Lognormal-BasedSamplingforFissionProductYields...

Research ArticleLognormal-Based Sampling for Fission Product YieldsUncertainty Propagation in Pebble-Bed HTGR

YizhenWang Menglei Cui Jiong Guo Jinlin Niu YingjieWu Baokun Liu and Fu Li

Institute of Nuclear and New Energy Technology Collaborative Innovation Center of Advance Nuclear Energy TechnologyKey Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education Tsinghua UniversityBeijing 100084 China

Correspondence should be addressed to Jiong Guo guojiong12tsinghuaeducn

Received 14 November 2019 Revised 29 July 2020 Accepted 12 August 2020 Published 25 September 2020

Academic Editor Alejandro Clausse

Copyright copy 2020 YizhenWang et al)is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Uncertainty analyses of fission product yields are indispensable in evaluating reactor burnup and decay heat calculationcredibility Compared with neutron cross section there are fewer uncertainty analyses conducted and it has been acontroversial topic by lack of properly estimated covariance matrix as well as adequate sampling method Specifically theconventional normal-based sampling method in sampling large uncertainty independent fission yields could inevitablygenerate nonphysical negative samples Cutting off these samples would introduce bias into uncertainty results Here weevaluate thermal neutron-induced U-235 independent fission yields covariance matrix by the Bayesian updating methodand then we use lognormal-based sampling method to overcome the negative fission yields samples issue Fission yieldsuncertainty contribution to effective multiplication factor and several fission productsrsquo atomic densities at equilibriumcore of pebble-bed HTGR are quantified and investigated )e results show that the lognormal-based sampling methodcould prevent generating negative yields samples and maintain the skewness of fission yields distribution Compared withthe zero cut-off normal-based sampling method the lognormal-based sampling method evaluates the uncertainty ofeffective multiplication factor and atomic densities are larger )is implies that zero cut-off effect in the normal-basedsampling method would underestimate the fission yields uncertainty contribution )erefore adopting the lognormal-based sampling method is crucial for providing reliable uncertainty analysis results in fission product yieldsuncertainty analysis

1 Introduction

Reactor design and safety analysis rely on accurate calcu-lations of system responses with properly evaluated un-certainties )ere has been an increasing need for evaluatingthe credibility of reactor safety Pebble-bed high temperaturegas-cooled reactor (pebble-bed HTGR) is a multiphysicsnonlinear coupled system including neutron transport andcomplex heat transfer hydraulics behaviour [1] In order tosystematically and thoroughly investigate the uncertaintiespropagation in pebble-bed HTGR an IAEA CoordinatedResearch Plan (CRP) [2 3] has been initiated after the startof OECDNEA UAM-LWR [4] Recent advances regardinguncertainty propagation analysis in pebble-bed HTGRmainly concern the nuclear cross section uncertainties

propagation in reactor neutronic calculations [5ndash9] Aspebble-bed HTGR allows fuels recirculation during fuelcycles and adopts higher fuel enrichment (85 wt) fuelsusually could achieve larger burnup values and then fissionproduct yields could be nonnegligible uncertainty sources inreactor burnup and decay heat calculations )eir uncer-tainties contributions to important reactor burnup re-sponses need to be considered properly for evaluating thecredibility of rector safety-related quantities of interest(QoI) eg maximum fuel pebble temperature

Fission product yield describes the fraction of a certainfission product produced per fission During the mea-surements of fission product yields correlated errors orcovariances may exist when using the same equipment ormethods [10] However they are ignored in evaluated

HindawiScience and Technology of Nuclear InstallationsVolume 2020 Article ID 8014521 21 pageshttpsdoiorg10115520208014521

nuclear data library Also self-consistent fission yieldsdata set should follow several physical constraints such asbinary fission mass conservation and charge conserva-tion [11] )ese constraints could introduce covariancesbetween fission yields data As the fission yieldsrsquo covari-ances in current releases of evaluated nuclear data librariesare still absent eg ENDFB-VII1 many methods aredeveloped to estimate these covariances informationbased on the imposed physical constraints )e Bayesianupdating method is widely used in data assimilation dataadjustment and model fitting problems It refines pa-rameters by taking both the prior information about thoseparameters and the likelihood which refers to new datainto consideration [10 12] It allows estimating the co-variance matrix of fission yields by sequentially intro-ducing the above physical constraints In the domain offission yields adjustment it is introduced by Kawano andChadwick [13] to update Pu-239 fission yields with chainyields to reduce the independent fission yields discrepancyin ENDFB-VII1 Pigni et al [14] expand it to involvecumulative fission yields into covariances estimation )edifference between chain yields-based updating and cu-mulative fission yields-based updating is further investi-gated by Fiorito et al [15 16] Based on the providedindependent and cumulative fission yields uncertaintiesinformation in ENDFB-VII1 this work adopts theBayesian updating method to estimate the independentfission yields covariances

Sampling-based methods for uncertainty analysis orstochastic UQ methods [17] require properly perturbedsamples to provide reliable uncertainty analysis results ofQoI As it is observed from the evaluated nuclear data li-brary independent fission yields generally have larger un-certainties Random sampling on these yields under normaldistribution could generate nonphysical negative samplesCutting off these negative yield samples by setting them tozero could artificially omit some information from theoriginal fission yields distribution resulting in biased un-certainty analysis results )is zero cut-off effect on quan-tified uncertainty has not been well studied )is raisesquestion whether normal distribution is sufficient for de-scribing inherently positive random variables with largeuncertainties only given their mean values and covariancematrix Smith et al [18] propose to replace normal distri-bution with lognormal distribution by the principle ofmaximum entropy [10] and Zerovnik et al [19 20] in-vestigate this method in the sampling resonance parameterswhere negative samples problem was encountered as infission yields )is work proposes an implementation of thelognormal-based sampling method in fission product yieldssampling

)e present work is organized as follows Section 2describes the nomenclature of fission product yields and theburnup calculation of pebble-bed HTGR An implementedstochastic UQ method for fission yields uncertainty prop-agation is described in Section 23 )e Bayesian updatingmethod and the lognormal-based sampling method aredetailed in Section 3 Finally results of fission yields un-certainty contributions to effective multiplication factor and

several important fission products atomic densities areprovided and discussed in Section 4

2 Model Description andUncertainty Propagation

21 ENDFB-VII1 Fission Product Yields SublibraryFission product yield characterizes the fraction of a par-ticular fission product nuclide produced per fission Acompound nucleus is formed when a fissile nucleus isbombarded by an incident neutron As its energy over-comes the fission barrier this compound nucleus couldundergo fission A brief description of the fission process isillustrated (see Figure 1) After the scission of compoundnucleus primary fission fragments are produced and theywould undergo deexcitation by releasing prompt neutronsdue to their high neutron to proton ratios After theemission of prompt neutrons the remaining fission frag-ments referred to as fission products would undergo βdecay isomeric transition or particle emission along theircorresponding decay chain and finally reach stable nu-clides Each fission product is identified by its mass numberA charge number Z and isomeric state I and is denoted asthe triplet (A Z I)

A detailed description about the nomenclature offission product yields could be found in [11] and they arebriefly summarized as follows IFYs and CFYs determinethe fraction of a fission product at different stages in thefission process IFYs denoted as y(A Z I) are the fractionof a fission product produced directly from one fissionafter the emission of prompt neutrons but prior to anyradioactive decays Because IFYs are produced before anyradioactive decay in the fission system they should besubject to the physical constraints of fission system egbinary fission conservation of mass and charge numberCFYs denoted as c(A Z I) determine the total fraction ofa fission product produced over all time after one fissionIt involves not only the direct production from fissionbut also the contributions from the decay of otherproducts

)e current releases of ENDFB-VII1 fission yieldsublibrary provide fission yields data for 31 fission actinidesfrom)-227 to Fm-255 )ough energy-dependence issueswithin fission spectrum are highlighted in current releasesof evaluated nuclear data library and neutron induced Pu-239 fission yields at 20MeV are supplemented to allowusers to linearly interpolate yields between 05MeV and20MeV for high accuracy purpose [22] other fission ac-tinide fission yields data are taken directly from ENDFB-VI evaluated by England and Rider [23] in 1993 )reefission systems for U-235 are evaluated with respect toincident neutron energy namely 00253 eV thermal en-ergy 05MeV fission spectrum energy and 140MeV highenergy IFYs and CFYs are evaluated for 1247 fissionproducts in thermal neutron induced U-235 fission yield(see Figure 2) )e relationship between IFYs and CFYs[11] is referred to as (1) where b(Aprime Zprime Iprime ⟶ A Z I) isthe branching ratio

2 Science and Technology of Nuclear Installations

c(A Z I) y(A Z I) + 1113944

AprimeZprime Iprime( )

b Aprime Zprime Iprime ⟶ A Z I( 1113857c Aprime Zprime Iprime( 1113857

(1)

It could be found that most IFYs appear in the upperregion of β-stability line and they are most likely toundergo βminus decay to reach a stable state As CFYsinvolve the production of a certain fission product fromthe decay of other fission products as shown in (1) thepeaks of CFYs distribution in neutron-charge numberfigure tend to be closer to the β-stability line (seeFigure 2)

)e evaluation of fission yields data requires a combinedwork of experimental measurements and theoretical modelpredictions It is natural for the evaluated fission yieldspossessing uncertainties originated from measurement er-rors and theoretical model parameters uncertainties Al-though England and Rider provide the uncertainties(standard deviation) of each fission product yield in theiroriginal work covariances information between fissionyields has not been provided since then )ose covariancesinformation is crucial for representing the physical con-straints imposed on IFYs and they should be estimatedproperly in order to generate self-consistent IFYs

Prescission

Release promptneutron

Deexcitation

β decay andrelease delayed

neutron

Neutron

Fissilenucleus

Compoundnucleus

Promptneutron Primary fission

fragments

Independent yields

Cumulative yields

Delayedneutron

Stablenuclide Long-lived

nuclide

Secondary fissionfragments

β

N (A + 1 Z)

NP (AP2 ZP2)

NP (AP1 ZP1)

NI (AI2 ZI2)

NI (AI1 ZI1)

NC (AC2 ZC2)

NC (AC1 ZC1)

N (A Z)

γ

Figure 1 Neutron induced fission process [21] )e fission products refer to the fission fragments after the emission of prompt neutronsIndependent fission yields (IFYs) characterize the fraction of a fission product produced before any radioactive decay whereas cumulativefission yields (CFYs) describe the fraction of that produced product over all time after a fission

Science and Technology of Nuclear Installations 3

perturbations )is work focuses on the propagation ofthermal neutron induced U-235 fission yields uncertaintiesto burnup simulation of pebble-bed HTGR based on ENDFB-VII1 )e estimation of covariances information will bedetailed in Section 31

22 Pebble-Bed HTGR Burnup Model and Built-In FissionYields Analysis Pebble-bed HTGR core (see Figure 3(a))consists of spherical fuel elements or fuel pebbles Each ofthese pebbles is composed of a spherical graphite matrix inthe centre where thousands of small coated particles (knownas TRISO particles) are embedded )ese particles containUO2 kernel in the centre with four structural coating layerssurrounding it (see Figure 3(b)) During reactor operationthese fuel pebbles are consistently flowing downward fromthe top of the core to the bottom and are irradiated atdifferent core spectrum regions randomly Fuel recirculationis a characterized fuel cycling procedure adopted in pebble-bed HTGR which is different with that applied in LightWater Reactor (LWR) Such recirculation allows fresh fuelpebbles being loaded into the core and spent fuel pebblesbeing discharged online without shutting reactor downMore importantly this recirculation permits fuel pebblesrunning through core multiple times before they are finallybeing discharged Because of the fuels recirculation thereexist running-in phase and equilibrium core states )e

equilibrium core state refers to the nuclei compositionsinside the core kept unchanged with time and thereforeeffective multiplication factor being stable at a certain value)is could give a more flattened power distribution acrossthe core and higher average discharge burnup value )eVSOP computer code system [26] is developed to performburnup calculation of pebble-bed HTGR by simulating thefuels recirculation process stepwise and conduct spectrumcalculation online at each spectrum region inside the core Adetailed description of this simulation process could befound in these articles [27]

)e built-in fission product chain in VSOP code in-volves 44 fission products and among these 44 fissionproductsrsquo data 14 are taken as IFYs while 30 are taken asCFYs )ese data are taken from ENDFB-IV and ENDFB-V An additional ldquononsaturatingrdquo fission product is evalu-ated to account for the sum of many lumped fission yieldswhich are not explicitly included in the chain [26] )ecomparison between these built-in fission yields andreplacing them with the current releases in ENDFB-VII1 isconducted to examine the availability of VSOP code forfission yields uncertainty propagation )is investigation isconducted on HTR-PM [28] with 87 fuel enrichment(while 85 enrichment is applied in actual design)

)e impact of each built-in fission yield on keff atequilibrium core state is investigated individually byreplacing them with ENDFB-VII1 It should be noted that

110

100

90

80

70

60

50

40

30

Charge number Z

Neu

tron

num

ber N

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

20 30 40 50 60 70

(a)

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

110

100

90

80

70

60

50

40

30

Charge number Z

Neu

tron

num

ber N

20 30 40 50 60 70

(b)

Figure 2 )ermal neutron induced U-235 fission yield data in ENDFB-VII1 )e natural logarithm values of fission yields are plotted(a) Independent yields (b) Cumulative yields


built-in fission yields library in VSOP includes a com-bination of IFYs and CFYs and they are presented separately(see Tables 1 and 2)

IFYs are evaluated by subtracting the total contributionsof its precursors from experimental measured CFYs Withthe improvement of CFYs measurements the evaluated IFYsbecome more precise It could be seen from the table thatIFYs in ENDFB-VII1 are lower than the built-in fissionyields used in VSOP Except the large discrepancy in thefission yield of Mo-95 all the impacts from replacing fissionyields are lower than 20 pcm )e overall impact is 67 pcm(see Table 3) when all the yields are replaced without FP-44)e difference is acceptable in effective multiplication factorcalculations when substituting built-in VSOP fissionyields with ENDFB-VII1 fission yields )e VSOPburnup model is further used to conduct fission productyields uncertainty propagation as described in Section 23

23 Uncertainty Quantification Scheme )e HTR-PM [28]reactor core is modelled in VSOP computer code systemto analyse the uncertainty propagation of fission yields inthis work 15 times recirculation of fuel is adopted and theaverage discharge burnup value is around 90 210MW middot dtUwith fresh fuel having 85 wt enrichment As fuel recir-culation tightly couples the neutronics and burnup calcu-lation spatially inside the core it is difficult to separate theuncertainty propagation step by step Stochastic UQmethodis used to investigate the uncertainty propagation in equi-librium core state An uncertainty propagation scheme isproposed in this work (see Figure 4)

Two sampling methods are implemented in this worknamely normal-based sampling and lognormal-basedsampling Different from normal-based sampling lognor-mal-based sampling requires a lognormal transformation ofthe original mean vector and covariance matrix When theIFYs samples are generated their corresponding CFYs arecalculated and combine them to form self-consistent fissionyield samples )ese combined IFYs and CFYs samples arepropagated to VSOP HTR-PM model for further uncer-tainty analysis Detailed Bayesian updating method de-scription and lognormal-based sampling procedures will beintroduced in Section 3

3 Fission Product Yield Perturbation

31 U-235 9ermal Neutron-Induced IFYs CovariancesEstimation Bayesian updating method or the generalizedleast square method (GLSM) is a data adjustment methodwhich allows the prior data being updated with combination ofnew knowledge about these data Such knowledge could bemeasured data or physical constraints imposed on these priordata )e present work applies Bayesian updating method toestimate the covariance matrix of IFYs based on ENDFB-VII1 thermal neutron induced U-235 fission yields sublibrary)e specification of this method is briefly recalled as follows

Consider a multivariate linear regression model shownin

c Xy + ε (2)

where c and y isin Rntimes1 are observables and parameters to beupdated or estimated respectively X isin Rntimesn is the design

Control rod drive

Fuelling lineTop reflector

Control rod

Pebble bedfuel core

Bottomreflector

Defuellingtube

Reactoroutlet

Core support

Core barrel

Side reflectir

Reactorpressure vessel

Top plate

SAS container

(a)

5mm graphite layer

Coated particles imbeddedin graphite matrix

Dia 60mm

Dia 092mm

Dia 05mmUranium dioxide

Fuel kernal

TRISOcoated particle

Section

Fuel spherePorous carbon buffer 051000mmInner pyrolytic carbon 401000mmSilicon carbon barrier coating 351000mmPyrolytic carbon 401000mm

(b)

Figure 3 Pebble-bed HTGR core (a) Core geometrical of PBR250 design [24] (b) Fuel pebbles [25]


matrix that represents linear mapping between estimatingparameters and observables ε isin Rntimes1 are the measurementerrors of observables with expectation E[ε] 0 isin Rntimes1 andvariance Var[ε] V isin Rntimesn By the principle of maximuminformation entropy it is objective and plausible to assign

Gaussian distribution on this error Similarly estimatingparameters y could also be assigned Gaussian distributiongiven their expectationE[y] y0 and variance Var[y] Z0)e generalized least square problem [29] is formulated bythe following minimization in the domain of estimating

Table 1 Comparison between VSOP built-in yields and ENDFB-VII1 in keff prediction (IFYs)

Index Fission productFission yields keff

VSOP ENDFB-VII1

Relative difference to ENDFB-VII1 () VSOP ENDF

B-VII1Difference to ENDF

B-VII1 (pcm)

1 Rh-103 18580Eminus 11 63796Eminus 13 9657 101027 101027 02 Pd-105 98300Eminus 13 00000E+ 00 10000 101027 101027 03 Xe-131 15400Eminus 08 14199Eminus 09 9078 101027 101027 04 Cs-133 50800Eminus 07 79194Eminus 09 9844 101027 101027 05 Cs-134 35700Eminus 07 38547Eminus 08 8920 101027 101027 06 Nd-143 95000Eminus 13 47997Eminus 14 9495 101027 101027 07 Pm-148m 74900Eminus 09 80994Eminus 11 9892 101027 101027 08 Pm-148g 57300Eminus 08 44497Eminus 11 9992 101027 101027 09 Sm-147 00000E+ 00 00000E+ 00 000 101027 101027 010 Sm-148 69500Eminus 13 16399Eminus 14 9764 101027 101027 011 Sm-149 00000E+ 00 17099Eminus 12 mdash 101027 101027 012 Sm-151 00000E+ 00 47497Eminus 09 mdash 101027 101027 013 Eu-154 16300Eminus 08 96993Eminus 10 9405 101027 101027 014 Gd-155 44100Eminus 11 40797Eminus 12 9075 101027 101027 0

Table 2 Comparison between VSOP built-in yields and ENDFB-VII1 in keff prediction (CFYs)


VSOP ENDFB-VII1 Relative difference to ENDFB-VII1 () VSOP ENDF


B-VII1 (pcm)

1 Xe-135 66023Eminus 02 65385Eminus 02 097 101027 101044 minus172 FP-44 94760Eminus 01 94760Eminus 01 000 101027 101027 03 Xe-136 62701Eminus 02 63127Eminus 02 minus068 101027 101027 04 Kr-83 53076Eminus 03 53620Eminus 03 minus102 101027 101027 05 Zr-95 64678Eminus 02 65027Eminus 02 minus054 101027 101027 06 Mo-95 16410Eminus 06 65029Eminus 02 minus396 times 1010 101027 100911 1167 Mo-97 59600Eminus 02 59968Eminus 02 minus062 101027 101027 08 Tc-99 61284Eminus 02 61087Eminus 02 032 101027 101028 minus19 Ru-101 50501Eminus 02 51725Eminus 02 minus242 101027 101026 110 Ru-103 31411Eminus 02 30309Eminus 02 351 101027 101042 minus1511 Rh-105 10199Eminus 02 96416Eminus 03 547 101027 101030 minus312 Pd-108 71032Eminus 04 54125Eminus 04 2380 101027 101028 minus113 Ag-109 29903Eminus 04 31221Eminus 04 minus441 101027 101027 014 Cd-113 12425Eminus 04 14038Eminus 04 minus1298 101027 101027 015 I-131 28325Eminus 02 28907Eminus 02 minus205 101027 101022 516 Xe-133 67859Eminus 02 66991Eminus 02 128 101027 101032 minus517 Pr-141 58929Eminus 02 58470Eminus 02 078 101027 101028 minus118 Pr-143 59710Eminus 02 59558Eminus 02 025 101027 101029 minus219 Nd-144 54523Eminus 02 54996Eminus 02 minus087 101027 101027 020 Nd-145 39339Eminus 02 39334Eminus 02 001 101027 101027 021 Nd-146 29912Eminus 02 29969Eminus 02 minus019 101027 101027 022 Pm-147 22701Eminus 02 22467Eminus 02 103 101027 101035 minus823 Pm-149 10888Eminus 02 10816Eminus 02 059 101027 101031 minus424 Sm-150 54130Eminus 06 29998Eminus 07 9446 101027 101027 025 Pm-151 42044Eminus 03 41877Eminus 03 040 101027 101028 minus126 Sm-152 27057Eminus 03 26691Eminus 03 135 101027 101029 minus227 Eu-153 16264Eminus 03 15828Eminus 03 268 101027 101029 minus228 Eu-155 33025Eminus 04 32136Eminus 04 269 101027 101028 minus129 Gd-156 13517Eminus 04 14853Eminus 04 minus988 101027 101027 030 Gd-157 64651Eminus 05 61506Eminus 05 486 101027 101027 0


Table 3 Reference calculation between built-in fission yields of VSOP and ENDFB-VII1 (all substitution without FP-44)

keff Difference to ENDFB-VII1 (pcm)Built-in VSOP fission yields ENDFB-VII1 fission product yields (without update)

101027 100960 67

Independent yieldCumulative yieldPrior uncertainties

(i)(ii)

(iii)

Neutron-inducedfission yield sublibrary

Decay datasublibrary

END

FB-

VII

1

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures

Nearest SPDsearching

Sequential bayesian method

Consistency with cumulative yieldConservation of total yieldConservation of mass numberConservation of charge number

Correlated sampling andexponential transformation

Independent yield samples1N

Cumulative yield samples1N

Mapping

Self-

cons

isten

tyi

eld

sam

ples

VSOP burnup simulation

Input

Upd

ated

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures

Zero-cutoff

Lognormal transformation Correlated sampling

Posterior independent yieldUpdated independent yieldEstimated covariance matrix

(i)(ii)

Logarithmic domaindistribution paratmeters

Log-domain meanLog-domain covariance matrix

(i)(ii)

Original samples parametersInvolving negative samples(i)

Truncated samples with zeros(i)Log-domain meanApproximated covariance matrix

(i)(ii)

Branching ratiosHalf-lifes

(i)(ii)

Figure 4 Flow chart of fission product yields uncertainty propagation


parameters to find the best least square estimated parametersas

miny

χ2 (Xy minus c)TV

minus1(Xy minus c) + y minus y0( 1113857

TZ

minus10 y minus y0( 11138571113960 1113961

(3)

)e above minimization process could also be inter-preted in the perspective of Bayesian updating Consider theestimated parameters have a prior of Gaussian distributionwith density p(y) in

p(y) 1

(2π)(n2)

Z01113868111386811138681113868

1113868111386811138681113868(12)

⎛⎝ ⎞⎠exp minus12

1113874 1113875 y minus y0( 1113857TZ

minus10 y minus y0( 11138571113882 1113883

(4)

And likelihood function determines the probability ofany candidate estimated parameters appearing in the ob-servables distribution )en likelihood function p(c | y | ) isgiven as

p(c | y) 1

(2π)(n2)

|V|(12)

1113888 1113889exp minus12(Xy minus c)

TV

minus1(Xy minus c)1113882 1113883 (5)

)e posterior distribution of estimated parameters y istherefore calculated by Bayesian theorem and it gives

p(y | c) p(c | y)

p(c)p(y)propp(c | y)p(y) (6)

Considering the conjugacy between Gaussian prior andlikelihood the posterior estimated parameters followsGaussian distribution as well Under quadratic loss theoptimal estimates of true values and their uncertainty are themean vector and covariance matrix of the posterior distri-bution It is worthwhile to mention that the estimated meanvector could maximize the exponential term in (3) and thiscould also lead to the solution of GLSM in (3)

)e posterior estimated parameters are obtained as

ypost y + Z1XTV

minus1c minus Xy01113858 1113859 (7)

where Z1 is the posterior covariance matrix of estimatedparameters and it is shown in (8) and after applyingWoodbury matrix identity it is reformed as (9)

Z1 Zminus10 + X

TV

minus1X1113872 1113873

minus1 (8)

Z1 Z0 minus Z0XT

V + XZ0XT

1113872 1113873minus1

XZ0 (9)

Here regarding IFYs as estimated parameters y with priorcovariance matrix Z0 (diagonal matrix with only consid-eration of each fission yields uncertainty in ENDFB-VII1)observables c represent the evaluated CFYs in ENDFB-VII1 total independent yields fission system total massnumber and charge number respectively )e corre-sponding design matrix could be formulated as follows

(1) Consistency with CFYs c My where M is the Q-matrix proposed in [11] It could be formulated fromthe linear mapping in (1) with the providedbranching ratios data in ENDFB-VII1 decay sub-library )is updating process follows Luca Fioritorsquosupdating procedures [15] on CFYs consistency inJEFF-312 Different than in previous work [14] thiswork explicitly constructs this design matrix withbranching ratios rather than obtaining each elementvia direct perturbations using a burnup code Such

procedures allow direct examination of consistencybetween IFYs and CFYs in the current releases ofENDFB-VII1 Total IFYs total mass number andtotal charge number conservations are implementedfollowing the procedures proposed in Pigni et alrsquoswork [14] )e updating results of IFYsrsquo covariancematrix are in

Z1 Z0 minus Z0MT

V + MZ0MT

1113872 1113873minus1

MZ0 (10)

(2) Conservation of binary fission Ty UTy whereU isin Rntimes1 is a unity vector)e sum of total yield Ty is20 with summation precision of σ2sum 10 times 10minus5)e updated covariance matrix subsequent to (10) islisted in (11) It should be noticed that ternary fissionsmay occur however they are not considered inENDFB-VII1 and these ternary fissions are not in-cluded in this updating process

Z2 Z1 minus Z1U σ2sum + UTZ1U1113872 1113873

minus1U

TZ1 (11)

(3) Conservation of fission system mass numberTM NTY where N isin Rntimes1 whose element corre-sponds to the mass number of each fission product)e total mass number of fission system is conservedto 23357915 (considering the average promptneutrons released at 00253 eV is 242085 recorded inENDFB-VII1 and mass defect of U-235 is notconsidered) )e assumed variance of total massnumber is 10 times 10minus5 )e updated covariance matrixsubsequent to (11) is shown in

Z3 Z2 minus Z2N σ2sum + NTZ2N1113872 1113873

minus1N

TZ2 (12)

(4) Conservation of fission system charge numberTC WTy where W isin Rntimes1 with each elementbeing the charge number of each fission productconsidered )e total charge number of fissionsystem is conserved as 9205318 )is total chargenumber is calculated from the charge numbers ofeach fission product weighted by their correspond-ing IFYs provided in ENDFB-VII1 It is observed in


this work that if we take the total charge number asexactly 920 the calculated CFYs calculated fromupdated IFYs will have large discrepancy with CFYsprovided in the library And this discrepancy will benarrowed when we take the decimal digits intoconsideration )e updated covariance subsequentto (12) is shown in

Z4 Z3 minus Z3W σ2sum + WTZ3W1113872 1113873

minus1W

TZ3 (13)

Correlation matrix of updated IFYs is plotted (seeFigure 5) )ese correlations are introduced sequentially tocooperate the consistency with CFYs conservation of binaryfission mass number and charge number of fission systemFigure 5(a) shows that there is a significantly two-humpedtendency in the correlation distribution )is tendency issimilar with the two-humped distribution of IFYs wheremany correlations are introduced from the conservationconstraints in fission system while fewer correlations areintroduced between humped part and valley part AndFigure 5(b) presents a close look of the correlations amongfission product index range from 65 to 245 It could benoticed that the diagonal of this correlation matrix is dividedinto several small groups regarding different decay chainsIFYs within each decay chain have negative correlation witheach other introduced from the consistency of CFYs

)e updated IFYs are compared with the prior fissionyields recorded in ENDFB-VII1 (see Figure 6) It could beseen that small adjustment is introduced to fission productyields in the two-humped part while larger adjustment isintroduced in the valley and two tail parts )is is mainlybecause IFYs in those parts have smaller prior fission yieldsand they are not as accurately evaluated as those larger onesin the two-humped part )erefore more adjustments areexpected in those regions )e updated and prior standarddeviations are presented and compared (see Figure 7) Itcould be seen that the adopted updating procedures couldreduce the uncertainty of updated IFYs)is is mainly due tothe introduced constraints that further constrain the un-certainty of these fission yields and introduce covariancesamong them

)e final updated covariance matrix of IFYs Z4 and theposterior IFYs mean vector y4 are applied to generate theperturbation samples of IFYs )e detailed sampling pro-cedures are further discussed in the following section

32 Lognormal-Based Sampling Procedures ConsideringIFYs are inherently positive random sampling under nor-mal distribution could draw unphysical negative samples)ese negative samples would appear significantly when thesampled parameters have large uncertainty (eg relativedifference σμgt 30) Smith et al concluded that when therelative uncertainty of a random variable exceeds 30 theprobability distribution of this parameter chosen to repre-sent its physical uncertain information tends to be skewednoticeably [18] and the drawn negative samples fractiontends to grow It could therefore be concluded that normaldistribution is not adequate to describe inherently positive

random variables whose uncertainties are large because itcould not capture the skewness of random variable distri-bution By the principle of maximum information entropylognormal distribution is suggested to be the optimal choicefor inherently positive parameter when only expectation andvariance are known about this parameter [10 29] Largerrelative uncertainty would result in a more skewed distri-bution (shown in Figure 8) Lognormal distribution isshifting to a normal-like distribution as its relative uncer-tainty becomes lower than 30 where skewness of thedistribution is not significant

)e updated posterior IFYs relative uncertainties arecompared with prior relative uncertainties (see Figure 9) inour previous work [30] Except for a few fission productswhich have their relative uncertainties increased most fis-sion products have their corresponding relative uncer-tainties decreased to around 42 )e increased relativeuncertainty fission products are Ag130m0 Cd129m0Sn127m1 Cd126m0 In126m0 Sb124m1 Zn123m1Ag115m0 Y93m1 Y93m0 Se85m1 and Ge77m0 )eirrelative uncertainties increased due to their updated smallerposterior mean values From Figure 10 it could be seen thatmost fission yields standard deviations have been reducedbecause of the updating process However the above fissionproducts have their mean value updated even smaller andthat makes their relative uncertainties increased Comparedwith the listed monitor fission products for fission of U-235in Fiorito et alrsquos work [15] they are not included and wemaythink they are less relevant to the reactor burnup and criticalcalculation When applying simple random sampling pro-cedures under normal distribution drawing samples inRntimesS

from the N(y4 Z4) where n is the number of fission yieldsand S is the sample size it is almost impossible to draw asample set with all positive yields as the yields domain is toolarge (eg ngt 900)

In this work lognormal random sampling proceduresare applied to generate IFYs perturbation samples )esampling follows the development of Zerovnik et al [19] andapplies it into the generation of IFYs samples Multivariatelognormal distribution is defined as

L ln(y) sim N μl Zl( 1113857 (14)

where y is the posterior IFYs with expectation y4 and co-variance matrix Z4 estimated by Bayesian updating methoddiscussed in Section 32 and L isin Rntimes1 is the natural loga-rithmic value of independent yields μl and Zl are the cor-respondingmean and covariancematrix of IFYs in the naturallogarithmic domain )e detailed derivation of their relationwith parameters in original domain (y4 and Z4) could befound in [20] )e basic idea is recapped in the following

Consider the preservation of probability the relationbetween random variables in original domain and loga-rithmic domain is formulated in

pL(l)dl pY(y)dy (15)

)e lognormal distribution density is therefore derivedas in


9008007006005004003002001000

900

800

700

600

500

400

300

200

100

0

Fission product index

Fiss

ion

prod

uct i

ndex

(a)

2402202001801601401201008060

240

220

200

180

160

140

120

100

80

60

Fiss

ion

prod

uct i

ndex


(b)

Figure 5 Estimated IFYs correlation matrix (a) All the estimated correlation information (b) Section of the estimated correlationinformation Red dot indicates the positive correlation and blue dot indicates the negative correlation )e fission product index refers toeach fission product identified by its charge number Z mass number A and isomeric state I (ZZAAAI) )ese indices are grouped by themass number and arranged in a descending manner For each mass group charge number is ordered in an ascending manner to cooperatethe βminus decay

0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield


0 200 600400 800 1000

Before updateAer update

Figure 6 IFYs distribution before and after sequential Bayesianupdating Natural logarithm is presented on the y-scale )e fissionproduct index refers to each fission product identified by its chargenumber Z mass number A and isomeric state I (ZZAAAI) )eseindices are grouped by the mass number and arranged in adescending manner

0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield

STD

Fission product index0 200 600400 800100 300 700500 900 1000


Figure 7 IFYs standard deviation (STD) distribution before andafter sequential Bayesian updating Natural logarithm is presentedon the y-scale )e fission product index refers to each fissionproduct identified by its charge number Z mass number A andisomeric state I (ZZAAAI) )ese indices are grouped by the massnumber and arranged in a descending manner


pY(y) 1

(2π)(n2)

Zl

11138681113868111386811138681113868111386811138681113868(12)

1113945n

i1yi

exp minus12

ln(y) minus μl1113858 1113859TZ

minus1l ln(y) minus μl1113858 11138591113882 1113883 (16)

03

025

02

015

01

005

0 2 4 6 8 10 12 14 16 18Random variable X

Prob

abili

ty d

ensit

y

R = 08R = 07R = 06R = 05

R = 04R = 03R = 02R = 01

Figure 8 Lognormal distribution of random variable X in terms of its relative uncertainty Relative uncertainty R (σXμX) is ranged from10 to 80 and μX 20 Dashed line shows the distribution with relative uncertainty lower than or equal to 40 whereas solid lineindicates the distribution with relative uncertainty larger than 40

18

16

14

12

1

08

06

04

02

0172 152 132 112 92 80 66

Mass number A

ENDFB-VII1 (prior)Updated (posterior)

Rela

tive u

ncer

tain

ty (σ

μ)

Figure 9 Relative uncertainties of prior and posterior independent yields in ENDFB-VII1 of fission products )ese fission products aregrouped with their correspondingmass number and the first mass numbers are labelled in this figure [30])e increased relative uncertaintyfission products are Ag130m0 Cd129m0 Sn127m1 Cd126m0 In126m0 Sb124m1 Zn123m1 Ag115m0 Y93m1 Y93m0 Se85m1 andGe77m0


With the logarithmic density function each element inμl and Zl is derived as

μ ln yi( 11138571113858 1113859 ln yi( 1113857 minusVar ln yi( 11138571113858 1113859

201113888 1113889 (17)

cov ln xi( 1113857 ln xj1113872 11138731113960 1113961 lncov yi yj1113872 1113873

μ yi1113858 1113859μ yj1113960 1113961+ 1⎡⎢⎣ ⎤⎥⎦ (18)

where cov(yi yj) and μ[yi] are retrieved from the posteriorupdated IFYs covariance matrix Z4 and updated IFYs meanvector y4With the calculated distribution parameters μl andZl the lognormal-based IFYs sampling procedures could beconducted as follows

(1) Obtain prior IFYs information including IFYs valuey0 as well as its covariance matrix Z0 from ENDFB-VII1 fission yield sublibrary Implement Bayesianupdating procedures detailed in Section 32 on theprior information and obtaining the updated IFYsmean vector y4 and the estimated covariance matrixZ4

(2) Consider IFYs follow lognormal distribution andtransform y4 and Z4 into natural logarithmic do-main with (17) and (18) )e normal distributionparameters of natural logarithmic yields are obtainedas mean vector μl and covariance Zl

(3) )e transformed logarithmic domain covariancecould not remain symmetric positive definite (SPD)due to the numerical error in the transformationprocedure A nearest-SPD searching algorithm [31]is therefore applied to search for the nearest SPDapproximation of the calculated covariance matrix inthe sense of least Frobenius norm difference )eapproximated SPD logarithmic domain covariancematrix is thus obtained as Zl

prime(4) Implement the simple random sampling procedures

in the logarithmic yield domain with distributionparameters mean μl and approximated SPD co-variance matrix Zl

prime And the generated logarithmicfission yields sample matrix PntimesS is obtained where n

denoted the number of fission products consideredand S is sample size

(5) Take the exponential transformation of each elementin sample matrix Pntimess and the sampled negative-freesamples are generated and denoted as YntimesS

)e nearest-SPD searching algorithm approximatesnon-SPD covariance matrix Zl by an approximated matrixZlprime with relative difference in Frobenius norm

(Zl minus ZlprimeFZl) 974 and their corresponding eigen-

values distributions are presented in Figure 11 )e nearest-SPD searching algorithm could approximate a non-SPDcovariance matrix while most of its eigenvalue unchanged

)e approximation that resides in the above samplingprocedures is the SPD approximation of calculated co-variance matrix )is approximation could affect consis-tency of each drawn IFYs sample with the physicalconstraints imposed on it )ere are 1000 IFYs samplesdrawn with the lognormal sampling procedures And thesample mean and standard deviation (STD) for each fissionproduct yield and Pearsonrsquos correlation coefficient betweenthese fission yields are calculated and justified by compar-ison with its corresponding population values in updated y4and Z4 (see Table 4)

Table 4 indicates that the proposed lognormal samplingprocedures could obtain an overall representation of IFYspopulation distribution considering the lower RMSEHowever there still exist a few fission products listed inFigure 12 having large biases compared with their corre-sponding population values considering the maximum ofabsolute relative difference After comparing these fissionproducts with the monitor fission products for thermalneutron induced U-235 fission listed in Fiorito et alrsquos work[15] they are not included and could be considered lessrelevant to reactor burnup and criticality calculations )eseoutliersrsquo appearance could result from the nearest-SPDprocedures and a further investigation regarding this will beconducted in future work Figure 13 presents the sampledPearsonrsquos correlation coefficients relative difference to theircorresponding population values It could be seen thatsimple random sampling procedure is not an efficientsampler for sampling low correlation fission yields (|ρ|lt 01)

as shown in the neighbour around 000 in this figureHowever these low correlations could have little impact onthe uncertainty quantification of fission yields comparedwith large correlations As for the larger correlations(|ρ|gt 025) 1000 samples are sufficient for maintaining theBayesian updated correlations and this discrepancy could befurther reduced when increasing the sample size A moreefficient sampler like Latin Hypercubic Sampler (LHS)could be adopted to guarantee more precise results whenusing 1000 samples and this will be adopted in future work

)e consistency of IFYs samples with these imposedphysical constraints is justified in Table 5 )e conservationparameters (eg total fission yields total mass number andtotal charge number) are calculated for each yield samplesand the mean and standard deviation are summarized tocompare with the target conservation value It is found thatalthough the consistency is not strictly restored as the

172 152 132 112 92 80 66Mass number A

10

08

06

04

02

00

Poste

rior t

o pr

ior fi

ssio

n yi

elds

stan

dard

dev

iatio

n ra

tio

Figure 10 Posterior to prior fission yields standard deviationratios )ese fission products are grouped with their correspondingmass number and the first mass numbers are labelled in this figure


standard deviation of the total yield is larger than the im-posed 10minus 5 their mean values are close enough to the targetvalue indicating the constraints are maintained )e largestandard deviation is originated from the approximationmentioned above

In order to examine the difference between normal-based sampling and lognormal-based sampling 1000samples are drawn from the IFYs distribution of Zr95m0Mo95m0 and Cs134m0 Notation m0 indicates these fis-sion products are at ground state )e IFYs of these threefission products are explicitly involved in VSOP burnupcalculation and are important for reactor decay heat releasecalculations Especially for Cs134m0 it is one of the maindecay heat contributors of UOX fuels in long-term afterreactor shutdown [32] )e updated relative uncertainty ofZr95m0 IFY is 161 while Cs134m0 and Mo95m0 havetheir relative uncertainties of 384 and 657 respec-tively From the sampled histogram of these fissionproducts IFYs samples (see Figures 14ndash16) lognormal-based sampling procedures (blue bars) could effectivelycapture the skewness of these fission yields and permitldquonegative-freerdquo samples It is also observed that theskewness of these fission products would become larger as

their relative uncertainties become larger (eg Mo95m0and Cs134m0)

4 Results and Discussion

41 Uncertainty Analysis of the EffectiveMultiplication Factorat Equilibrium Core )e unperturbed burnup calculationis conducted with VSOP built-in fission yields libraryand ENDFB-VII1 posterior fission yields Figure 17 showsthat reactor achieved the equilibrium state after operatinglonger than 2500 days Effective multiplication factorcalculated from ENDFB-VII1 posterior fission yields iscompared with that calculated from VSOP built-in fis-sion yields and the total discrepancy at equilibrium corestate (which is at the end point of fuel cycle time in Fig-ure 17) is lower than 50 pcm which is small enough for thefollowing fission product yields uncertainty propagationanalysis )e comparison between ENDFB-VII1 posteriorfission yields predicated keff (black dashed line) and built-in yields predicted keff (orange dashed line) are shown inFigure 18(b) )is discrepancy is within the samplingdistribution of keff

1000 fission yields samples are generated with normal-based sampling procedures and lognormal-based samplingprocedures and they are propagated to VSOP burnupcalculation to obtain keff samples under equilibrium corestate (3049 days) )e sample distributions from these twosampling procedures are drawn and compared (see Fig-ure 18) It is obvious from the comparison that normal-based samples contain fewer distribution informationcompared with lognormal samples as its distribution range issmaller than that in lognormal samples )is is due to thezero cut-off procedure of the uncontrolled negative samplesSuch procedure artificially omits certain information in theoriginal fission yields distributions and could not provide a

003

002

001

000

Log

(eig

enva

lue)

0 200 400 600 800 1000Eigenvalue index

0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()

Original log-domain relative covariance matrixSearched log-domain relative covariance matrix

Figure 11 Eigenvalue distribution of relative covariance matrix Blue dots show the eigenvalue distribution of transformed logarithmicrelative covariance Red dots show the eigenvalue distribution of approximated relative covariance matrix searched by nearest-SPD al-gorithm Grey lines show the relative difference of these eigenvalues

Table 4 Independent yield samples justification

Relativedifference Mean RMSE Max of absolute

Sample mean 8244 times 10minus4 1604 times 10minus2 1456 times 10minus1

Sample STD 6492 times 10minus3 6016 times 10minus2 8926 times 10minus1

Sample ρ minus9494 times 10minus4 1212 times 10minus4 7933 times 100

Comment

STD sample standard deviation ρ Pearsonrsquoscorrelation coefficient

RMSE root mean square errorMax of absolute the maximum absolute value

of relative difference


correspondingly reasonable and satisfied sampling distri-bution of keff In this sense lognormal sampling proceduresovercome this problem by imposing a more plausible dis-tribution on fission yields and allow the generation of

smaller perturbed samples )erefore it leads to a negativeskewness (long tail in left) of effective multiplication factordistribution and permits a more rational and persuasivesampling distribution

)e uncertainty analysis results are presented (see Ta-ble 6) )e propagated sampled distribution of keff fromnormal-based sampling method passes the normality testwith p value 03737 and the quantified relative uncertainty isaround 109 times 10minus 4 Lognormal samples provide a skewedkeff distribution and fails the normality test with p valuesmaller than 005 )e quantified relative uncertainty from

015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD

Figure 12 Relative difference of lognormal-based sampled independent fission yields mean values (blue dots) and STD (orange dots) toBayesian updated values )e outlier fission products are (mean values) Ag130m0 Sn127m1 Cd126m0 In126m0 In118m1 Br86m1 andGe75m0 and (STD values) Sb131m0 Ag130m0 Cd126m0 Sn122m0 and Br86m1

100

075

025

000

050

ndash025

ndash050

ndash075

ndash100075025000 050ndash025ndash050ndash075ndash100

Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent

Independent fission yields Personrsquos correlation coefficient

Sample size 1000Sample size 10000

Figure 13 Relative difference of lognormal-based sampled independent fission yields Pearsonrsquos correlation coefficients to Bayesian updatedvalues against Bayesian updated correlation coefficients Blue dots represent the values obtained from 1000 samples and orange dotsrepresent the values obtained from 10000 samples

Table 5 Independent yield physical constraints consistency

Constraint Target Mean STDBinary fission 200000 200062 41960 times 10minus3

Mass number 23357915 23364866 46685 times 10minus1

Charge number 9205318 9207647 18432 times 10minus1


this distribution is 258 times 10minus4 )e keff quantified fromlognormal-based sampling method is larger than that fromnormal-based samplingmethod and this shows that the zerocut-off effect in normal-based sampling method could causeunderestimation of fission product yields uncertainty con-tribution to QoIs

42 Uncertainty Analysis of Certain Fission Products AtomicDensities In this section fission products Zr95m0Mo95mo and Cs134m0 atomic densities uncertaintiescontributed from fission products yields are quantifiedSpecifically their uncertainties differences from differentsampling methods are compared and discussed From the

00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts

μ = 1282946 times 10ndash3

σ = 2132231 times 10ndash4

Lognormal-based sampledindependent fission yields of Zr95m0

(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts



Normal-based sampledindependent fission yields of Zr95m0

(b)

Figure 14 Histogram comparison of lognormal sampled (a) and normal sampled (b) independent fission yields of Zr95m0 )is fissionproduct has relative uncertainty of 161)e text presents the sampledmean value and STD and the populationmean and STD for Zr95m0are 1271856 times 10minus3 and 2029263 times 10minus4

200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts

Lognormal-based sampledindependent fission yields of Cs134m0



(a)

175

150

125

100

75

50

25

0

Cou

nts

00 02 04 06 08 10Normal-based sampled

independent fission yields of Cs134m0



1e ndash 7

(b)

Figure 15 Histogram comparison of lognormal sampled (a) and normal sampled (b) independent fission yields of Cs134m0 )is fissionproduct has relative uncertainty of 384 )e text presents the sampled mean value and STD and the population mean and STD forCs134m0 are 38544056 times 10minus8 and 1456322 times 10minus8


discussions in Section 32 Zr95m0 Cs134m0 and Mo95m0have their IFYs relative uncertainties of 161 384 and657 respectively And their IFYs sampling results shownin Figures 14ndash16 (blue bars) indicate that the skewness of thesampled distribution becomes significant with their fissionyields relative uncertainties increasing When we adoptnormal-based sampling procedures to a skewed distributionthere will be more negative samples values and the zero cut-off would deliver more underestimation into the uncertaintyanalysis results

Figures 19ndash21 track the atomic density of Zr95m0Mo95m0 and Cs134m0 in the loaded 98 kg fresh fuel (with

enrichment 85 wt) in HTR-PM along with their irradi-ation )e horizontal axis indicates the average burnupvalues of these fuels As HTR-PM allows recirculation offuels 15 times recirculation is adopted in this analysis whichindicates these fresh fuels will be reloaded into the core 15times before they are finally discharged )e dischargedburnup value or the end point of the horizontal axis is9021044 MWmiddotdtU )roughout the burnup process thethermal power of reactor core is kept at 250MW

)e atomic densities of Zr-95m0 fluctuate along with theincreases of fuels burnup value )is fluctuation is due to thefuel recirculation procedures adopted in VSOP burnup

Cou

nts

00ndash05 05 10 15 20Lognormal-based sampled

independent fission yields of Mo95m0

300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts

00ndash05 05 10 15 20Normal-based sampled




1e ndash 1

(b)

Figure 16 Histogram comparison of lognormal sampled (a) and normal sampled (b) independent fission yields samples of Mo95m0 )isfission product has relative uncertainty of 657)e text presents the sampled mean value and STD and the population mean and STD forMo95m0 are 4939650 times 10minus12 and 3161380 times 10minus12

11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor

0 500 1000 1500 2000 2500 3000Fuel cycle time (days)

100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)

VSOP built-in yieldsENDFB-VIII yieldsDifference

Figure 17 Multiplication factor predictions with operation time Operation time involves a running-phase (0ndash2500 days) and equilibriumcore state (3049 days) )e multiplication factor prediction differences of VSOP built-in yields and ENDFB-VII1 posterior yields areplotted


calculations)ere are total 14 lower valleys that appeared indashed line of Figure 19 which corresponds to the 14 timesreloading of the fuels from the bottom of the core to the topFor each reloading the fuels will be irradiated again duringtheir passes through the core As it could be seen from

Figures 19ndash21 except for the atomic densitiesrsquo decrease ofZr95m0 along with the increase of average burnup valuesMo95m0 and Cs134m0 have their atomic densities accu-mulated throughout the whole burnup process During theburnup process their atomic densities relative uncertainties

250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts

Sampled multiplication factor

(a)

250

200

150

100

50

0

Cou

nts

1 10005 1001 10015 1002Sampled multiplication factor

Posterior Built-in

(b)

Figure 18 Histogram of 1000 multiplication factor samples obtained at equilibrium core state (a) Samples obtained by normal-basedsampling procedures (b) Samples obtained by lognormal-based samples procedures Red line indicates the superimposed fitted normaldensity from the sampled data

Table 6 Uncertainty analysis results and comparison of multiplication factor at equilibrium core state

Sampling procedures

Nominal predictionVSOP with ENDF

B-VII1posterior fission

yields

Fission yield uncertainty analysis results

Samplesmean

Relativeuncertainty 95 CI Normality test(2)

Normal 100106 100105 109Eminus 04 [105Eminus 04 114Eminus 04] Passedp 3737E minus 01

Lognormal 100106 100105 244Eminus 04 [244Eminus 04 276Eminus 04](1) Failed p 1103E minus 04

Comment (1) 95 confidence interval is estimated by bootstrapping method with 100000 bootstrap samples(2) Normality test is conducted on the z-scores of multiplication factor samples with the K-S test


contributed from thermal neutron induced U-235 fissionproducts yields are investigated

)e atomic density relative uncertainties of the abovethree fission products varying with the average burnupvalues of fuels are plotted in Figures 19ndash21 (blue and orange

00055

00050

00045

00040

00035

00030

00025

0 20000 40000 60000 80000Average burnup value (MWmiddotdtu)

Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)

Zr95m0 lognormal-based relative uncertaintyZr95m0 normal-based relative uncertaintyZr95m0 atomic density

times10ndash7

20

25

30

15

10

05

00

Figure 19 Relative uncertainties of Zr95m0 against fuels average burnup values Lognormal-based sampling results (blue line) and normal-based sampling results (orange line) are plotted )e shades in this figure are the 95 confidence interval of relative uncertainties

00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

Mo95m0 lognormal-based relative uncertaintyMo95m0 normal-based relative uncertaintyMo95m0 atomic density

times10ndash6

Figure 20 Relative uncertainties of Mo95m0 against fuels average burnup values Lognormal-based sampling results (blue line) andnormal-based sampling results (orange line) are plotted )e shades in this figure are the 95 confidence interval of relative uncertainties


solid line) )e blue and orange shadings in these figures arethe 95 confidence interval of relative uncertainty com-puted by bootstrap method From these figures it is worth tomention that lognormal-based sampling quantified atomicdensity relative uncertainties are larger than that quantifiedfrom normal-based sampling for all of these three fissionproducts )is is reasonable as zero cut-off adopted innormal-based sampling method would artificially omit someinformation provided by fission yields distributions and thiswould result in an underestimated atomic density relativeuncertainty quantification result After closely comparingthe atomic density relative uncertainties underestimation forZr95m0 and Cs34m0 it could be seen that this underesti-mation effect will be enlarged when the fission products IFYshave larger relative uncertainties (Zr95m0 161 andCs134m0 384) )is is because lognormal distributionwould resemble normal distribution when the randomvariate has smaller relative uncertainty as discussed inSection 32 And in this case lognormal-based samplingresults would be in agreement with those calculated from

normal-based sampling )erefore this underestimationwould be narrowed

Besides another interesting phenomenon is observedhere )is underestimation seems not positively correlatedwith the relative uncertainty of random variates as it is seenfrom comparison between Mo95m0 and Cs134m0 Al-though Mo95m0 has its relative uncertainty (657) largerthan Cs134m0 (384) the underestimation effect observedfrom Figures 20 and 21 shows that the underestimation effectof Mo95m0 is smaller than that of Cs134m0 One possiblereason could be the decay of these fission products AsMo95m0 is the direct descendant of Zr95m0 whose half-lifeis around 64 days its atomic density relative uncertainty iscontributed both from its own fission yields uncertainty andthe atomic density uncertainty of Zr95m0 As Zr95m0atomic density uncertainty is less underestimated theatomic density relative uncertainty underestimation inMo95m0 is therefore counterbalanced While Cs134m0 istreated as stable fission products in VSOP burnup fissionproduct chains its atomic density relative uncertainty isdirectly related to its fission yields uncertainty and large

20000 40000 60000 80000Average burnup value (MWmiddotdtu)

0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00

Cs134m0 lognormal-based relative uncertaintyCs134m0 normal-based relative uncertaintyCs134m0 atomic density

times10ndash7

Figure 21 Relative uncertainties of Cs134m0 against fuels average burnup values Lognormal-based sampling results (blue line) andnormal-based sampling results (orange line) are plotted )e shades in this figure are the 95 confidence interval of relative uncertainties

Table 7 Uncertainty analysis results of Zr95m0 Mo95m0 and Cs134m0

Fissionproducts

Atomic density(atomsbarnmiddotcm)

Fission yield uncertainty analysis resultsNormal-based relative

uncertainty ()Lognormal-based relative

uncertainty ()Relative difference to lognormal-based

relative uncertainty ()Zr95m0 13532 times 10minus7 0242 0244 minus08Mo95m0 41478 times 10minus6 0456 0458 minus04Cs134m0 23575 times 10minus7 0244 0313 minus220Comment )ese results show atomic densities in 98 kg heavy metal irradiated up to 9021044MW(dtU)


atomic density relative uncertainty underestimation couldbe seen )e atomic density relative uncertainties of all thesethree fission products quantified at 9021044 MWmiddotdtU aresummarized in Table 7

5 Conclusions

)e present work proposed a stochastic UQ method forpropagation fission products yields uncertainties VSOPcode [26] is used to conduct the burnup calculation of HTR-PM reactor core with allowing 15 times recirculation of fuelpebbles [30] Uncertainties of thermal neutron inducedU-235 IFYs are investigated in this work based on ENDFB-VII1 Bayesian updating method is applied to estimate thecovariance matrix of IFYs Lognormal-based samplingmethod is implemented to generate perturbations of yieldssamples )e differences of quantified uncertainties betweenconventional normal-based sampling method and lognor-mal-based method are addressed and investigated Specifi-cally the effect of zero cut-off procedures used in normal-based sampling method is studied and discussed From theabove investigation conclusions are summarized as follows

(1) Lognormal-based sampling method could effectivelyovercome the negative samples generation caused bythe large relative uncertainties in fission yields dataCompared with normal-based sampling method itcould provide reasonable and negative-free fissionyields samples to permit a more plausible and rea-sonable QoI sampling distribution for further un-certainty analysis

(2) )e contribution of thermal neutron induced U-235fission yields uncertainties in ENDFB-VII1 to keff ofpebble-bed HTGR at equilibrium core is 00258)is contribution is smaller than that from neutroncross section 048 at equilibrium core [33]

(3) )e zero cut-off procedures used in conventionalnormal-based sampling method to overcome thenegative fission yields samples appearance would un-derestimate the uncertainty analysis results For relativeuncertainty of effective multiplication factor it wouldunderestimate the results by 00149 which is around42 of results obtained from lognormal-based sam-pling method For atomic density relative uncertaintythe underestimations are also observed and especiallyfor Cs134m0 this zero cut-off effect would underes-timate the atomic density relative uncertainty by 22compared with lognormal-based quantified results

It is worth to mention that there are several approxi-mations and simplifications made during the Bayesianupdating process and implementing of lognormal-basedsampling methods in this work )e considered constraintsfor Bayesian updating independent yields covariance matrixare preliminary in this work and a more complete andcomprehensive study regarding this will be conducted infuture work Also the effect of using nearest SPD algorithmin implementing lognormal-based sampling method will beinvestigated in the future For the following work additional

fission systems will be investigated with the proposed un-certainty propagation scheme And a sensitivity analysis ofeffective multiplication factor to fission yields should beconducted to determine the reason behind the formation ofeffective multiplication factor skewed distribution

Nomenclature

IFYs or y(A Z I) Independent fission yieldsCFYs or c(A Z I) Cumulative fission yieldsA Nuclide mass numberZ Nuclide charge numberI Nuclide isomeric stateb(Aprime Zprime Iprime ⟶ A Z I) Branching ratiokeff Effective multiplication factory4 Bayesian updated IFYs mean

vectorZ4 Bayesian updated IFYs covariance

matrixμl Natural logarithmic value of IFYs

mean vectorZl Natural logarithmic value of IFYs

covariance matrixZlprime Nearest-SPD approximated Zl

SPD Symmetric positive definiteμ Meanσ Standard deviationρ Pearsonrsquos correlation coefficientRntimes1 n-dimension real vectorRntimesn n-dimension real matrix

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the Chinese National NaturalScience Foundation Project nos 11505102 and 11375099Chinese National SampT Major Project 2018ZX06902013 andIAEA CRP I31020

References

[1] H Zhang J Guo J Lu J Niu F Li and Y Xu ldquo)ecomparison between nonlinear and linear preconditioningJFNK method for transient neutronicsthermal-hydraulicscoupling problemrdquo Annals of Nuclear Energy vol 132pp 357ndash368 2019

[2] International Atomic Energy Agency (IAEA) HTGR ReactorPhysics 9ermal-Hydraulics and Depletion UncertaintyAnalysis International Atomic Energy Agency (IAEA)Vienna Austria 2020 httpswwwiaeaorgprojectscrpi31020


[3] B Tyobeka F Resitsma and K Ivanov ldquoHTGR reactorphysics thermal-hydraulics and depletion uncertanty analy-sis a proposed IAEA coordinated research projectrdquo in Pro-ceedings of the International Conference on Mathematics andComputational Methods Applied to Nuclear Science and En-gineering (MampC 2011) Rio de Janeiro Brazil 2011

[4] K Ivanov C Parisi and O Cabellos ldquoUncertainty analysis inreactor physics modelingrdquo Science and Technology of NuclearInstallations vol 2013 Article ID 697057 2 pages 2013

[5] L Wang J Guo and Li Fu ldquoDifference of graphite capturecross sections in ENDFB librariesrdquo in Proceedings of theInternational Conference on Nuclear Engineering (ICONE23)Chiba Japan 2015

[6] F Bostelmann G Strydom F Reitsma and K Ivanov ldquo)eIAEA coordinated research programme on HTGR uncer-tainty analysis phase I status and Ex I-1 prismatic referenceresultsrdquo Nuclear Engineering and Design vol 306 pp 77ndash882016

[7] F Bostelmann and G Strydom ldquoNuclear data uncertainty andsensitivity analysis of the VHTRC benchmark using SCALErdquoAnnals of Nuclear Energy vol 110 pp 317ndash329 2017

[8] P Rouxelin G Strydom A Alfonsi and K Ivanov ldquo)eIAEA CRP on HTGR uncertainties sensitivity study ofPHISICSRELAP5-3D MHTGR-350 core calculations usingvarious SCALENEWT cross-section sets for Ex II-1ardquo Nu-clear Engineering and Design vol 329 pp 156ndash166 2018

[9] C Hao Y Cheng and Q Teng ldquoQuantification and mech-anism analysis of the kinf uncertainty propagated from nu-clear data for the TRISO particle fuel pebblerdquo Annals ofNuclear Energy vol 127 pp 248ndash256 2019

[10] F H Frohner ldquoAssigning uncertainties to scientific datardquoNuclear Science and Engineering vol 126 no 1 pp 1ndash18 1997

[11] M F James RWMills and D RWeaver ldquoA new evaluationof fission product yields and the production of a new library(UKFY2) of independent and cumulative yieldsrdquo Progress inNuclear Energy vol 26 no 1 pp 1ndash29 1991

[12] X Wu T Kozlowski H Meidani and K Shirvan ldquoInverseuncertainty quantification using the modular Bayesian ap-proach based on Gaussian process part 1 theoryrdquo NuclearEngineering and Design vol 335 no 15 pp 339ndash355 2018

[13] T Kawano and M B Chadwick ldquoEstimation of 239Pu in-dependent and cumulative fission product yields from thechain yield data using a Bayesian techniquerdquo Journal ofNuclear Science and Technology vol 50 no 10 pp 1034ndash10422013

[14] M T Pigni M W Francis and I C Gauld ldquoInvestigation ofinconsistent ENDFB-VII1 independent and cumulativefission product yields with proposed revisionsrdquo Nuclear DataSheets vol 123 pp 231ndash236 2015

[15] L Fiorito C J Diez O Cabellos A StankovskiyG Van den Eynde and P E Labeau ldquoFission yield covariancegeneration and uncertainty propagation through fission pulsedecay heat calculationrdquo Annals of Nuclear Energy vol 69pp 331ndash343 2014

[16] L Fiorito A Stankovskiy G Van den Eynde C J DiezO Cabellos and P E Labeau ldquoGeneration of fission yieldcovariances to correct discrepancies in the nuclear data li-brariesrdquo Annals of Nuclear Energy vol 88 pp 12ndash23 2016

[17] J C Helton J D Johnson C J Sallaberry and C B StorlieldquoSurvey of sampling-based methods for uncertainty andsensitivity analysisrdquo Reliability Engineering amp System Safetyvol 91 no 10-11 pp 1175ndash1209 2006

[18] D L Smith D G Naberejnev and L A VanWormer ldquoLargeerrors and sever conditionsrdquo Nuclear Instruments and

Methods in Physics Research A vol 488 no 1-2 pp 342ndash3612002

[19] G Zerovnik A Trkov and I A Kodeli ldquoCorrelated randomsampling for multivariate normal and log-normal distribu-tionsrdquo Nuclear Instruments and Methods in Physics ResearchSection A Accelerators Spectrometers Detectors and Associ-ated Equipment vol 690 pp 75ndash78 2012

[20] G Zerovnik A Trkov D L Smith and R Capote ldquoTrans-formation of correlation coefficients between normal andlognormal distribution and implications for nuclear appli-cationsrdquo Nuclear Instruments and Methods in Physics Re-search Section A Accelerators Spectrometers Detectors andAssociated Equipment vol 727 pp 33ndash39 2013

[21] T K Shin Okumura P Jaffke P Talou T Yoshida and S ChibaldquoFission product yield calculations by the Hauser-Feshbach sta-tistical decay and beta decayrdquo 2019 httpsindicocernchevent675816contributions2905172attachments16754732689797Okumurapdf

[22] M B Chadwick M Herman P Oblozinsky et al ldquoENDFB-VII1 nuclear data for science and technology cross sectionscovariances fission product yields and decay datardquo NuclearData Sheets vol 112 no 12 pp 2887ndash2996 2011

[23] T R England and B F Rider Evaluation and Compilation ofFission Product Yields Los Alamos National Laboratory LosAlamos NM USA 1994

[24] F Resitsma Gerhard Strydom B Tyobeka and K Ivanovldquo)e IAEA coordinated research program on HTGR reactorphysics thermal-hydraulics and depletion uncertainty anal-ysis description of the benchmark test cases and phasesrdquo inProceedings of the HTR 2012 pp 1ndash16 Tokyo Japan 2012

[25] G Brahler M Hartung J Fachinger K-H Grosse andR Seemann ldquoImprovements in the fabrication of HTR fuelelementsrdquo Nuclear Engineering and Design vol 251pp 239ndash243 2012

[26] H J Rutten K A Haas H Brockmann and W SchererldquoVSOP (9905) computer code system for reactor physics andfuel cycle simulationrdquo Forschungszentrum Julich GmbH ISRvol 4189 2005

[27] QWang D She B Xia and L Shi ldquoEvaluation of pebble-bedhomogenized cross sections in HTGR fuel cycle simulationsrdquoProgress in Nuclear Energy vol 117 Article ID 103041 2019

[28] Z Zhang Y Dong F Li et al ldquo)e Shandong shidao bay 200MW e high-temperature gas-cooled reactor pebble-bedmodule (HTR-PM) demonstration power plant an engi-neering and technological innovationrdquo Engineering vol 2no 1 pp 112ndash118 2016

[29] L Donald ldquoSmith probability statistics and data uncer-tainties in nuclear science and technologyrdquo 1991

[30] Y Wang M Cui J Guo and Li Fu ldquoFission yield uncertaintypropagation in multi-pass refueling pebble-bed HTGRrdquo inProceedings of the PHYSOR 2020 Transition to a ScalableNuclear Future Cambridge UK March 2020

[31] N J Higham ldquoComputing a nearest symmetric positivesemidefinite matrixrdquo Linear Algebra and its Applicationsvol 103 pp 103ndash118 1988

[32] Y Bilodid E Fridman D Kotlyar and E ShwagerausldquoExplicit decay heat calculation in the nodal diffusion codeDYN3Drdquo Annals of Nuclear Energy vol 121 pp 374ndash3812018

[33] L Wang ldquoNuclear data uncertainty and sensitivity analysis inpebble-bed HTRrdquo Institute of Nuclear and New EnergyTechnology Tsinghua University Beijing China Doctor ofphilosophy 2016


nuclear data library Also self-consistent fission yieldsdata set should follow several physical constraints such asbinary fission mass conservation and charge conserva-tion [11] )ese constraints could introduce covariancesbetween fission yields data As the fission yieldsrsquo covari-ances in current releases of evaluated nuclear data librariesare still absent eg ENDFB-VII1 many methods aredeveloped to estimate these covariances informationbased on the imposed physical constraints )e Bayesianupdating method is widely used in data assimilation dataadjustment and model fitting problems It refines pa-rameters by taking both the prior information about thoseparameters and the likelihood which refers to new datainto consideration [10 12] It allows estimating the co-variance matrix of fission yields by sequentially intro-ducing the above physical constraints In the domain offission yields adjustment it is introduced by Kawano andChadwick [13] to update Pu-239 fission yields with chainyields to reduce the independent fission yields discrepancyin ENDFB-VII1 Pigni et al [14] expand it to involvecumulative fission yields into covariances estimation )edifference between chain yields-based updating and cu-mulative fission yields-based updating is further investi-gated by Fiorito et al [15 16] Based on the providedindependent and cumulative fission yields uncertaintiesinformation in ENDFB-VII1 this work adopts theBayesian updating method to estimate the independentfission yields covariances

Sampling-based methods for uncertainty analysis orstochastic UQ methods [17] require properly perturbedsamples to provide reliable uncertainty analysis results ofQoI As it is observed from the evaluated nuclear data li-brary independent fission yields generally have larger un-certainties Random sampling on these yields under normaldistribution could generate nonphysical negative samplesCutting off these negative yield samples by setting them tozero could artificially omit some information from theoriginal fission yields distribution resulting in biased un-certainty analysis results )is zero cut-off effect on quan-tified uncertainty has not been well studied )is raisesquestion whether normal distribution is sufficient for de-scribing inherently positive random variables with largeuncertainties only given their mean values and covariancematrix Smith et al [18] propose to replace normal distri-bution with lognormal distribution by the principle ofmaximum entropy [10] and Zerovnik et al [19 20] in-vestigate this method in the sampling resonance parameterswhere negative samples problem was encountered as infission yields )is work proposes an implementation of thelognormal-based sampling method in fission product yieldssampling

)e present work is organized as follows Section 2describes the nomenclature of fission product yields and theburnup calculation of pebble-bed HTGR An implementedstochastic UQ method for fission yields uncertainty prop-agation is described in Section 23 )e Bayesian updatingmethod and the lognormal-based sampling method aredetailed in Section 3 Finally results of fission yields un-certainty contributions to effective multiplication factor and

several important fission products atomic densities areprovided and discussed in Section 4

2 Model Description andUncertainty Propagation

21 ENDFB-VII1 Fission Product Yields SublibraryFission product yield characterizes the fraction of a par-ticular fission product nuclide produced per fission Acompound nucleus is formed when a fissile nucleus isbombarded by an incident neutron As its energy over-comes the fission barrier this compound nucleus couldundergo fission A brief description of the fission process isillustrated (see Figure 1) After the scission of compoundnucleus primary fission fragments are produced and theywould undergo deexcitation by releasing prompt neutronsdue to their high neutron to proton ratios After theemission of prompt neutrons the remaining fission frag-ments referred to as fission products would undergo βdecay isomeric transition or particle emission along theircorresponding decay chain and finally reach stable nu-clides Each fission product is identified by its mass numberA charge number Z and isomeric state I and is denoted asthe triplet (A Z I)

A detailed description about the nomenclature offission product yields could be found in [11] and they arebriefly summarized as follows IFYs and CFYs determinethe fraction of a fission product at different stages in thefission process IFYs denoted as y(A Z I) are the fractionof a fission product produced directly from one fissionafter the emission of prompt neutrons but prior to anyradioactive decays Because IFYs are produced before anyradioactive decay in the fission system they should besubject to the physical constraints of fission system egbinary fission conservation of mass and charge numberCFYs denoted as c(A Z I) determine the total fraction ofa fission product produced over all time after one fissionIt involves not only the direct production from fissionbut also the contributions from the decay of otherproducts

)e current releases of ENDFB-VII1 fission yieldsublibrary provide fission yields data for 31 fission actinidesfrom)-227 to Fm-255 )ough energy-dependence issueswithin fission spectrum are highlighted in current releasesof evaluated nuclear data library and neutron induced Pu-239 fission yields at 20MeV are supplemented to allowusers to linearly interpolate yields between 05MeV and20MeV for high accuracy purpose [22] other fission ac-tinide fission yields data are taken directly from ENDFB-VI evaluated by England and Rider [23] in 1993 )reefission systems for U-235 are evaluated with respect toincident neutron energy namely 00253 eV thermal en-ergy 05MeV fission spectrum energy and 140MeV highenergy IFYs and CFYs are evaluated for 1247 fissionproducts in thermal neutron induced U-235 fission yield(see Figure 2) )e relationship between IFYs and CFYs[11] is referred to as (1) where b(Aprime Zprime Iprime ⟶ A Z I) isthe branching ratio


c(A Z I) y(A Z I) + 1113944



(1)



Prescission


Deexcitation


neutron

Neutron

Fissilenucleus

Compoundnucleus


fragments

Independent yields

Cumulative yields

Delayedneutron


nuclide


β

N (A + 1 Z)

NP (AP2 ZP2)

NP (AP1 ZP1)

NI (AI2 ZI2)

NI (AI1 ZI1)

NC (AC2 ZC2)

NC (AC1 ZC1)

N (A Z)

γ








110

100

90

80

70

60

50

40

30

Charge number Z

Neu

tron

num

ber N

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

20 30 40 50 60 70

(a)

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

110

100

90

80

70

60

50

40

30

Charge number Z

Neu

tron

num

ber N

20 30 40 50 60 70

(b)










c Xy + ε (2)


Control rod drive


Control rod

Pebble bedfuel core

Bottomreflector

Defuellingtube

Reactoroutlet

Core support

Core barrel

Side reflectir


Top plate

SAS container

(a)

5mm graphite layer


Dia 60mm

Dia 092mm


Fuel kernal


Section


(b)







VSOP ENDFB-VII1



B-VII1 (pcm)






B-VII1 (pcm)





101027 100960 67


(i)(ii)

(iii)



END

FB-

VII

1

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures







Mapping

Self-

cons

isten

tyi

eld

sam

ples


Input

Upd

ated

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures

Zero-cutoff



(i)(ii)



(i)(ii)



(i)(ii)


(i)(ii)




miny

χ2 (Xy minus c)TV


TZ

minus10 y minus y0( 11138571113960 1113961

(3)


p(y) 1

(2π)(n2)

Z01113868111386811138681113868

1113868111386811138681113868(12)


1113874 1113875 y minus y0( 1113857TZ

minus10 y minus y0( 11138571113882 1113883

(4)


p(c | y) 1

(2π)(n2)

|V|(12)


TV

minus1(Xy minus c)1113882 1113883 (5)


p(y | c) p(c | y)




ypost y + Z1XTV

minus1c minus Xy01113858 1113859 (7)


Z1 Zminus10 + X

TV

minus1X1113872 1113873

minus1 (8)

Z1 Z0 minus Z0XT

V + XZ0XT

1113872 1113873minus1

XZ0 (9)




Z1 Z0 minus Z0MT

V + MZ0MT

1113872 1113873minus1

MZ0 (10)



minus1U

TZ1 (11)



minus1N

TZ2 (12)





minus1W

TZ3 (13)









L ln(y) sim N μl Zl( 1113857 (14)






9008007006005004003002001000

900

800

700

600

500

400

300

200

100

0


Fiss

ion

prod

uct i

ndex

(a)

2402202001801601401201008060

240

220

200

180

160

140

120

100

80

60

Fiss

ion

prod

uct i

ndex


(b)


0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield


0 200 600400 800 1000



0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield

STD





pY(y) 1

(2π)(n2)

Zl

11138681113868111386811138681113868111386811138681113868(12)

1113945n

i1yi

exp minus12

ln(y) minus μl1113858 1113859TZ


03

025

02

015

01

005


Prob

abili

ty d

ensit

y

R = 08R = 07R = 06R = 05

R = 04R = 03R = 02R = 01


18

16

14

12

1

08

06

04

02

0172 152 132 112 92 80 66

Mass number A


Rela

tive u

ncer

tain

ty (σ

μ)





201113888 1113889 (17)


μ yi1113858 1113859μ yj1113960 1113961+ 1⎡⎢⎣ ⎤⎥⎦ (18)

















172 152 132 112 92 80 66Mass number A

10

08

06

04

02

00

Poste

rior t

o pr

ior fi

ssio

n yi

elds

stan

dard

dev

iatio

n ra

tio









003

002

001

000

Log

(eig

enva

lue)


0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()








Comment








015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References





































c(A Z I) y(A Z I) + 1113944



(1)



Prescission


Deexcitation


neutron

Neutron

Fissilenucleus

Compoundnucleus


fragments

Independent yields

Cumulative yields

Delayedneutron


nuclide


β

N (A + 1 Z)

NP (AP2 ZP2)

NP (AP1 ZP1)

NI (AI2 ZI2)

NI (AI1 ZI1)

NC (AC2 ZC2)

NC (AC1 ZC1)

N (A Z)

γ








110

100

90

80

70

60

50

40

30

Charge number Z

Neu

tron

num

ber N

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

20 30 40 50 60 70

(a)

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

110

100

90

80

70

60

50

40

30

Charge number Z

Neu

tron

num

ber N

20 30 40 50 60 70

(b)










c Xy + ε (2)


Control rod drive


Control rod

Pebble bedfuel core

Bottomreflector

Defuellingtube

Reactoroutlet

Core support

Core barrel

Side reflectir


Top plate

SAS container

(a)

5mm graphite layer


Dia 60mm

Dia 092mm


Fuel kernal


Section


(b)







VSOP ENDFB-VII1



B-VII1 (pcm)






B-VII1 (pcm)





101027 100960 67


(i)(ii)

(iii)



END

FB-

VII

1

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures







Mapping

Self-

cons

isten

tyi

eld

sam

ples


Input

Upd

ated

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures

Zero-cutoff



(i)(ii)



(i)(ii)



(i)(ii)


(i)(ii)




miny

χ2 (Xy minus c)TV


TZ

minus10 y minus y0( 11138571113960 1113961

(3)


p(y) 1

(2π)(n2)

Z01113868111386811138681113868

1113868111386811138681113868(12)


1113874 1113875 y minus y0( 1113857TZ

minus10 y minus y0( 11138571113882 1113883

(4)


p(c | y) 1

(2π)(n2)

|V|(12)


TV

minus1(Xy minus c)1113882 1113883 (5)


p(y | c) p(c | y)




ypost y + Z1XTV

minus1c minus Xy01113858 1113859 (7)


Z1 Zminus10 + X

TV

minus1X1113872 1113873

minus1 (8)

Z1 Z0 minus Z0XT

V + XZ0XT

1113872 1113873minus1

XZ0 (9)




Z1 Z0 minus Z0MT

V + MZ0MT

1113872 1113873minus1

MZ0 (10)



minus1U

TZ1 (11)



minus1N

TZ2 (12)





minus1W

TZ3 (13)









L ln(y) sim N μl Zl( 1113857 (14)






9008007006005004003002001000

900

800

700

600

500

400

300

200

100

0


Fiss

ion

prod

uct i

ndex

(a)

2402202001801601401201008060

240

220

200

180

160

140

120

100

80

60

Fiss

ion

prod

uct i

ndex


(b)


0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield


0 200 600400 800 1000



0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield

STD





pY(y) 1

(2π)(n2)

Zl

11138681113868111386811138681113868111386811138681113868(12)

1113945n

i1yi

exp minus12

ln(y) minus μl1113858 1113859TZ


03

025

02

015

01

005


Prob

abili

ty d

ensit

y

R = 08R = 07R = 06R = 05

R = 04R = 03R = 02R = 01


18

16

14

12

1

08

06

04

02

0172 152 132 112 92 80 66

Mass number A


Rela

tive u

ncer

tain

ty (σ

μ)





201113888 1113889 (17)


μ yi1113858 1113859μ yj1113960 1113961+ 1⎡⎢⎣ ⎤⎥⎦ (18)

















172 152 132 112 92 80 66Mass number A

10

08

06

04

02

00

Poste

rior t

o pr

ior fi

ssio

n yi

elds

stan

dard

dev

iatio

n ra

tio









003

002

001

000

Log

(eig

enva

lue)


0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()








Comment








015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References










































110

100

90

80

70

60

50

40

30

Charge number Z

Neu

tron

num

ber N

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

20 30 40 50 60 70

(a)

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

110

100

90

80

70

60

50

40

30

Charge number Z

Neu

tron

num

ber N

20 30 40 50 60 70

(b)










c Xy + ε (2)


Control rod drive


Control rod

Pebble bedfuel core

Bottomreflector

Defuellingtube

Reactoroutlet

Core support

Core barrel

Side reflectir


Top plate

SAS container

(a)

5mm graphite layer


Dia 60mm

Dia 092mm


Fuel kernal


Section


(b)







VSOP ENDFB-VII1



B-VII1 (pcm)






B-VII1 (pcm)





101027 100960 67


(i)(ii)

(iii)



END

FB-

VII

1

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures







Mapping

Self-

cons

isten

tyi

eld

sam

ples


Input

Upd

ated

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures

Zero-cutoff



(i)(ii)



(i)(ii)



(i)(ii)


(i)(ii)




miny

χ2 (Xy minus c)TV


TZ

minus10 y minus y0( 11138571113960 1113961

(3)


p(y) 1

(2π)(n2)

Z01113868111386811138681113868

1113868111386811138681113868(12)


1113874 1113875 y minus y0( 1113857TZ

minus10 y minus y0( 11138571113882 1113883

(4)


p(c | y) 1

(2π)(n2)

|V|(12)


TV

minus1(Xy minus c)1113882 1113883 (5)


p(y | c) p(c | y)




ypost y + Z1XTV

minus1c minus Xy01113858 1113859 (7)


Z1 Zminus10 + X

TV

minus1X1113872 1113873

minus1 (8)

Z1 Z0 minus Z0XT

V + XZ0XT

1113872 1113873minus1

XZ0 (9)




Z1 Z0 minus Z0MT

V + MZ0MT

1113872 1113873minus1

MZ0 (10)



minus1U

TZ1 (11)



minus1N

TZ2 (12)





minus1W

TZ3 (13)









L ln(y) sim N μl Zl( 1113857 (14)






9008007006005004003002001000

900

800

700

600

500

400

300

200

100

0


Fiss

ion

prod

uct i

ndex

(a)

2402202001801601401201008060

240

220

200

180

160

140

120

100

80

60

Fiss

ion

prod

uct i

ndex


(b)


0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield


0 200 600400 800 1000



0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield

STD





pY(y) 1

(2π)(n2)

Zl

11138681113868111386811138681113868111386811138681113868(12)

1113945n

i1yi

exp minus12

ln(y) minus μl1113858 1113859TZ


03

025

02

015

01

005


Prob

abili

ty d

ensit

y

R = 08R = 07R = 06R = 05

R = 04R = 03R = 02R = 01


18

16

14

12

1

08

06

04

02

0172 152 132 112 92 80 66

Mass number A


Rela

tive u

ncer

tain

ty (σ

μ)





201113888 1113889 (17)


μ yi1113858 1113859μ yj1113960 1113961+ 1⎡⎢⎣ ⎤⎥⎦ (18)

















172 152 132 112 92 80 66Mass number A

10

08

06

04

02

00

Poste

rior t

o pr

ior fi

ssio

n yi

elds

stan

dard

dev

iatio

n ra

tio









003

002

001

000

Log

(eig

enva

lue)


0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()








Comment








015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References












































c Xy + ε (2)


Control rod drive


Control rod

Pebble bedfuel core

Bottomreflector

Defuellingtube

Reactoroutlet

Core support

Core barrel

Side reflectir


Top plate

SAS container

(a)

5mm graphite layer


Dia 60mm

Dia 092mm


Fuel kernal


Section


(b)







VSOP ENDFB-VII1



B-VII1 (pcm)






B-VII1 (pcm)





101027 100960 67


(i)(ii)

(iii)



END

FB-

VII

1

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures







Mapping

Self-

cons

isten

tyi

eld

sam

ples


Input

Upd

ated

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures

Zero-cutoff



(i)(ii)



(i)(ii)



(i)(ii)


(i)(ii)




miny

χ2 (Xy minus c)TV


TZ

minus10 y minus y0( 11138571113960 1113961

(3)


p(y) 1

(2π)(n2)

Z01113868111386811138681113868

1113868111386811138681113868(12)


1113874 1113875 y minus y0( 1113857TZ

minus10 y minus y0( 11138571113882 1113883

(4)


p(c | y) 1

(2π)(n2)

|V|(12)


TV

minus1(Xy minus c)1113882 1113883 (5)


p(y | c) p(c | y)




ypost y + Z1XTV

minus1c minus Xy01113858 1113859 (7)


Z1 Zminus10 + X

TV

minus1X1113872 1113873

minus1 (8)

Z1 Z0 minus Z0XT

V + XZ0XT

1113872 1113873minus1

XZ0 (9)




Z1 Z0 minus Z0MT

V + MZ0MT

1113872 1113873minus1

MZ0 (10)



minus1U

TZ1 (11)



minus1N

TZ2 (12)





minus1W

TZ3 (13)









L ln(y) sim N μl Zl( 1113857 (14)






9008007006005004003002001000

900

800

700

600

500

400

300

200

100

0


Fiss

ion

prod

uct i

ndex

(a)

2402202001801601401201008060

240

220

200

180

160

140

120

100

80

60

Fiss

ion

prod

uct i

ndex


(b)


0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield


0 200 600400 800 1000



0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield

STD





pY(y) 1

(2π)(n2)

Zl

11138681113868111386811138681113868111386811138681113868(12)

1113945n

i1yi

exp minus12

ln(y) minus μl1113858 1113859TZ


03

025

02

015

01

005


Prob

abili

ty d

ensit

y

R = 08R = 07R = 06R = 05

R = 04R = 03R = 02R = 01


18

16

14

12

1

08

06

04

02

0172 152 132 112 92 80 66

Mass number A


Rela

tive u

ncer

tain

ty (σ

μ)





201113888 1113889 (17)


μ yi1113858 1113859μ yj1113960 1113961+ 1⎡⎢⎣ ⎤⎥⎦ (18)

















172 152 132 112 92 80 66Mass number A

10

08

06

04

02

00

Poste

rior t

o pr

ior fi

ssio

n yi

elds

stan

dard

dev

iatio

n ra

tio









003

002

001

000

Log

(eig

enva

lue)


0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()








Comment








015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References









































VSOP ENDFB-VII1



B-VII1 (pcm)






B-VII1 (pcm)





101027 100960 67


(i)(ii)

(iii)



END

FB-

VII

1

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures







Mapping

Self-

cons

isten

tyi

eld

sam

ples


Input

Upd

ated

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures

Zero-cutoff



(i)(ii)



(i)(ii)



(i)(ii)


(i)(ii)




miny

χ2 (Xy minus c)TV


TZ

minus10 y minus y0( 11138571113960 1113961

(3)


p(y) 1

(2π)(n2)

Z01113868111386811138681113868

1113868111386811138681113868(12)


1113874 1113875 y minus y0( 1113857TZ

minus10 y minus y0( 11138571113882 1113883

(4)


p(c | y) 1

(2π)(n2)

|V|(12)


TV

minus1(Xy minus c)1113882 1113883 (5)


p(y | c) p(c | y)




ypost y + Z1XTV

minus1c minus Xy01113858 1113859 (7)


Z1 Zminus10 + X

TV

minus1X1113872 1113873

minus1 (8)

Z1 Z0 minus Z0XT

V + XZ0XT

1113872 1113873minus1

XZ0 (9)




Z1 Z0 minus Z0MT

V + MZ0MT

1113872 1113873minus1

MZ0 (10)



minus1U

TZ1 (11)



minus1N

TZ2 (12)





minus1W

TZ3 (13)









L ln(y) sim N μl Zl( 1113857 (14)






9008007006005004003002001000

900

800

700

600

500

400

300

200

100

0


Fiss

ion

prod

uct i

ndex

(a)

2402202001801601401201008060

240

220

200

180

160

140

120

100

80

60

Fiss

ion

prod

uct i

ndex


(b)


0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield


0 200 600400 800 1000



0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield

STD





pY(y) 1

(2π)(n2)

Zl

11138681113868111386811138681113868111386811138681113868(12)

1113945n

i1yi

exp minus12

ln(y) minus μl1113858 1113859TZ


03

025

02

015

01

005


Prob

abili

ty d

ensit

y

R = 08R = 07R = 06R = 05

R = 04R = 03R = 02R = 01


18

16

14

12

1

08

06

04

02

0172 152 132 112 92 80 66

Mass number A


Rela

tive u

ncer

tain

ty (σ

μ)





201113888 1113889 (17)


μ yi1113858 1113859μ yj1113960 1113961+ 1⎡⎢⎣ ⎤⎥⎦ (18)

















172 152 132 112 92 80 66Mass number A

10

08

06

04

02

00

Poste

rior t

o pr

ior fi

ssio

n yi

elds

stan

dard

dev

iatio

n ra

tio









003

002

001

000

Log

(eig

enva

lue)


0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()








Comment








015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References







































101027 100960 67


(i)(ii)

(iii)



END

FB-

VII

1

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures







Mapping

Self-

cons

isten

tyi

eld

sam

ples


Input

Upd

ated

Logn

orm

al-b

ased

Sam

plin

g pr

oced

ures

Zero-cutoff



(i)(ii)



(i)(ii)



(i)(ii)


(i)(ii)




miny

χ2 (Xy minus c)TV


TZ

minus10 y minus y0( 11138571113960 1113961

(3)


p(y) 1

(2π)(n2)

Z01113868111386811138681113868

1113868111386811138681113868(12)


1113874 1113875 y minus y0( 1113857TZ

minus10 y minus y0( 11138571113882 1113883

(4)


p(c | y) 1

(2π)(n2)

|V|(12)


TV

minus1(Xy minus c)1113882 1113883 (5)


p(y | c) p(c | y)




ypost y + Z1XTV

minus1c minus Xy01113858 1113859 (7)


Z1 Zminus10 + X

TV

minus1X1113872 1113873

minus1 (8)

Z1 Z0 minus Z0XT

V + XZ0XT

1113872 1113873minus1

XZ0 (9)




Z1 Z0 minus Z0MT

V + MZ0MT

1113872 1113873minus1

MZ0 (10)



minus1U

TZ1 (11)



minus1N

TZ2 (12)





minus1W

TZ3 (13)









L ln(y) sim N μl Zl( 1113857 (14)






9008007006005004003002001000

900

800

700

600

500

400

300

200

100

0


Fiss

ion

prod

uct i

ndex

(a)

2402202001801601401201008060

240

220

200

180

160

140

120

100

80

60

Fiss

ion

prod

uct i

ndex


(b)


0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield


0 200 600400 800 1000



0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield

STD





pY(y) 1

(2π)(n2)

Zl

11138681113868111386811138681113868111386811138681113868(12)

1113945n

i1yi

exp minus12

ln(y) minus μl1113858 1113859TZ


03

025

02

015

01

005


Prob

abili

ty d

ensit

y

R = 08R = 07R = 06R = 05

R = 04R = 03R = 02R = 01


18

16

14

12

1

08

06

04

02

0172 152 132 112 92 80 66

Mass number A


Rela

tive u

ncer

tain

ty (σ

μ)





201113888 1113889 (17)


μ yi1113858 1113859μ yj1113960 1113961+ 1⎡⎢⎣ ⎤⎥⎦ (18)

















172 152 132 112 92 80 66Mass number A

10

08

06

04

02

00

Poste

rior t

o pr

ior fi

ssio

n yi

elds

stan

dard

dev

iatio

n ra

tio









003

002

001

000

Log

(eig

enva

lue)


0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()








Comment








015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References






































miny

χ2 (Xy minus c)TV


TZ

minus10 y minus y0( 11138571113960 1113961

(3)


p(y) 1

(2π)(n2)

Z01113868111386811138681113868

1113868111386811138681113868(12)


1113874 1113875 y minus y0( 1113857TZ

minus10 y minus y0( 11138571113882 1113883

(4)


p(c | y) 1

(2π)(n2)

|V|(12)


TV

minus1(Xy minus c)1113882 1113883 (5)


p(y | c) p(c | y)




ypost y + Z1XTV

minus1c minus Xy01113858 1113859 (7)


Z1 Zminus10 + X

TV

minus1X1113872 1113873

minus1 (8)

Z1 Z0 minus Z0XT

V + XZ0XT

1113872 1113873minus1

XZ0 (9)




Z1 Z0 minus Z0MT

V + MZ0MT

1113872 1113873minus1

MZ0 (10)



minus1U

TZ1 (11)



minus1N

TZ2 (12)





minus1W

TZ3 (13)









L ln(y) sim N μl Zl( 1113857 (14)






9008007006005004003002001000

900

800

700

600

500

400

300

200

100

0


Fiss

ion

prod

uct i

ndex

(a)

2402202001801601401201008060

240

220

200

180

160

140

120

100

80

60

Fiss

ion

prod

uct i

ndex


(b)


0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield


0 200 600400 800 1000



0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield

STD





pY(y) 1

(2π)(n2)

Zl

11138681113868111386811138681113868111386811138681113868(12)

1113945n

i1yi

exp minus12

ln(y) minus μl1113858 1113859TZ


03

025

02

015

01

005


Prob

abili

ty d

ensit

y

R = 08R = 07R = 06R = 05

R = 04R = 03R = 02R = 01


18

16

14

12

1

08

06

04

02

0172 152 132 112 92 80 66

Mass number A


Rela

tive u

ncer

tain

ty (σ

μ)





201113888 1113889 (17)


μ yi1113858 1113859μ yj1113960 1113961+ 1⎡⎢⎣ ⎤⎥⎦ (18)

















172 152 132 112 92 80 66Mass number A

10

08

06

04

02

00

Poste

rior t

o pr

ior fi

ssio

n yi

elds

stan

dard

dev

iatio

n ra

tio









003

002

001

000

Log

(eig

enva

lue)


0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()








Comment








015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References







































minus1W

TZ3 (13)









L ln(y) sim N μl Zl( 1113857 (14)






9008007006005004003002001000

900

800

700

600

500

400

300

200

100

0


Fiss

ion

prod

uct i

ndex

(a)

2402202001801601401201008060

240

220

200

180

160

140

120

100

80

60

Fiss

ion

prod

uct i

ndex


(b)


0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield


0 200 600400 800 1000



0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield

STD





pY(y) 1

(2π)(n2)

Zl

11138681113868111386811138681113868111386811138681113868(12)

1113945n

i1yi

exp minus12

ln(y) minus μl1113858 1113859TZ


03

025

02

015

01

005


Prob

abili

ty d

ensit

y

R = 08R = 07R = 06R = 05

R = 04R = 03R = 02R = 01


18

16

14

12

1

08

06

04

02

0172 152 132 112 92 80 66

Mass number A


Rela

tive u

ncer

tain

ty (σ

μ)





201113888 1113889 (17)


μ yi1113858 1113859μ yj1113960 1113961+ 1⎡⎢⎣ ⎤⎥⎦ (18)

















172 152 132 112 92 80 66Mass number A

10

08

06

04

02

00

Poste

rior t

o pr

ior fi

ssio

n yi

elds

stan

dard

dev

iatio

n ra

tio









003

002

001

000

Log

(eig

enva

lue)


0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()








Comment








015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References





































9008007006005004003002001000

900

800

700

600

500

400

300

200

100

0


Fiss

ion

prod

uct i

ndex

(a)

2402202001801601401201008060

240

220

200

180

160

140

120

100

80

60

Fiss

ion

prod

uct i

ndex


(b)


0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield


0 200 600400 800 1000



0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash30

ndash35

ndash40

ndash45

ndash50

Loga

rthm

ic v

alue

of i

ndep

ende

nt y

ield

STD





pY(y) 1

(2π)(n2)

Zl

11138681113868111386811138681113868111386811138681113868(12)

1113945n

i1yi

exp minus12

ln(y) minus μl1113858 1113859TZ


03

025

02

015

01

005


Prob

abili

ty d

ensit

y

R = 08R = 07R = 06R = 05

R = 04R = 03R = 02R = 01


18

16

14

12

1

08

06

04

02

0172 152 132 112 92 80 66

Mass number A


Rela

tive u

ncer

tain

ty (σ

μ)





201113888 1113889 (17)


μ yi1113858 1113859μ yj1113960 1113961+ 1⎡⎢⎣ ⎤⎥⎦ (18)

















172 152 132 112 92 80 66Mass number A

10

08

06

04

02

00

Poste

rior t

o pr

ior fi

ssio

n yi

elds

stan

dard

dev

iatio

n ra

tio









003

002

001

000

Log

(eig

enva

lue)


0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()








Comment








015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References





































pY(y) 1

(2π)(n2)

Zl

11138681113868111386811138681113868111386811138681113868(12)

1113945n

i1yi

exp minus12

ln(y) minus μl1113858 1113859TZ


03

025

02

015

01

005


Prob

abili

ty d

ensit

y

R = 08R = 07R = 06R = 05

R = 04R = 03R = 02R = 01


18

16

14

12

1

08

06

04

02

0172 152 132 112 92 80 66

Mass number A


Rela

tive u

ncer

tain

ty (σ

μ)





201113888 1113889 (17)


μ yi1113858 1113859μ yj1113960 1113961+ 1⎡⎢⎣ ⎤⎥⎦ (18)

















172 152 132 112 92 80 66Mass number A

10

08

06

04

02

00

Poste

rior t

o pr

ior fi

ssio

n yi

elds

stan

dard

dev

iatio

n ra

tio









003

002

001

000

Log

(eig

enva

lue)


0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()








Comment








015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References







































201113888 1113889 (17)


μ yi1113858 1113859μ yj1113960 1113961+ 1⎡⎢⎣ ⎤⎥⎦ (18)

















172 152 132 112 92 80 66Mass number A

10

08

06

04

02

00

Poste

rior t

o pr

ior fi

ssio

n yi

elds

stan

dard

dev

iatio

n ra

tio









003

002

001

000

Log

(eig

enva

lue)


0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()








Comment








015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References











































003

002

001

000

Log

(eig

enva

lue)


0

ndash20

ndash40

ndash60

ndash80

ndash100

Rela

tive d

iffer

ence

()








Comment








015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References








































015

010

005

000

ndash005

ndash010

ndash015

Relat

ive d

iffer

ence

of s

ampl

ed m

ean

valu

e

66 80 92 112 132 152 172Mass number A

06

04

02

00

ndash02

ndash04

ndash06

Relat

ive d

iffer

ence

of s

ampl

ed S

TD


100

075

025

000

050

ndash025

ndash050

ndash075


Relat

ive d

iffer

ence

of l

ogno

rmal

-bas

ed sa

mpl

edPe

arso

nrsquos co

rrel

atio

n co

effici

ent











00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References







































00008 00010 00012 00014 00016 00018 00020

175

150

125

100

75

50

25

0

Cou

nts




(a)

00008 00010 00012 00014 0001600018 00020

175

150

125

100

75

50

25

0

Cou

nts




(b)


200

175

150

125

100

75

50

25

000 02 04 06 08 10

1e ndash 7

Cou

nts




(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 7

(b)







Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References









































Cou

nts



300

250

200

150

100

50

0



1e ndash 1

(a)

175

150

125

100

75

50

25

0

Cou

nts





1e ndash 1

(b)


11

1075

105

1025

1

0975

095

0925

09

Mul

tiplic

atio

n fa

ctor


100

75

50

25

0

ndash25

ndash50

ndash75

ndash100

Diff

eren

ce (p

cm)






250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References







































250

200

150

100

50

01 10005 1001 10015 1002

Cou

nts


(a)

250

200

150

100

50

0

Cou

nts


Posterior Built-in

(b)



Sampling procedures



yields


Samplesmean








00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References







































00055

00050

00045

00040

00035

00030

00025


Relat

ivre

unc

erta

inty

(ndash)

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash7

20

25

30

15

10

05

00


00056

00054

00052

00050

00048

00046

00044


Relat

ivre

unc

erta

inty

(ndash)

4

3

2

1

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)


times10ndash6







0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References









































0007

0006

0005

0004

0003

Relat

ivre

unc

erta

inty

(ndash)

0

Atom

ic d

ensit

y (a

tom

sba

rncm

)

20

15

10

05

00


times10ndash7



Fissionproducts








5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References






































5 Conclusions







Nomenclature







Data Availability




Acknowledgments


References





































Lognormal-BasedSamplingforFissionProductYields...

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Transcript of Lognormal-BasedSamplingforFissionProductYields...