determination of an effective detector position for pulsed...

Research ArticleDetermination of an Effective Detector Position forPulsed-Neutron-Source Alpha Measurement by Time-DependentMonte Carlo Neutron Transport Simulations

Sang Hoon Jang and Hyung Jin Shim

Seoul National University 1 Gwanak-ro Gwanak-gu Seoul 08826 Republic of Korea

Correspondence should be addressed to Hyung Jin Shim shimhjsnuackr

Received 31 January 2018 Accepted 15 March 2018 Published 2 May 2018

Academic Editor Eugenijus Uspuras

Copyright copy 2018 Sang Hoon Jang and Hyung Jin Shim This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A simple method using the time-dependent Monte Carlo (TDMC) neutron transport calculation is presented to determine aneffective detector position for the prompt neutron decay constant (120572) measurement through the pulsed-neutron-source (PNS)experiment In the proposed method the optimum detector position is searched by comparing amplitudes of detector signalsat different positions when their 120572 estimates by the slope fitting are converged The developed method is applied to the Pb-Bi-zoned ADS experimental benchmark at Kyoto University Critical Assembly The 120572 convergence time estimated by the TDMC PNSsimulation agrees well with the experimental results The 120572 convergence time map and the corresponding signal amplitude mappredicted by the developed method show that polyethylene moderator regions adjacent to fuel region are better positions thanother candidates for the PNS 120572measurement

1 Introduction

Since the early 1990s accelerator-driven subcritical systems(ADS) for transmutation of radioactive wastes and energyproduction have been proposed and designed throughoutthe world with their advantages of high flexibility of fuelcompositions and the enhanced safety concept [1ndash3] Theneutronic characteristics of the subcritical reactor have beenextensively studied theoretically [4 5] and experimentally [6ndash8] The prompt neutron decay constant (hereafter referred toas 120572) of a subcritical system is a fundamental kinetics param-eter which represents its asymptotic behavior ignoring thedelayed neutron effect Moreover 120572 can be directly measured[9 10] by injecting a short burst of neutrons in the systemcalled the pulsed-neutron-source (PNS) experiment SinceSimmons and King [9] applied an exponential regression toneutron detector signals from the PNS experiment this 120572measurement method has been popularly employed becauseit can provide 120572 results independent of the positioning andenergy characteristics of the detector and neutron source

[9 11 12] by reducing higher-mode contaminations on theexponential fitting [13 14]

In practice however the PNS 120572 measurement may yieldconsiderably different results at different detector positionsand neutron sources as reported in the experimental bench-marks on an ADS at Kyoto University Critical Assembly(KUCA) [15 16] This measurement dependency on thedetector position and the neutron source can be attributedmostly to the signal contamination [11 16] by the higher-mode components of the prompt neutron flux which iscaused by taking detector signals before the higher-modecomponents fully decay out It is difficult however to obtainconfident detector signals after the prompt neutron fluxconverges to the fundamental mode in a deep subcriticalsystem where the prompt neutron flux decreases rapidlyTherefore it is necessary to determine effective detectorpositions where the prompt neutron flux converges fast withlarger signal strength than other candidate positions

The objective of this paper is to devise a simple butpractical way to determine an optimum detector position

HindawiScience and Technology of Nuclear InstallationsVolume 2018 Article ID 2350458 7 pageshttpsdoiorg10115520182350458

2 Science and Technology of Nuclear Installations

for the 120572 measurement through the PNS experiment usingthe time-dependentMonte Carlo (TDMC) neutron transportanalyses [17ndash19] In the TDMC calculations the combingalgorithm [17 20] is applied to maintain the time-bin-wiseneutron population because an exponential decrease of theneutron population in an analog TDMC calculation of asubcritical system causes large statistical uncertainties In theproposedmethod the optimum detector position is searchedby comparing the strength of detector signal at each spatialposition when the 120572 estimate at the position is convergedThe position-dependent 120572 convergence is diagnosed by aslope fitting to the detector signals obtained from the TDMCcalculations The proposed methods are implemented in aSeoul National University continuous-energy Monte Carlo(MC) code McCARD [21] and applied to the Pb-Bi-zonedADS experimental benchmark at KUCA [22]

2 Determination of an Optimum DetectorPosition through the TDMC Analysis

21 TDMC PNS Simulation The population of prompt neu-trons induced froma fast neutron burst in a subcritical systemdecreases exponentially Thus a special population controltechnique is necessary for an efficient TDMC calculationHere we adopt an analog MC simulation of the branchingprocess in which extra neutrons from fission are sampledand tracked accompanied with the combing technique [17]In the TDMC simulations with the combing techniquethe time domain is split into time bins and each neutronis simulated time-bin-by-time-bin with updating its timevariable whenever its track is sampled by [19]

119905119894119895119896= 119905119894119895119896minus1

+ 119897119894119895119896radic2119864119894119895119896119898119899 (1)

where 1199051198941198951198961015840

(1198961015840 = 119896 or 119896 minus 1) 119897119894119895119896 and 119864119894119895

119896are the time after

the 1198961015840th flight the length and the neutron energy of the 119896thtrack of history 119895 at time bin 119894 119898119899 is the neutron mass Ifthe sampled time is greater than the upper time bound of the119894th time bin that is 119905119894119895

119896gt 119879119894+1 then the track length of and

time after the last flight119870 of history 119895 denoted by 119897119894119895119870 and 119905119894119895119870 respectively become

119897119894119895119870 = (119879119894+1 minus 119905119894119895119870minus1) sdot radic 2119864119894119895119870119898119899 119905119894119895119870 = 119879119894+1

(2)

where 119864119894119895119870 means the neutron energy of the last flight 119870of history 119895 at time bin 119894 After the 119894th time-bin TDMCsimulations for all histories the number of neutrons for thenext time-bin simulations is increased to be the user-inputtednumber of histories by splitting according to the number ofsurviving neutrons at 119879119894+1 with conserving the total weight

22 120572 Estimation by the Slope Fitting The time-dependentdetector signals from prompt neutrons can be represented

by MC responses of the reaction rate in the detector volume119881119863 at r during time interval (1199051198941015840 minus Δ1199052 1199051198941015840 + Δ1199052) 119877119863(r 1199051198941015840)defined as

119877119863 (r 1199051198941015840)= sum119898

sum119903

int119881119863

int119864int1199051198941015840+Δ11990521199051198941015840minusΔ1199052

Σ119898119903 (r 119864) 120601119901 (r 119864 119905) 119889r 119889119864119889119905 (3)

where 1198941015840119898 and 119903 are the time-step isotope and reaction typeindex 120601119901 denotes the prompt neutron flux

Then 120572 corresponding to the detector position r can beestimated by an exponential fitting to the TDMC results of119877119863(r 1199051198941015840) as [13]

119877119863 (r 119905) = 1198621 sdot exp [minus120572est (r | 119905119904) sdot (119905 minus 119905119904)] + 1198622 (4)

where1198621 and1198622 are fitting constants and 119905 and 119905119904 are the timeafter the neutron burst and the beginning time of the fittinginterval respectively 120572est(r | 119905119904) indicates an estimate of 120572from a neutron detector located at r using 119905119904 In this study120572est(r | 119905119904) are calculated with increasing 119905119904 from 00ms to39ms by 01ms and setting the fitting interval to 10ms

An onset time of the convergence of 120572est(r) 1199050(r) isdetermined when the relative error of a mean value of 120572est(r |119905119904) comparing to its reference denoted by 120572ref becomes lessthan a prescribed value 120576 as

1199050 (r) = min119905119904 100381610038161003816100381610038161003816100381610038161003816120572est (r | 119905119904) minus 120572ref120572ref

100381610038161003816100381610038161003816100381610038161003816 lt 120576 (5)

120572est (r | 119905119904) = 1119873119873sum119899=1

120572est119899 (r | 119905119904) (6)

where 119873 is the number of replicas with different randomnumber sequences 120572est119899(r | 119905119904) is an 120572 estimate of the 119899threplica calculation 120576 of 005 is used for this convergencediagnosis

Here 120572ref is calculated by theMC 120572-iterationmethod [23]which is developed to solve the 120572-mode eigenvalue equationexpressed as

119878119905 = 120572R119878119905 (7)

R119878119905 equiv 1V (119864) Σ119905 (r 119864)

sdot infinsum119896=0

int119889r1015840 int1198891198640 int119889Ω0119870119901119896 (r1015840 1198640Ω0 997888rarr r 119864Ω)times int119889r0119879 (r0 997888rarr r1015840 | 1198640Ω0) 119878119905 (r0 1198640Ω0)

(8)

119870119901119896 (r1015840 1198640Ω0 997888rarr r 119864Ω) = int119889r1 int1198891198641 int119889Ω1sdot sdot sdot int 119889r119896minus1 int119889119864119896minus1 int119889Ω119896minus1 times 119870119901 (r119896minus1 119864119896minus1Ω119896minus1997888rarr r 119864Ω) sdot sdot sdot 119870119901 (r1015840 1198640Ω0 997888rarr r1 1198641Ω1)

(9)

Science and Technology of Nuclear Installations 3

119870119901 (r1015840 1198641015840Ω1015840 997888rarr r 119864Ω) = 119879 (r1015840 997888rarr r | 119864Ω)sdot 119862119901 (1198641015840Ω1015840 997888rarr 119864Ω | r1015840) (10)

119862119901 (1198641015840Ω1015840 997888rarr 119864Ω | r1015840)= sum119903 =fis

]119903Σ119903 (r1015840 1198641015840)Σ119905 (r1015840 1198641015840) 119891119903 (1198641015840Ω1015840 997888rarr 119864Ω)+ ]119901Σ119891 (r1015840 1198641015840)Σ119905 (r1015840 1198641015840) sdot 120594119901 (119864)4120587

(11)

119879 (r1015840 997888rarr r | 119864Ω) = Σ119905 (r 119864)1003816100381610038161003816r minus r101584010038161003816100381610038162sdot exp[minusint|rminusr1015840|

0Σ119905 (r minus 119904 r minus r10158401003816100381610038161003816r minus r10158401003816100381610038161003816 119864) 119889119904] 120575(Ω

sdot r minus r10158401003816100381610038161003816r minus r10158401003816100381610038161003816 minus 1) (12)

119878119905 equiv 120572V (119864)120601119901 (r 119864Ω) (13)

where the subscript 119901 indicates prompt neutron 119878119905 is namedthe time source [23] V(119864) is a neutron speed correspondingto its energy 119864 ]119903 and ]119901 denote the average numbers ofneutrons emitted from reaction type 119903 and prompt fissionneutrons respectively 119891119903(1198641015840Ω1015840 rarr EΩ)119889119864119889Ω is theprobability that a collision of type 119903 by a neutron of directionΩ1015840 and energy 1198641015840 will produce a neutron in direction interval119889Ω about Ω with energy in 119889119864 about 119864 Other notations

follow convention By directly applying the power iterationmethod [24] for (8) it is demonstrated [23] to stably estimate120572 even for a deep subcritical system

23 Determination of an Optimum Detector Position Theamplitude of neutron signals used for the exponential regres-sion when 120572est(r) is converged can be defined as

119877119863 (r)= sum119898

sum119903

int119881119863

int119864int1199050(r)+Δ1198791199050(r)

Σ119898119903 (r 119864) 120601119901 (r 119864 119905) 119889r 119889119864119889119905 (14)

where Δ119879 denotes the fitting time intervalThen the optimum detector position for the PNS 120572

measurement can be determined as a position r where 119877119863(r)becomes maximized because the statistical uncertainty of thedetector signals during [1199050(r) 1199050(r) + Δ119879] is assumed to beinversely proportional to the signal amplitude at the positionby following the Poisson distribution

3 Application Results

31 Pb-Bi-Zoned Experimental Benchmark The developedmethod to determine the optimum detector position for thePNS 120572 measurement is applied for the Pb-Bi-zoned ADSexperimental benchmark at KUCA [22] The benchmark

p p p p p p p p pp p

p p pp p p pp p p

p p pp p p

F F F

p pp pp pp p

pp p

p p p Fp p p Fp p Fp p p Fp p p Fp p p

F F Ff f ff f ff fF FF F

p pF pp pF p

pF pp pF pp pF p

pp p

p p p pp p p pp p p pp p p pp p p pp p p p

p pp pp pp pp pp p

p pp pp pp pp pp pp pp pp pp pp pp p

S6

C1

S5

C2

S4

C3

F

f

C

S

Fuel (36 EU)

Fuel (EU + Pb-Bi)

Aluminum Sheath

p Polyethylenemoderator

BF3 detector(12 diam)

Optical fiber 1

Optical fiber 3

Optical fiber 2

Pb-Bi targetOptical fiber

Safety rod

Control rod

Figure 1 Core configuration of the PNS experiment (Case 6)

provides 6 different subcritical cores comprised of Pb-Biloaded enriched uranium fuel and polyethylene moderatorand reflector The spallation neutron source is generated inthe center of the core by injecting 100MeV protons to the Pb-Bi target The PNS 120572 measurement is conducted with threeoptical fiber detectors in different positions Case 6 among thesix cores is chosen for an application of the developedmethodand its core configuration is shown in Figure 1

311 Spallation Source Treatment The spallation neutronsource information is obtained from MCNPX260 [25] pro-ton source simulations The spallation neutron spectra fromthe Pb-Bi target are tallied with respect to angle betweenthe outgoing direction of neutrons and proton beam Theangle bin is equally divided by 15 degrees The MCNPXcalculation is done with 10000000 histories and la150hproton library provided The neutron spectra and relativeangular flux distribution are given in Figure 2 One can seethat the neutron spectra tend to be more hardened as itsdirection is more forwarded and the overall neutron yield isbiased to the forward direction The direction and energy ofthe spallation neutrons are inputted in the formof histogramsand uniformly sampled in each bin at the beginning of theMcCARD TDMC simulations


150ndash165∘120ndash135∘90ndash105∘60ndash75∘30ndash45∘0ndash15∘


1E minus 7

1E minus 6

1E minus 5

1E minus 4

1E minus 3

001

01

1

10N

orm

aliz

ed F

lux

(1c

m2le

thar

gys

ourc

e)

01 1 10 100001Energy (MeV)

(a)

000001002003004005006007

0

10

20

30

4050

60708090100110

120130

140

150

160

170

180

190

200

210

220230

240250 260 270 280 290

300310

320

330

340

350001002003004005006007

relative flux(b)

Figure 2 Spallation neutron spectra (a) and relative angular flux distribution (b)

312 Detector Modeling The neutron detector used in theexperiment is a small-sized optical fiber detector [26] with1mm diameter which makes it available to be inserted intogaps between assembliesThedetector consists of amixture of6LiF neutron converter and ZnS scintillator of which signalsare induced by charged particles emitted from (119899 120572) and(119899 119901) reactions Since the real size of the detector is too smallto obtain confident tally results in the TDMC simulationsthe detector size is enlarged to cover the active core region ateach intersection of air gapsThe tally region of the detector isshown in Figure 3 In the detector regions the detector signalsare tallied as a sum of (119899 120572) and (119899 119901) reaction rates while theneutron simulation is conducted as if the detectors are filledwith air to prevent them distorting the MC neutron tracking

32 Searching the Optimum Detector Positions To verify thefeasibility of the devised method 120572est(r | 119905119904) is estimated atthe two detector positions which are marked as optical fibers1 and 2 in Figure 1 120572est(r | 119905119904) at each detector positionis compared with 120572ref calculated by the 120572-iteration methodand 120572exp(r | 119905119904) which is estimated by the exponential regres-sion of experimental detector signals Note that comparisonresults for optical fiber 3 are omitted because its detectorsignals might be contaminated with gamma-ray induced byhigh energy neutron sources 120572est(r | 119905119904) is estimated with100 replicas of TDMC simulation using 1000000 historiesand 01ms time bin up to 5ms 120572ref is calculated by the120572-iteration method with 100000 histories and 100 activeiterations ENDFB-VII1 cross section libraries are used forboth calculations

Figure 4 shows comparison results for the two detectorpositions The solid lines and the dashed lines are the TDMC

and experimental results at detector positions The value of120572ref is estimated to be 19500 with its standard deviationof 20 From the figure one can see that the 120572 estimatesconverge to the reference value with different convergencerates depending on their positions Also one can see adiscrepancy of the initial convergence trends of detector2 between the TDMC and experimental results whichcan be attributed to a difference of detector signal yieldssensitive to neutron energy range The convergence timesdetected by (5) using the TDMC tally results are 08ms fordetector 1 and 04ms for detector 2 whereas those fromthe experiments are 15ms for detector 1 and 11ms fordetector 2 Although the convergence times estimated fromthe TDMC calculation differ from the experimentsrsquo by 07msfor both detectors due to initial effects of the higher-modecomponents it is noteworthy that their differences betweenthe two detector positions are the same as 04msThis impliesthat the proposed method based on the TDMC calculationcan predict quitewell the sensitivity of the120572 convergence timedepending on detector positions

The TDMC 120572 estimations are conducted for all possibledetector positions in the air gaps between assemblies tosearch the optimum detector positions Figure 5 showsthe convergence time and amplitude of neutron signal ateach candidate position The white colored positions in theconvergence time map are where the 120572 estimates do notconverge until 40ms Both the convergence time and theneutron signal map show that the polyethylene moderatorregions adjacent to fuel region converge faster and give higherneutron signals than other regions It is expected to obtainmore reliable detector signals for the PNS 120572measurement atthese optimum detector positions


times30

unit

35cm cells

times15

unitcells

times15

unitcells

70cm

f

f f

f

(8526

6+5080

ＧＧ)

18p

times27

+10

ptimes2

(2305

34+25

40

ＧＧ)

18p

times73+10

p

Figure 3 Tally region of the optical fiber detector (red region)

TDMC_det1TDMC_det2

Experiment_det1Experiment_det2

0

500

1000

1500

2000

2500

3000

3500

alph

a (1

sec)

0001 0002 0003 00040000starting time of fitting (sec)

Ｌ 1950

Figure 4 Comparisons of 120572 estimates from TDMC and experimental data


30

27

24

21

18

16

13

10

07

04

01

convergence time (ms) relative neutron signal

57

51

46

40

34

28

23

17

11

06

00

Figure 5 Convergence time and amplitude of neutron signal maps

4 Conclusions

A simple method to determine an effective detector positionfor the PNS 120572measurement is proposed by comparing signalamplitudes at different detector positions estimated by theTDMCneutron transport calculationswhen their120572 estimatesby the slope fitting are converged The developed method isimplemented in McCARD and applied to case 6 core in theKUCA Pb-Bi-zoned ADS experimental benchmarks Fromthe comparisons with experimental results it is shown thatthe TDMC calculation predicts the 120572 convergence time quitewell The proposed method provides the 120572 convergence timemap and the corresponding signal amplitude map for case6 core which can be used to determine effective detectorpositions and to validate experimental results in the PNS 120572measurement

Disclosure

The authors have presented an earlier version of this researchto RPHA17 (the Reactor Physics Asia 2017) conferenceChengdu China August 24-25 2017

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by the Brain Korea 21 Plus Project(no 21A20130012821)

References

[1] H A Abderrahim J Galambos Y Gohar et al ldquoAccelerator andTarget Technology for Accelerator Driven Transmutation and

Energy Productionrdquo DOE white paper on ADS vol 1 no 1 pp1ndash23 2010

[2] D De Bruyn et al in Proceedings of the ICAPP 2016 SanFrancisco CA USA 2016

[3] X Yan L Yang X Zhang and W Zhan ldquoConcept of anaccelerator-driven advanced nuclear energy systemrdquo Energiesvol 10 no 7 article no 944 2017

[4] A Gandini and M Salvatores ldquoThe physics of subcriticalmultiplying systemsrdquo Journal of Nuclear Science and Technologyvol 39 no 6 pp 673ndash686 2002

[5] K Kobayashi and K Nishihara ldquoDefinition of subcriticalityusing the importance function for the production of fissionneutronsrdquo Nuclear Science and Engineering vol 136 no 2 pp272ndash281 2000

[6] R Soule W Assal P Chaussonnet et al ldquoNeutronic Studiesin Support of Accelerator-Driven Systems the MUSE Experi-ments in theMASURCAFacilityrdquoNuclear Science and Engineer-ing vol 148 no 1 pp 124ndash152 2004

[7] C H Pyeon M Hervault T Misawa H Unesaki T Iwasakiand S Shiroya ldquoStatic and kinetic experiments on accelerator-driven system with 14MeV neutrons in Kyoto university criticalassemblyrdquo Journal of Nuclear Science and Technology vol 45 no11 pp 1171ndash1182 2008

[8] C H Pyeon T Misawa J-Y Lim et al ldquoFirst injection ofspallation neutrons generated by high-energy protons into thekyoto university critical assemblyrdquo Journal of Nuclear Scienceand Technology vol 46 no 12 pp 1091ndash1093 2009

[9] B E Simmons and J S King ldquoA Pulsed Neutron Techniquefor Reactivity DeterminationrdquoNuclear Science and Engineeringvol 3 no 5 pp 595ndash608 1958

[10] G R Keepin Physics of nuclear kinetics Addison-Wesley PubCo MA USA 1965

[11] F R N McDonnell and M J Harris ldquoPulsed-source experi-ments in a reflected coupled-core reactor-I reactivity measure-mentsrdquo Journal of Nuclear Energy vol 26 no 3 pp 113ndash1281972


[12] E A Stumbur A G Shokodrsquoko V I Zhuravlev I PMatveenkoand Z N Milyutina ldquoCombined pulsed method of measuringhigh reactivities for reactors with reflectorsrdquo Soviet AtomicEnergy vol 36 no 3 pp 224ndash228 1974

[13] T Suzaki ldquoSubcriticality Determination of Low-Enriched UO2Lattices in Water by Exponential Experimentrdquo Journal ofNuclear Science and Technology vol 28 no 12 pp 1067ndash10771991

[14] C-M Persson P Seltborg A Ahlander et al ldquoAnalysis ofreactivity determination methods in the subcritical experimentYalinardquo Nuclear Instruments and Methods in Physics ResearchSection A Accelerators Spectrometers Detectors and AssociatedEquipment vol 554 no 1-3 pp 374ndash383 2005

[15] C H Pyeon Experimental Benchmarks on Thorium-LoadedAccelerator-Driven System at Kyoto University Critical AssemblyKURR-TR(CD)-48 Research Reactor Institute Kyoto Univer-sity 2015

[16] C H Pyeon M Yamanaka T Endo W F G van Rooijen andG Chiba ldquoExperimental benchmarks on kinetic parametersin accelerator-driven system with 100 MeV protons at KyotoUniversity Critical Assemblyrdquo Annals of Nuclear Energy vol105 pp 346ndash354 2017

[17] D E Cullen C J Clouse R Procassini and R C LittleldquoStatic andDynamicCriticality AreTheyDifferentrdquo Tech RepUCRL-TR-201506 2003

[18] B L Sjenitzer and J Eduard Hoogenboom ldquoDynamic montecarlo method for nuclear reactor kinetics calculationsrdquo NuclearScience and Engineering vol 175 no 1 pp 94ndash107 2013

[19] N Shaukat M Ryu and H J Shim ldquoDynamic MonteCarlo transient analysis for the Organization for EconomicCo-operation and Development Nuclear Energy Agency(OECDNEA) C5G7-TD benchmarkrdquo Nuclear Engineering andTechnology vol 49 no 5 pp 920ndash927 2017

[20] T E Booth ldquoA Weight (Charge) Conserving Importance-Weighted Comb for Monte Carlordquo LA-URndash96-0051 LosAlamos National Laboratory NM USA 1996

[21] H J Shim B S Han J S Jung H J Park and C H KimldquoMcCARD Monte Carlo code for advanced reactor design andanalysisrdquoNuclear Engineering and Technology vol 44 no 2 pp161ndash176 2012

[22] C H Pyeon Experimental Benchmarks of Neutronics on SolidPb-Bi in Accelerator-Driven System with 100 MeV Protons atKyoto University Critical Assembly KURRI-TR-447 ResearchReactor Institute Kyoto University 2017

[23] H J Shim S H Jang and S M Kang ldquoMonte Carlo AlphaIteration Algorithm for a Subcritical System Analysisrdquo Scienceand Technology of Nuclear Installations vol 2015 pp 1ndash7 2015

[24] S Nakamura Computational methods in engineering and sci-ence Wiley-Interscience [John Wiley amp Sons] New York-London-Sydney 1977

[25] J S Hendricks Tech Rep LA-UR-08-2216 Los AlamosNational Laboratory NM 2008

[26] T Yagi H Unesaki T Misawa et al ldquoDevelopment of a smallscintillation detector with an optical fiber for fast neutronsrdquoApplied Radiation and Isotopes vol 69 no 2 pp 539ndash544 2011

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for the 120572 measurement through the PNS experiment usingthe time-dependentMonte Carlo (TDMC) neutron transportanalyses [17ndash19] In the TDMC calculations the combingalgorithm [17 20] is applied to maintain the time-bin-wiseneutron population because an exponential decrease of theneutron population in an analog TDMC calculation of asubcritical system causes large statistical uncertainties In theproposedmethod the optimum detector position is searchedby comparing the strength of detector signal at each spatialposition when the 120572 estimate at the position is convergedThe position-dependent 120572 convergence is diagnosed by aslope fitting to the detector signals obtained from the TDMCcalculations The proposed methods are implemented in aSeoul National University continuous-energy Monte Carlo(MC) code McCARD [21] and applied to the Pb-Bi-zonedADS experimental benchmark at KUCA [22]

2 Determination of an Optimum DetectorPosition through the TDMC Analysis

21 TDMC PNS Simulation The population of prompt neu-trons induced froma fast neutron burst in a subcritical systemdecreases exponentially Thus a special population controltechnique is necessary for an efficient TDMC calculationHere we adopt an analog MC simulation of the branchingprocess in which extra neutrons from fission are sampledand tracked accompanied with the combing technique [17]In the TDMC simulations with the combing techniquethe time domain is split into time bins and each neutronis simulated time-bin-by-time-bin with updating its timevariable whenever its track is sampled by [19]

119905119894119895119896= 119905119894119895119896minus1

+ 119897119894119895119896radic2119864119894119895119896119898119899 (1)

where 1199051198941198951198961015840

(1198961015840 = 119896 or 119896 minus 1) 119897119894119895119896 and 119864119894119895

119896are the time after

the 1198961015840th flight the length and the neutron energy of the 119896thtrack of history 119895 at time bin 119894 119898119899 is the neutron mass Ifthe sampled time is greater than the upper time bound of the119894th time bin that is 119905119894119895

119896gt 119879119894+1 then the track length of and

time after the last flight119870 of history 119895 denoted by 119897119894119895119870 and 119905119894119895119870 respectively become

119897119894119895119870 = (119879119894+1 minus 119905119894119895119870minus1) sdot radic 2119864119894119895119870119898119899 119905119894119895119870 = 119879119894+1

(2)

where 119864119894119895119870 means the neutron energy of the last flight 119870of history 119895 at time bin 119894 After the 119894th time-bin TDMCsimulations for all histories the number of neutrons for thenext time-bin simulations is increased to be the user-inputtednumber of histories by splitting according to the number ofsurviving neutrons at 119879119894+1 with conserving the total weight

22 120572 Estimation by the Slope Fitting The time-dependentdetector signals from prompt neutrons can be represented

by MC responses of the reaction rate in the detector volume119881119863 at r during time interval (1199051198941015840 minus Δ1199052 1199051198941015840 + Δ1199052) 119877119863(r 1199051198941015840)defined as

119877119863 (r 1199051198941015840)= sum119898

sum119903

int119881119863

int119864int1199051198941015840+Δ11990521199051198941015840minusΔ1199052

Σ119898119903 (r 119864) 120601119901 (r 119864 119905) 119889r 119889119864119889119905 (3)

where 1198941015840119898 and 119903 are the time-step isotope and reaction typeindex 120601119901 denotes the prompt neutron flux

Then 120572 corresponding to the detector position r can beestimated by an exponential fitting to the TDMC results of119877119863(r 1199051198941015840) as [13]

119877119863 (r 119905) = 1198621 sdot exp [minus120572est (r | 119905119904) sdot (119905 minus 119905119904)] + 1198622 (4)

where1198621 and1198622 are fitting constants and 119905 and 119905119904 are the timeafter the neutron burst and the beginning time of the fittinginterval respectively 120572est(r | 119905119904) indicates an estimate of 120572from a neutron detector located at r using 119905119904 In this study120572est(r | 119905119904) are calculated with increasing 119905119904 from 00ms to39ms by 01ms and setting the fitting interval to 10ms

An onset time of the convergence of 120572est(r) 1199050(r) isdetermined when the relative error of a mean value of 120572est(r |119905119904) comparing to its reference denoted by 120572ref becomes lessthan a prescribed value 120576 as

1199050 (r) = min119905119904 100381610038161003816100381610038161003816100381610038161003816120572est (r | 119905119904) minus 120572ref120572ref

100381610038161003816100381610038161003816100381610038161003816 lt 120576 (5)

120572est (r | 119905119904) = 1119873119873sum119899=1

120572est119899 (r | 119905119904) (6)

where 119873 is the number of replicas with different randomnumber sequences 120572est119899(r | 119905119904) is an 120572 estimate of the 119899threplica calculation 120576 of 005 is used for this convergencediagnosis

Here 120572ref is calculated by theMC 120572-iterationmethod [23]which is developed to solve the 120572-mode eigenvalue equationexpressed as

119878119905 = 120572R119878119905 (7)

R119878119905 equiv 1V (119864) Σ119905 (r 119864)

sdot infinsum119896=0

int119889r1015840 int1198891198640 int119889Ω0119870119901119896 (r1015840 1198640Ω0 997888rarr r 119864Ω)times int119889r0119879 (r0 997888rarr r1015840 | 1198640Ω0) 119878119905 (r0 1198640Ω0)

(8)

119870119901119896 (r1015840 1198640Ω0 997888rarr r 119864Ω) = int119889r1 int1198891198641 int119889Ω1sdot sdot sdot int 119889r119896minus1 int119889119864119896minus1 int119889Ω119896minus1 times 119870119901 (r119896minus1 119864119896minus1Ω119896minus1997888rarr r 119864Ω) sdot sdot sdot 119870119901 (r1015840 1198640Ω0 997888rarr r1 1198641Ω1)

(9)



119862119901 (1198641015840Ω1015840 997888rarr 119864Ω | r1015840)= sum119903 =fis


(11)




119878119905 equiv 120572V (119864)120601119901 (r 119864Ω) (13)




119877119863 (r)= sum119898

sum119903

int119881119863

int119864int1199050(r)+Δ1198791199050(r)

Σ119898119903 (r 119864) 120601119901 (r 119864 119905) 119889r 119889119864119889119905 (14)






p p pp p p pp p p

p p pp p p

F F F

p pp pp pp p

pp p



p pF pp pF p

pF pp pF pp pF p

pp p


p pp pp pp pp pp p


S6

C1

S5

C2

S4

C3

F

f

C

S

Fuel (36 EU)

Fuel (EU + Pb-Bi)

Aluminum Sheath



Optical fiber 1

Optical fiber 3

Optical fiber 2


Safety rod

Control rod







1E minus 7

1E minus 6

1E minus 5

1E minus 4

1E minus 3

001

01

1

10N

orm

aliz

ed F

lux

(1c

m2le

thar

gys

ourc

e)

01 1 10 100001Energy (MeV)

(a)

000001002003004005006007

0

10

20

30

4050

60708090100110

120130

140

150

160

170

180

190

200

210

220230

240250 260 270 280 290

300310

320

330

340

350001002003004005006007

relative flux(b)








times30

unit

35cm cells

times15

unitcells

times15

unitcells

70cm

f

f f

f

(8526

6+5080

ＧＧ)

18p

times27

+10

ptimes2

(2305

34+25

40

ＧＧ)

18p

times73+10

p


TDMC_det1TDMC_det2


0

500

1000

1500

2000

2500

3000

3500

alph

a (1

sec)


Ｌ 1950



30

27

24

21

18

16

13

10

07

04

01


57

51

46

40

34

28

23

17

11

06

00


4 Conclusions


Disclosure




Acknowledgments


References









































Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018




119862119901 (1198641015840Ω1015840 997888rarr 119864Ω | r1015840)= sum119903 =fis


(11)




119878119905 equiv 120572V (119864)120601119901 (r 119864Ω) (13)




119877119863 (r)= sum119898

sum119903

int119881119863

int119864int1199050(r)+Δ1198791199050(r)

Σ119898119903 (r 119864) 120601119901 (r 119864 119905) 119889r 119889119864119889119905 (14)






p p pp p p pp p p

p p pp p p

F F F

p pp pp pp p

pp p



p pF pp pF p

pF pp pF pp pF p

pp p


p pp pp pp pp pp p


S6

C1

S5

C2

S4

C3

F

f

C

S

Fuel (36 EU)

Fuel (EU + Pb-Bi)

Aluminum Sheath



Optical fiber 1

Optical fiber 3

Optical fiber 2


Safety rod

Control rod







1E minus 7

1E minus 6

1E minus 5

1E minus 4

1E minus 3

001

01

1

10N

orm

aliz

ed F

lux

(1c

m2le

thar

gys

ourc

e)

01 1 10 100001Energy (MeV)

(a)

000001002003004005006007

0

10

20

30

4050

60708090100110

120130

140

150

160

170

180

190

200

210

220230

240250 260 270 280 290

300310

320

330

340

350001002003004005006007

relative flux(b)








times30

unit

35cm cells

times15

unitcells

times15

unitcells

70cm

f

f f

f

(8526

6+5080

ＧＧ)

18p

times27

+10

ptimes2

(2305

34+25

40

ＧＧ)

18p

times73+10

p


TDMC_det1TDMC_det2


0

500

1000

1500

2000

2500

3000

3500

alph

a (1

sec)


Ｌ 1950



30

27

24

21

18

16

13

10

07

04

01


57

51

46

40

34

28

23

17

11

06

00


4 Conclusions


Disclosure




Acknowledgments


References









































Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018





1E minus 7

1E minus 6

1E minus 5

1E minus 4

1E minus 3

001

01

1

10N

orm

aliz

ed F

lux

(1c

m2le

thar

gys

ourc

e)

01 1 10 100001Energy (MeV)

(a)

000001002003004005006007

0

10

20

30

4050

60708090100110

120130

140

150

160

170

180

190

200

210

220230

240250 260 270 280 290

300310

320

330

340

350001002003004005006007

relative flux(b)








times30

unit

35cm cells

times15

unitcells

times15

unitcells

70cm

f

f f

f

(8526

6+5080

ＧＧ)

18p

times27

+10

ptimes2

(2305

34+25

40

ＧＧ)

18p

times73+10

p


TDMC_det1TDMC_det2


0

500

1000

1500

2000

2500

3000

3500

alph

a (1

sec)


Ｌ 1950



30

27

24

21

18

16

13

10

07

04

01


57

51

46

40

34

28

23

17

11

06

00


4 Conclusions


Disclosure




Acknowledgments


References









































Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



times30

unit

35cm cells

times15

unitcells

times15

unitcells

70cm

f

f f

f

(8526

6+5080

ＧＧ)

18p

times27

+10

ptimes2

(2305

34+25

40

ＧＧ)

18p

times73+10

p


TDMC_det1TDMC_det2


0

500

1000

1500

2000

2500

3000

3500

alph

a (1

sec)


Ｌ 1950



30

27

24

21

18

16

13

10

07

04

01


57

51

46

40

34

28

23

17

11

06

00


4 Conclusions


Disclosure




Acknowledgments


References









































Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



30

27

24

21

18

16

13

10

07

04

01


57

51

46

40

34

28

23

17

11

06

00


4 Conclusions


Disclosure




Acknowledgments


References









































Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018






























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018


determination of an effective detector position for pulsed...

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