainties may be introduced into Monte-Carlo sensitivity calculations by choosing distributions of sample cross-sections to have the same variances and covariances as given by the TENDL data.

22Uncertainty from sensitivity calculation reference run + S inventory calculations independent { ; i = 1,...,I; s = 1,...,S} dependent { ; j = 1,...,J; s = 1,...,S} independent variables selected using random numbersnormal, log-normal, uniform, log-uniform means Xi and standard deviations Xi compute summary results: means standard deviationsPearson correlation coefficients output full data for post-processing

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Monte Carlo approach to sensitivity analysisoutput mean and standard deviation

Pearson correlation coefficient

controlled by keywords SENSITIVITY, MCSAMPLE, MCSEED, COVARIANCE

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On a series of S runs

Pearson product-moment correlation coefficient

24Uncertainty from reduced models EAF-2010 decay - 2233 nuclides, TENDL-2013 decay - 3873 nuclidescalculation includes all nuclides in master indexINDEXPATH generates reduced master index from pathwaystypically few 10s of nuclidesnumber adjustable by pathway parametersreduced master index run vs full run to validate discardsMonte-Carlo sensitivity for reduced master index runs faster + comparable answers

25EASY-II Verification and Validation V&VDifferentialC/E with the latest EXFOR, C/C with EAFs, ENDFs filesSACS: Statistical Analysis of Cross Section; Kopecky & Forrest legacy systematics. Integral CriticalityTRIPOLI-4.9 ICSBEP suite (130 cases)MCNP6 SvdM NRG ICSBEP suite (1900 cases)Transport, shieldingMCNP6 SvdM NRG Sinbad and LLNL suites Activation-transmutation; activity, gamma, decay heatEASY-II validation suite (500 reaction rates, thousands of integral E, time dependent, fusion orientated)MACS and RI: Maxwellian-averaged cross sections and astrophysical reaction rates, resonance integrals

26JAEA FNS Assembly; decay heat14 MeV neutrons are generated by a 2 mA deuteron beam impinging on a stationary tritium bearing titanium target; Fusion Neutron Source

FNS Neutron spectra, neutron fluence monitored by 27Al(n,a)Na24

Two experimental campaigns: 1996 and 2000; 74 materials27Decay power: FNS JAERI Ni28random-walk

ProductPathways T Path % E/CCo62 Ni62(n,p)Co62 1.5m 99.8 0.90 Co62mNi62(n,p)Co62m 13.9m 100.0 0.89

Decay power: FNS JAERI Ni29

Decay power simulation with EASY-II30

ConclusionsFISPACT-II:A powerful activation-transmutation prediction tool Identifies and quantifies important reactions and decays Uses full TENDL-2013 covariance dataUncertainty estimates: pathways to dominant nuclidesMonte-Carlo sensitivityreduced model + Monte-Carlo sensitivity Uncertainty on all responses: number density, activity, decay heat, dose rate, inhalation and ingestion indices, .http://www.ccfe.ac.uk/EASY.aspx31

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