Research ArticleMonte Carlo Alpha Iteration Algorithm fora Subcritical System Analysis

Hyung Jin Shim Sang Hoon Jang and Soo Min Kang

Seoul National University 1 Gwanak-ro Gwanak-gu Seoul 151-744 Republic of Korea

Correspondence should be addressed to Hyung Jin Shim shimhjsnuackr

Received 14 April 2015 Revised 21 June 2015 Accepted 29 June 2015

Academic Editor Valerio Giusti

Copyright copy 2015 Hyung Jin Shim et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The 120572-k iteration method which searches the fundamental mode alpha-eigenvalue via iterative updates of the fission sourcedistribution has been successfully used for theMonte Carlo (MC) alpha-static calculations of supercritical systems However the 120572-k iterationmethod for the deep subcritical system analysis suffers from a gigantic number of neutron generations or a huge neutronweight which leads to an abnormal termination of the MC calculations In order to stably estimate the prompt neutron decayconstant (120572) of prompt subcritical systems regardless of subcriticality we propose a newMCalpha-static calculationmethod namedas the 120572 iteration algorithmThe newmethod is derived by directly applying the power method for the 120572-mode eigenvalue equationand its calculation stability is achieved by controlling the number of time source neutrons which are generated in proportion to 120572divided by neutron speed in MC neutron transport simulations The effectiveness of the 120572 iteration algorithm is demonstrated fortwo-group homogeneous problems with varying the subcriticality by comparisons with analytic solutions The applicability of theproposed method is evaluated for an experimental benchmark of the thorium-loaded accelerator-driven system

1 Introduction

The prompt neutron decay constant (referred to as 120572) insubcritical systems is a basic neutronics parameter whichcan be directly measured from reactor experiments In theMonte Carlo (MC) neutron transport analysis 120572 has beenestimated by two approaches [1] dynamicMC simulations todirectly solve the time-dependent neutron transport equationand the alpha-static MC calculations to solve the 120572-modeeigenvalue equation The dynamic MC calculations coverthe 120572 estimation from numerical simulations of the pulsedneutron source (PNS) experiment by fitting time-dependenttally results of a neutron detector to the exponential functionHowever it is well known that a starting time of fittingshould be carefully determined by ensuring the convergenceof estimated values [2 3] The TART code [4] is equippedwith a unique method to measure 120572 in time-stepwise MCsimulations with controlling the neutron population by thecombing algorithm [1]

Differently from theMC dynamic simulations the alpha-static MC methods calculate the fundamental mode or

higher-order solutions of the 120572-mode eigenvalue equation forprompt neutron which can be expressed in operator notationas

LΦ = F119901Φ + 119878119905 (1)

LΦ = Ω sdot nablaΦ (r 119864Ω) + Σ119905 (r 119864)Φ (r 119864Ω) minus int11986410158401198891198641015840

sdot int

4120587

119889Ω1015840Σ119904 (r 119864

1015840997888rarr 119864Ω

1015840997888rarrΩ)Φ (r 1198641015840Ω1015840)

(2)

F119901Φ =

120594119901 (119864)

4120587

int

11986410158401198891198641015840int

4120587

119889Ω1015840]119901Σ119891 (r 119864

1015840)

sdot Φ (r 1198641015840Ω1015840) (3)

119878119905 =120572

V (119864)Φ (r 119864Ω) (4)

where Φ is the neutron angular flux and the subscript 119901indicates ignorance of the delayed fission neutron 119878119905 isnamed as the time source distribution V(119864) is a neutron

Hindawi Publishing CorporationScience and Technology of Nuclear InstallationsVolume 2015 Article ID 859242 7 pageshttpdxdoiorg1011552015859242

2 Science and Technology of Nuclear Installations

speed corresponding to its energy 119864 Other notations followconvention

For the alpha-static MC calculations the 120572-k iterationmethods [5ndash9] searching the fundamental mode alpha-eigenvalue in the middle of iterative fission source updatesby the MC power iteration method (k iteration methodhereafter) [10] have been mainly used since Brockway et al[5] developed an MC algorithm with considering the timeabsorption [1] of minus120572V for prompt supercritical systems (120572 lt

0) The 120572-k iteration methods for prompt subcritical systems(120572 gt 0) treat a negative value of the time absorption as thetime production (120572V) by which the time source neutrons aregenerated in the MC neutron simulations to avoid negativeabsorption reactions Yamamoto [11] suggested a neutronsimulation algorithm with correcting the neutron weight ineach track by the time production reaction However thetime production strategy of the conventional 120572-k iterationmethod [1] and Yamamotorsquos weight correction method forhighly subcritical systems are reported [6 8 12] to cause agigantic number of source neutron generations or a hugevalue of the neutron weight which leads to an abnormaltermination of theMC calculations To reduce this instabilityproblem Ye et al [6] introduced a pseudo neutron absorptionterm of minus120578120572VΦ (120578 gt 0) into both sides of (1) and itsslightly modified formulation [8] is used in Tripoli-4 [13]However an appropriate adjustment parameter 120578 is requiredto ensure a stable running without a halt in the pseudoneutron absorption approaches

To overcome this instability problem in the 120572-k iterationmethods we propose a new alpha-static MC method namedas the 120572 iteration algorithm for the prompt subcritical systemanalysis In Section 2 the 120572 iteration algorithm is derived byapplying the power method [14] for the 120572-mode eigenvalueequation of (1) with a normalization scheme to stably controlthe number of time source neutrons at each iteration Thenew 120572 iteration and the existing 120572-k iteration methods havebeen implemented in the Seoul National University MCcode McCARD [15] The effectiveness of the new methodis examined by comparing 120572rsquos calculated by the 120572 iterationand the 120572-k iteration methods with analytic solutions fortwo-group infinite homogeneous problemsThe applicabilityof the proposed method is tested for the thorium-loadedaccelerator-driven system [16] at Kyoto University CriticalAssembly (KUCA) by comparing 120572rsquos calculated by the 120572

iteration algorithm with those from McCARD dynamicsimulations as well as measurements of the PNS experi-ments

2 120572 Iteration Algorithm

21 Derivation In order to obtain a MC neutron trackingalgorithm from an integrodifferential form of the 120572-modeeigenvalue equation written by (1) it is required to transform(1) into its integral form By integrating (1) along with acharacteristic curve [17] in the sameway to derive the integralformof the neutron transport equation [17 18] and expressingthe resulting integral equation for the collision density 120595

defined by Σ119905Φ [19] one can obtain the collision densityequation for the 120572-mode eigenvalue equation

120595 (r 119864Ω) = int119889r1015840119879 (r1015840 997888rarr r | 119864Ω) 119878119905 (r1015840 119864Ω)

+ int119889r1015840 int11986410158401198891198641015840int

4120587

119889Ω1015840

sdot 119870119901 (r1015840 1198641015840Ω1015840997888rarr r 119864Ω)

sdot 120595 (r1015840 1198641015840Ω1015840)

(5)

where119870119901 is the transition kernel ignoring the delayed fissionneutrons defined as a product of the collision kernel withoutthe delayed fission neutron 119862119901 and the transport kernel 119879as follows

119870119901 (r1015840 1198641015840Ω1015840997888rarr r 119864Ω) = 119879 (r1015840 997888rarr r | 119864Ω)

sdot 119862119901 (1198641015840Ω1015840997888rarr 119864Ω | r1015840)

(6)

119862119901 (1198641015840Ω1015840997888rarr 119864Ω | r1015840)

= sum

119903 =fis

]119903Σ119903 (r1015840 1198641015840)Σ119905 (r1015840 1198641015840)

119891119903 (1198641015840Ω1015840997888rarr 119864Ω)

+

]119901Σ119891 (r1015840 1198641015840)Σ119905 (r1015840 1198641015840)

sdot

120594119901 (119864)

4120587

(7)

119879 (r1015840 997888rarr r | 119864Ω) = Σ119905 (r 119864)1003816100381610038161003816r minus r1015840100381610038161003816

1003816

2

sdot exp[minusint|rminusr1015840|

0

Σ119905 (r minus 119904r minus r10158401003816100381610038161003816r minus r1015840100381610038161003816

1003816

119864) 119889119904]

sdot 120575(Ω sdotr minus r10158401003816100381610038161003816r minus r1015840100381610038161003816

1003816

minus 1)

(8)

where 119903 is used to index the neutron reaction except fission ]119903is the average number of neutrons produced from a reactiontype 119903 and 119891119903 (119864

1015840Ω1015840rarr 119864Ω)119889119864119889Ω is the probability that

a collision of type 119903 by a neutron of direction Ω1015840 and energy1198641015840 will produce a neutron in direction interval 119889Ω about Ω

with energy in 119889119864 about 119864Then the Neumann series solution [19] to (5) which

describes the MC neutron simulations can be written by

120595 (r 119864Ω) =infin

sum

119895=0

int119889r1015840 int1198891198640 int119889Ω0

sdot 119870119901119895 (r1015840 1198640Ω0 997888rarr r 119864Ω) int119889r0

sdot 119879 (1198640Ω0 r0 997888rarr r1015840) 119878119905 (r0 1198640Ω0)

(9)

Science and Technology of Nuclear Installations 3

119870119901119895 (r1015840 1198640Ω0 997888rarr r 119864Ω) = int119889r1 int1198891198641

sdot int 119889Ω1 sdot sdot sdot int 119889r119895minus1 int119889119864119895minus1 int119889Ω119895minus1

sdot 119870119901 (r119895minus1 119864119895minus1Ω119895minus1 997888rarr r 119864Ω) sdot sdot sdot 119870119901 (r1015840 1198640Ω0

997888rarr r1 1198641Ω1)

(10)

Multiplying 120572Vminus1Σ119905minus1 on both sides of (9) one can

obtain the 120572-mode eigenvalue equation for the time sourcedistribution as

119878119905 = 120572R119878119905 (11)

R119878119905 =1

V (119864) Σ119905 (r 119864)

infin

sum

119895=0

int119889r1015840 int1198891198640 int119889Ω0

sdot 119870119901119895 (r1015840 1198640Ω0 997888rarr r 119864Ω) int119889r0

sdot 119879 (1198640Ω0 r0 997888rarr r1015840) 119878119905 (r0 1198640Ω0)

(12)

To calculate the fundamental mode alpha-eigenvaluefrom (11) we directly apply the power method [14] to (11)

119878119894

119905= 120572119894R119878119894minus1119905 (13)

120572119894=

1

int 119889rint119889119864int119889ΩR119878119894minus1119905 (14)

where 119894 is the iteration index Equation (13) implies that thefundamental-mode alpha-eigenvalue and 119878119905 are calculatedby iterative updates of the times source distribution untilthey converge It is notable that the transition kernels inoperator R defined by (12) are the same as the ones usedin the fixed source mode MC calculations except that thedelayed fission neutrons are not considered which meansthat all the prompt fission neutrons should be simulatedwithin an iteration Note that an expected number of promptfission neutrons generated within an iteration is less than 1 forprompt subcritical systems of which the prompt criticality 119896119901is less than 1

22 Application to Monte Carlo Calculations The derived 120572iteration method can be implemented by slightly modifyingthe 119896 iteration algorithm [10 20] used in existing MC codesThe main difference between the 120572 iteration and the 119896

iteration methods is that the time source distribution isupdated in the 120572 iteration algorithm while the fission sourcedistribution in the 119896 iteration oneWhen a collision occurs inthe course of the 120572 iteration at iteration 119894 (= 1 2 ) whichis governed by (12) and (13) the time source neutrons for thenext iteration can be sampled at the collision site as many as

119872119894119895119896 = lfloor120572119894minus1

sdot

119908119894119895119896

V (119864119894119895119896) Σ119905 (r119894119895119896 119864119894119895119896)+ 120585rfloor (15)

where 119895 and 119896 are indices of time source neutrons andcollisions respectively lfloor119909rfloor denotes the largest integer not

exceeding 119909 119908119894119895119896 is the neutron weight for the 119896th collisionfrom the 119895th time source neutron at iteration 119894 120585 is a uniformrandom number on the interval of (0 1) 120572119894minus1 is an alpha-eigenvalue estimated at iteration 119894 minus 1 When 119894 = 1 1205720indicates an initial 120572 value guessed by a code user When119872119894119895119896 is greater than or equal to one the location energyand direction of the collision neutron (r119894119895119896 119864119894119895119896Ω119894119895119896) areset to the sampled119872119894119895119896 sourcesrsquo parameters It is meaningfulto compare (15) with a typical fission source site samplingformulation of lfloor1119896

119894minus1

sdot((119908119894119895119896sdot]Σ119891(r119894119895119896 119864119894119895119896))Σ119905(r119894119895119896 119864119894119895119896))+120585rfloor[20] where 119896

119894minus1

denotes a 119896-eigenvalue estimated at cycle 119894minus1Note that the direction of the collision neutron Ω119894119895119896 shouldbe stored in the 120572 iteration algorithm while the directionof a fission source neutron may not be banked in the 119896

iteration algorithm because of an assumption of its isotropicdistributionThemultiplication of 120572119894minus1 in the right hand sideof (15) plays an important role in controlling the total numberof time source neutrons per iteration like the division by 119896

119894minus1

in the 119896 iteration algorithm As noted in the previous sectionthe fission reactions should be sampled in a routine of thecollision type selection from the collision kernel defined by(7) in the 120572 iteration algorithm whereas the fission reactionis not allowed to occur in a common implementation of the119896 iteration algorithm [20]

At the beginning of the next iteration 119894 + 1 the initialweight of the time source neutrons 119908119894+1 is determined fromthe number of the sampled sources at iteration 119894 by

119908119894+1 =119872

sum119895sum119896119872119894119895119896

(16)

where 119872 is a number of time source neutrons per iterationinputted by a code user

To estimate 120572119894 (119894 = 1 2 ) by (14) a collision estimatorfor an inverse of 120572

119894 that is int119889rint119889119864int119889ΩR119878119894minus1119905

sumsup 119908119894119895119896V(119864119894119895119896)Σ119905(r119894119895119896 119864119894119895119896) at each collision Then afterfinishing each iteration for the 119872 time source neutrons120572119894(119894 = 1 2 ) can be calculated from a mean of119872 history

results as

120572119894=

119872

sum119895sum119896 119908119894119895119896V (119864119894119895119896)minus1

Σ119905 (r119894119895119896 119864119894119895119896)minus1 (17)

In the same reason to apply the inactive cycle runs in the119896 iteration method the fundamental-mode alpha-eigenvalueshould be estimated by averaging 120572119894rsquos at active iterations afterproper number of inactive iterations to converge the timesource distribution

3 Numerical Results

31 Two-Group Infinite Homogeneous Problems In order toinvestigate the effectiveness of the proposed method the 120572and the 120572-k iteration algorithms implemented in McCARD[15] are tested for two-group infinite homogeneous mediumproblems Table 1 shows two-group cross sections varying theprompt criticality 119896119901 The differential scattering cross section

4 Science and Technology of Nuclear Installations

Table 1 Two-group cross sections for the infinite homogeneousproblem

Cross section First group (119892 = 1) Second group (119892 = 2)Σ119905119892 050 050Σ119891119892 0025 0175]119901119892 20 20Σ119904119892119892 010 020Σ1199041198921015840119892 (119892 = 119892

1015840) Variable 000120594119901119892 10 001V119892 [seccm] 228626 times 10minus10 129329 times 10minus6

of the first group Σ11990421 is set to 0265714 0197143 01285710060000 or 0008571 which correspond to 119896119901 of 09 07 0503 or 015

The MC 120572 calculations are performed for 1000 activeiterations on 10000 sources per iteration Table 2 showscomparisons of 120572rsquos calculated by the new algorithm andthe 120572-k iteration method applying the pseudo absorptionadjustment [6] with analytic solutions In the table thepseudo absorption adjustment parameter 120578 of zero meansthe conventional 120572-k iterationmethod [1] without the pseudoabsorption adjustment From the table one can see that theMC results from the 120572 iteration method agree well with theanalytic references within 95 confidence intervals while the120572-k iteration method fails when 119896119901 are 03 and 015 due toabnormal terminations

These abnormal terminations of the 120572-k iterationmethodcan be explained by counting the number of time sourceneutrons generated from a source neutron In the 120572-k iter-ation algorithm with the 120578 parameter a time source neutronis generated at each collision site with the probability of 119901120572119892(=(1 + 120578)120572V119892(Σ119905119892 + 120578120572V119892)) while the neutron is absorbedwith the probability of119901119886119892 (=Σ119905119892+120578120572V119892minusΣ119904119892(Σ119905119892+120578120572V119892))Considering that the time source generation is dominated bythe second-group neutron by1199011205722 ≫ 1199011205721 for a small value of 120578let us calculate the expected number of time source neutrons119873time2 during theMCneutron transport simulations startingfrom a second-group source until it is absorbed BecauseΣ11990412 = 0 as given in Table 1119873time2 can be written as

119873time2 = 1199011205722 + (1 minus 1199011198862) 1199011205722 + (1 minus 1199011198862)21199011205722 + sdot sdot sdot

=

1199011205722

1 minus (1 minus 1199011198862)

=

(1 + 120578) 120572

V2 (Σ1199052 minus Σ1199042) + 120578120572

(18)

When 119873time2 is greater than or equal to 1 the totalnumber of time source neutrons generated in a source historybecomes infinity which leads to the abnormal terminationBy inserting the analytic solution of the fundamental modealpha-eigenvalue in Table 2 and the constants given in Table 1into (18)119873time2 is found to be closer to 1 with values of 011034 057 080 and 097 when 120578 = 0 and 028 061 080092 and 099 when 120578 = 2 as the subcriticality is deeper from090 to 015 Because the 120572 estimate from the 120572-k iterationmethod is updated iteratively and fluctuated due to a finite

100000

120000

140000

160000

180000

200000

220000

240000

260000

280000

120572limit

120572-k iteration method with 120578 = 0

120572-k iteration method with 120578 = 2

120572

Iteration index0 1 2 3 4 5 6 7 8 9

Figure 1 Iteration-wise 120572 values estimated by the 120572-k iterationmethod for the two-group infinite homogeneous problem of 119896119901 =03

number of MC history simulations a stability condition canbe expressed from (18) as

120572119894

120572-119896 lt 120572limit = V2 (Σ1199052 minus Σ1199042) (19)

where 120572119894120572-119896 is the fundamental mode alpha-eigenvalue esti-

mated at the 119894th fission source iteration From Table 1120572limit for the two-group infinite homogeneous problems isdetermined to be 231967 Figure 1 shows 120572119894

120572-119896 for the cases of119896119901 of 03 with different 120578 values of 0 and 2 on 10

7 histories periteration starting from an initially guessed 120572 1205720

120572-119896 of 185568From the figure we can observe that 120572119894

120572-119896 becomes close to120572limit of 231967 at cycle 8 in the run with 120578 of 0 and at cycle6 with 120578 of 2 which yield the abnormal terminations at thevery next iteration calculations

The 120572-k iteration method with the pseudo absorptionterm may reduce occurrence of the infinite number oftime source generations by suppressing fluctuations of 120572119894

120572-119896for MC calculations with continuous cross section data asreported in [12] However it is noteworthy that the abnormalterminations in the 120572-k iteration calculations can happen byinevitable statistical fluctuations of 120572 estimates

32 Application to the Th-ADS Experimental BenchmarksThe proposed 120572 iteration method is applied for the exper-imental benchmarks on thorium-loaded accelerator-drivensystem (Th-ADS) [16] The Th-ADS experiments were per-formed using the solid-moderated and solid-reflected type-A core of KUCA for seven core configurations combinedwith a 14MeV pulsed neutron generator or a synchrotrontype proton acceleratorThehighly enriched uranium (HEU)thorium (Th) and natural uranium (NU) fuel was loadedtogether with the reflectors including polyethylene (PE)graphite (Gr) and beryllium (Be) Figure 2 shows a coreconfiguration with three 3He detectors used for Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PE cores and one with two

Science and Technology of Nuclear Installations 5

Table 2 Comparisons of 120572 estimates for the infinite homogeneous problem

119896119901 Ref 120572 120572-119896 iteration with 120578 = 0 (SD) 120572-119896 iteration with 120578 = 2 (SD) 120572 iteration (SD)090 265071 265001 (128) 265456 (148) 265244 (140)070 795234 795746 (338) 794557 (387) 795226 (181)050 1325440 1325180 (708) 1324310 (944) 1325390 (209)030 1855680 Fail Fail 1855540 (266)015 2253380 Fail Fail 2253280 (313)

F

F

F

F

F

F F F

F F F

F F F

F F F

F F F

F

F

F

F

F

Tritium targetDeuteron beams Aluminum sheath

PolyethyleneF Fuel

He detector

3He 1

3He 2 3He 3

(a) Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PE

F

F

F

F

F

F F F

F F F

F F F

F F F

F F F

F

F

F

F

F

Tritium targetDeuteron beams Aluminum sheath

PolyethyleneF Fuel

He detector

3He 1 3He 2

(b) Th-HEU-5PE andTh-HEU-Gr-PE

Figure 2 Th-ADS core configurations with 14MeV neutrons

3He detectors for Th-HEU-5PE and Th-HEU-Gr-PE coreshaving the 14MeV pulsed neutron generator The effectivemultiplication factors 119896eff rsquos of the Th-PE Th-Gr Th-BeTh-HEU-PE NU-PE Th-HEU-5PE and Th-HEU-Gr-PEconfigurations are reported to be 000613 000952 000765058754 050867 085121 and 035473 respectively [16]

The McCARD calculations are performed with contin-uous-energy cross section libraries produced from JENDL-40 for the seven cores with the 14MeV pulsed neutrongenerator The alpha-static calculations by the 120572 iterationalgorithm are conducted with 5000 active iterations on10000 sources per iteration For the comparison McCARDsimulations for the PNS experiments are performed on 20billion of 14MeV neutron sources Figure 3 shows flux tallyresults of the three detectors for the Th-Gr core and the twodetectors for the Th-HEU-5PE core in 1 120583s time bins from00ms to 40ms A time decay constant 120572 can be obtained byfitting the tally results to the exponential function 120601 in thetime range when the higher-mode fluxes sufficiently decay as[2]

120601 (119905) = 119860119890minus120572(119905minus119905

119904)+ 119861 (20)

where 119860 and 119861 are fitting constants and 119905 and 119905119904 are thetime after the neutron burst and the starting time of fittingrespectively Figure 4 shows the convergence of 120572 to a stableone with increasing 119905119904 for the fixed fitting intervals of 800120583sFrom these convergence diagnoses 119905119904 for the Th-PE Th-GrTh-Be Th-HEU-PE NU-PE Th-HEU-5PE and Th-HEU-Gr-PE cores is determined to be 13 16 16 17 17 16 and13ms respectively

Table 3 shows comparisons of 120572rsquos estimated by the 120572

iteration method with the measurements given in [16] andthe 120572 values from the McCARD PNS simulations which arecalculated by averaging fitted values of 3He detectors 2 and3 for the Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PEcores and 3He detectors 1 and 2 for the Th-HEU-5PE andTh-HEU-Gr-PE cores From the comparisons between the 120572values calculated by the120572 iterationmethod and theMcCARDPNS simulations we can see that the absolute values of therelative difference for the Th-PE Th-Be Th-HEU-5PE andTh-HEU-Gr-PE cores are less than 10 while being 1321 and 34 for theTh-GrTh-HEU-PE andNU-PE coresrespectively From the comparisons with measurements wecan see that the 120572 iteration results are quite comparable with

6 Science and Technology of Nuclear Installations

Table 3 Comparisons of 120572 estimates for the Th-ADS experimental benchmark

Core Measurements with 14MeV neutrons (SD) MC PNS simulation (SD) MC 120572 iteration (SD)3He 1 3He 2 3He 3

Th-PE 6642 (11) 6224 (27) 5751 (25) 52454 (49) 52433 (13)Th-Gr 6451 (12) 5945 (15) 5701 (17) 53101 (77) 52397 (13)Th-Be 6515 (8) 6111 (17) 5746 (20) 52242 (72) 52401 (14)Th-HEU-PE 5692 (11) 5275 (7) 5231 (9) 50256 (63) 51312 (28)NU-PE 5748 (11) 6592 (15) 5010 (11) 49951 (64) 51638 (35)Th-HEU-5PE 3110 (11) 3104 (10) mdash 29771 (14) 29541 (12)Th-HEU-Gr-PE 4980 (40) 4939 (50) mdash 47610 (29) 47667 (102)

0 1 2 3 4

He number 1He number 2

He number 3

Neu

tron

flux

Time (ms)

101001

1E minus 31E minus 41E minus 51E minus 61E minus 71E minus 81E minus 91E minus 101E minus 111E minus 121E minus 131E minus 141E minus 151E minus 161E minus 17

(a) Th-Gr core

He number 1He number 2

0 1 2 3 4

Neu

tron

flux

Time (ms)

1

01

001

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

1E minus 8

1E minus 9

1E minus 10

1E minus 11

1E minus 12

1E minus 13

(b) Th-HEU-5PE core

Figure 3 Time-dependent flux for the Th-ADS benchmark prob-lems

5000

5200

5400

5600

5800

6000

Starting time of fitting (ms)02 04 06 08 10 12 14 16 18

He number 1He number 2

He number 3

120572

(a) Th-Gr core

1800

2000

2200

2400

2600

2800

3000

3200

He number 1He number 2

Starting time of fitting (ms)02 04 06 08 10 12 14 16 18

120572

(b) Th-HEU-5PE core

Figure 4 Fitted 120572 values according to the starting time of fitting forthe Th-ADS benchmark problems

Science and Technology of Nuclear Installations 7

the experimental results for core configurations where twoestimates from different detectors show good agreementsthat is the Ph-HEU-PE Th-HEU-5PE and Ph-HEU-Gr-PEcores

4 Conclusion

A new MC alpha-static calculation method is developed tostably estimate the fundamental-mode alpha-eigenvalue ofprompt subcritical systems While the fission source distri-bution is updated in the existing 120572-k iteration method thenew method named as the 120572 iteration algorithm directlyupdates the time source distribution iteration-by-iterationThe calculation stability of the 120572 iteration algorithm isachieved by controlling the number of time source neutronsgenerated at each iteration by the normalization scheme forthe time source distribution

It is demonstrated that the 120572 iteration algorithm doesnot suffer from the instability problem in the two-grouphomogeneous problems with deep subcriticalities where the120572-k iteration method fails to obtain the 120572 value due to thehuge number of neutron productions From the comparisonswithmeasurements in theTh-ADS experimental benchmarks[16] it is observed that the 120572 results calculated by the 120572

iterationmethod are quite comparable with each other for thecases where the experiments provide reliable estimates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Nuclear RampDProgram through theNational Research Foundation of Korea(NRF) funded by MSIP (Ministry of Science ICT and FuturePlanning) (no 2010-0018758)

References

[1] D E Cullen C J Clouse R Procassini and R C Little ldquoStaticand dynamic criticality are they differentrdquo Report UCRL-TR-201506 Lawrence Livermore National Laboratory LivermoreCalif USA 2003

[2] T Suzaki ldquoSubcritically determination of low-enriched UO2lattices in water by exponential experimentrdquo Journal of NuclearScience and Technology vol 28 no 12 pp 1067ndash1077 1991

[3] T Yamamoto K Sakurai T Arakawa and Y Naito ldquoAccu-rate estimation of subcriticality using indirect bias estimationmethod (I) theoryrdquo Journal of Nuclear Science and Technologyvol 34 no 5 pp 454ndash460 1997

[4] D E Cullen ldquoTART 2002 a coupled neutron-photon 3-Dcombinatorial geometry time dependent Monte Carlo trans-port coderdquo Tech Rep UCRL-ID-126455 Rev 4 LawrenceLivermore National Laboratory 2002

[5] D Brockway P Soran and P Whalen ldquoMonte Carlo 120572 calcu-lationrdquo Tech Rep LA-UR-85-1224 Los Alamos National Labo-ratory 1985

[6] T Ye C Chen W Sun B Zhang and D Tian ldquoPrompt timeconstants of a reflected reactorrdquo in Proceedings of the Symposiumon Nuclear Data T Fukahori Ed Ibaraki-ken Japan January2007

[7] S D Nolen T R Adams and J E Sweezy ldquoIntegral criticalityestimators in MCATK(U)rdquo Tech Rep LA-UR-12-25458 LosAlamos National Laboratory Los Alamos NM USA 2012

[8] A Zoia E Brun and F Malvagi ldquoAlpha eigenvalue calculationswith TRIPOLI-4rdquo Annals of Nuclear Energy vol 63 pp 276ndash284 2014

[9] A Zoia E Brun and F Malvagi ldquoA Monte Carlo method forprompt and delayed alpha eigenvalue calculationsrdquo in Proceed-ings of the International Conference on the Role of Reactor PhysicsToward a Sustainable Future (PHYSOR rsquo14) CD-ROM KyotoJapan September-October 2014

[10] J Lieberoth ldquoAMonte Carlo technique to solve the static eigen-value problemof the boltzmann transport equationrdquoNukleonikvol 11 article 213 1968

[11] T Yamamoto and Y Miyoshi ldquoAn algorithm of 120572- and 120574-modeeigenvalue calculations by Monte Carlo methodrdquo in Proceed-ings of the 7th International Conference on Nuclear CriticalitySafety (ICNC rsquo03) JAERI-Conf 2003-019 Japan Atomic EnergyResearch Institute Tokai Japan October 2003

[12] T Yamamoto ldquoHigher order 120572mode eigenvalue calculation byMonte Carlo power iterationrdquo Progress in Nuclear Science andTechnology vol 2 pp 826ndash835 2011

[13] O Petit F-X Hugot Y-K Lee C Jouanne and A MazzoloTRIPOLI-4 Version 4 User Guide CEA-R-6169 CEA Saclay2008

[14] S Nakamura Computational Methods in Engineering and Sci-ence With Applications to Fluid Dynamics and Nuclear SystemsWiley-Interscience 1977

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

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

[17] G I Bell and S Glasstone Nuclear Reactor Theory VanNostrand Reinhold Company New York NY USA 1970

[18] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976

[19] I Lux andLKoblingerMonteCarlo Particle TransportMethodsNeutron and Photon Calculations CRC Press Boca Raton FlaUSA 1991

[20] J F Briesmeister ldquoMCNP A general Monte Carlo N-particletransport code version 4Brdquo Tech Rep LA-13181 Los AlamosNational Laboratory Los Alamos NM USA 1997

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

2 Science and Technology of Nuclear Installations

speed corresponding to its energy 119864 Other notations followconvention

For the alpha-static MC calculations the 120572-k iterationmethods [5ndash9] searching the fundamental mode alpha-eigenvalue in the middle of iterative fission source updatesby the MC power iteration method (k iteration methodhereafter) [10] have been mainly used since Brockway et al[5] developed an MC algorithm with considering the timeabsorption [1] of minus120572V for prompt supercritical systems (120572 lt

0) The 120572-k iteration methods for prompt subcritical systems(120572 gt 0) treat a negative value of the time absorption as thetime production (120572V) by which the time source neutrons aregenerated in the MC neutron simulations to avoid negativeabsorption reactions Yamamoto [11] suggested a neutronsimulation algorithm with correcting the neutron weight ineach track by the time production reaction However thetime production strategy of the conventional 120572-k iterationmethod [1] and Yamamotorsquos weight correction method forhighly subcritical systems are reported [6 8 12] to cause agigantic number of source neutron generations or a hugevalue of the neutron weight which leads to an abnormaltermination of theMC calculations To reduce this instabilityproblem Ye et al [6] introduced a pseudo neutron absorptionterm of minus120578120572VΦ (120578 gt 0) into both sides of (1) and itsslightly modified formulation [8] is used in Tripoli-4 [13]However an appropriate adjustment parameter 120578 is requiredto ensure a stable running without a halt in the pseudoneutron absorption approaches

To overcome this instability problem in the 120572-k iterationmethods we propose a new alpha-static MC method namedas the 120572 iteration algorithm for the prompt subcritical systemanalysis In Section 2 the 120572 iteration algorithm is derived byapplying the power method [14] for the 120572-mode eigenvalueequation of (1) with a normalization scheme to stably controlthe number of time source neutrons at each iteration Thenew 120572 iteration and the existing 120572-k iteration methods havebeen implemented in the Seoul National University MCcode McCARD [15] The effectiveness of the new methodis examined by comparing 120572rsquos calculated by the 120572 iterationand the 120572-k iteration methods with analytic solutions fortwo-group infinite homogeneous problemsThe applicabilityof the proposed method is tested for the thorium-loadedaccelerator-driven system [16] at Kyoto University CriticalAssembly (KUCA) by comparing 120572rsquos calculated by the 120572

iteration algorithm with those from McCARD dynamicsimulations as well as measurements of the PNS experi-ments

2 120572 Iteration Algorithm

21 Derivation In order to obtain a MC neutron trackingalgorithm from an integrodifferential form of the 120572-modeeigenvalue equation written by (1) it is required to transform(1) into its integral form By integrating (1) along with acharacteristic curve [17] in the sameway to derive the integralformof the neutron transport equation [17 18] and expressingthe resulting integral equation for the collision density 120595

defined by Σ119905Φ [19] one can obtain the collision densityequation for the 120572-mode eigenvalue equation

120595 (r 119864Ω) = int119889r1015840119879 (r1015840 997888rarr r | 119864Ω) 119878119905 (r1015840 119864Ω)

+ int119889r1015840 int11986410158401198891198641015840int

4120587

119889Ω1015840

sdot 119870119901 (r1015840 1198641015840Ω1015840997888rarr r 119864Ω)

sdot 120595 (r1015840 1198641015840Ω1015840)

(5)

where119870119901 is the transition kernel ignoring the delayed fissionneutrons defined as a product of the collision kernel withoutthe delayed fission neutron 119862119901 and the transport kernel 119879as follows

119870119901 (r1015840 1198641015840Ω1015840997888rarr r 119864Ω) = 119879 (r1015840 997888rarr r | 119864Ω)

sdot 119862119901 (1198641015840Ω1015840997888rarr 119864Ω | r1015840)

(6)

119862119901 (1198641015840Ω1015840997888rarr 119864Ω | r1015840)

= sum

119903 =fis

]119903Σ119903 (r1015840 1198641015840)Σ119905 (r1015840 1198641015840)

119891119903 (1198641015840Ω1015840997888rarr 119864Ω)

+

]119901Σ119891 (r1015840 1198641015840)Σ119905 (r1015840 1198641015840)

sdot

120594119901 (119864)

4120587

(7)

119879 (r1015840 997888rarr r | 119864Ω) = Σ119905 (r 119864)1003816100381610038161003816r minus r1015840100381610038161003816

1003816

2

sdot exp[minusint|rminusr1015840|

0

Σ119905 (r minus 119904r minus r10158401003816100381610038161003816r minus r1015840100381610038161003816

1003816

119864) 119889119904]

sdot 120575(Ω sdotr minus r10158401003816100381610038161003816r minus r1015840100381610038161003816

1003816

minus 1)

(8)

where 119903 is used to index the neutron reaction except fission ]119903is the average number of neutrons produced from a reactiontype 119903 and 119891119903 (119864

1015840Ω1015840rarr 119864Ω)119889119864119889Ω is the probability that

a collision of type 119903 by a neutron of direction Ω1015840 and energy1198641015840 will produce a neutron in direction interval 119889Ω about Ω

with energy in 119889119864 about 119864Then the Neumann series solution [19] to (5) which

describes the MC neutron simulations can be written by

120595 (r 119864Ω) =infin

sum

119895=0

int119889r1015840 int1198891198640 int119889Ω0

sdot 119870119901119895 (r1015840 1198640Ω0 997888rarr r 119864Ω) int119889r0

sdot 119879 (1198640Ω0 r0 997888rarr r1015840) 119878119905 (r0 1198640Ω0)

(9)

Science and Technology of Nuclear Installations 3

119870119901119895 (r1015840 1198640Ω0 997888rarr r 119864Ω) = int119889r1 int1198891198641

sdot int 119889Ω1 sdot sdot sdot int 119889r119895minus1 int119889119864119895minus1 int119889Ω119895minus1

sdot 119870119901 (r119895minus1 119864119895minus1Ω119895minus1 997888rarr r 119864Ω) sdot sdot sdot 119870119901 (r1015840 1198640Ω0

997888rarr r1 1198641Ω1)

(10)

Multiplying 120572Vminus1Σ119905minus1 on both sides of (9) one can

obtain the 120572-mode eigenvalue equation for the time sourcedistribution as

119878119905 = 120572R119878119905 (11)

R119878119905 =1

V (119864) Σ119905 (r 119864)

infin

sum

119895=0

int119889r1015840 int1198891198640 int119889Ω0

sdot 119870119901119895 (r1015840 1198640Ω0 997888rarr r 119864Ω) int119889r0

sdot 119879 (1198640Ω0 r0 997888rarr r1015840) 119878119905 (r0 1198640Ω0)

(12)

To calculate the fundamental mode alpha-eigenvaluefrom (11) we directly apply the power method [14] to (11)

119878119894

119905= 120572119894R119878119894minus1119905 (13)

120572119894=

1

int 119889rint119889119864int119889ΩR119878119894minus1119905 (14)

where 119894 is the iteration index Equation (13) implies that thefundamental-mode alpha-eigenvalue and 119878119905 are calculatedby iterative updates of the times source distribution untilthey converge It is notable that the transition kernels inoperator R defined by (12) are the same as the ones usedin the fixed source mode MC calculations except that thedelayed fission neutrons are not considered which meansthat all the prompt fission neutrons should be simulatedwithin an iteration Note that an expected number of promptfission neutrons generated within an iteration is less than 1 forprompt subcritical systems of which the prompt criticality 119896119901is less than 1

22 Application to Monte Carlo Calculations The derived 120572iteration method can be implemented by slightly modifyingthe 119896 iteration algorithm [10 20] used in existing MC codesThe main difference between the 120572 iteration and the 119896

iteration methods is that the time source distribution isupdated in the 120572 iteration algorithm while the fission sourcedistribution in the 119896 iteration oneWhen a collision occurs inthe course of the 120572 iteration at iteration 119894 (= 1 2 ) whichis governed by (12) and (13) the time source neutrons for thenext iteration can be sampled at the collision site as many as

119872119894119895119896 = lfloor120572119894minus1

sdot

119908119894119895119896

V (119864119894119895119896) Σ119905 (r119894119895119896 119864119894119895119896)+ 120585rfloor (15)

where 119895 and 119896 are indices of time source neutrons andcollisions respectively lfloor119909rfloor denotes the largest integer not

exceeding 119909 119908119894119895119896 is the neutron weight for the 119896th collisionfrom the 119895th time source neutron at iteration 119894 120585 is a uniformrandom number on the interval of (0 1) 120572119894minus1 is an alpha-eigenvalue estimated at iteration 119894 minus 1 When 119894 = 1 1205720indicates an initial 120572 value guessed by a code user When119872119894119895119896 is greater than or equal to one the location energyand direction of the collision neutron (r119894119895119896 119864119894119895119896Ω119894119895119896) areset to the sampled119872119894119895119896 sourcesrsquo parameters It is meaningfulto compare (15) with a typical fission source site samplingformulation of lfloor1119896

119894minus1

sdot((119908119894119895119896sdot]Σ119891(r119894119895119896 119864119894119895119896))Σ119905(r119894119895119896 119864119894119895119896))+120585rfloor[20] where 119896

119894minus1

denotes a 119896-eigenvalue estimated at cycle 119894minus1Note that the direction of the collision neutron Ω119894119895119896 shouldbe stored in the 120572 iteration algorithm while the directionof a fission source neutron may not be banked in the 119896

iteration algorithm because of an assumption of its isotropicdistributionThemultiplication of 120572119894minus1 in the right hand sideof (15) plays an important role in controlling the total numberof time source neutrons per iteration like the division by 119896

119894minus1

in the 119896 iteration algorithm As noted in the previous sectionthe fission reactions should be sampled in a routine of thecollision type selection from the collision kernel defined by(7) in the 120572 iteration algorithm whereas the fission reactionis not allowed to occur in a common implementation of the119896 iteration algorithm [20]

At the beginning of the next iteration 119894 + 1 the initialweight of the time source neutrons 119908119894+1 is determined fromthe number of the sampled sources at iteration 119894 by

119908119894+1 =119872

sum119895sum119896119872119894119895119896

(16)

where 119872 is a number of time source neutrons per iterationinputted by a code user

To estimate 120572119894 (119894 = 1 2 ) by (14) a collision estimatorfor an inverse of 120572

119894 that is int119889rint119889119864int119889ΩR119878119894minus1119905

sumsup 119908119894119895119896V(119864119894119895119896)Σ119905(r119894119895119896 119864119894119895119896) at each collision Then afterfinishing each iteration for the 119872 time source neutrons120572119894(119894 = 1 2 ) can be calculated from a mean of119872 history

results as

120572119894=

119872

sum119895sum119896 119908119894119895119896V (119864119894119895119896)minus1

Σ119905 (r119894119895119896 119864119894119895119896)minus1 (17)

In the same reason to apply the inactive cycle runs in the119896 iteration method the fundamental-mode alpha-eigenvalueshould be estimated by averaging 120572119894rsquos at active iterations afterproper number of inactive iterations to converge the timesource distribution

3 Numerical Results

31 Two-Group Infinite Homogeneous Problems In order toinvestigate the effectiveness of the proposed method the 120572and the 120572-k iteration algorithms implemented in McCARD[15] are tested for two-group infinite homogeneous mediumproblems Table 1 shows two-group cross sections varying theprompt criticality 119896119901 The differential scattering cross section

4 Science and Technology of Nuclear Installations

Table 1 Two-group cross sections for the infinite homogeneousproblem

Cross section First group (119892 = 1) Second group (119892 = 2)Σ119905119892 050 050Σ119891119892 0025 0175]119901119892 20 20Σ119904119892119892 010 020Σ1199041198921015840119892 (119892 = 119892

1015840) Variable 000120594119901119892 10 001V119892 [seccm] 228626 times 10minus10 129329 times 10minus6

of the first group Σ11990421 is set to 0265714 0197143 01285710060000 or 0008571 which correspond to 119896119901 of 09 07 0503 or 015

The MC 120572 calculations are performed for 1000 activeiterations on 10000 sources per iteration Table 2 showscomparisons of 120572rsquos calculated by the new algorithm andthe 120572-k iteration method applying the pseudo absorptionadjustment [6] with analytic solutions In the table thepseudo absorption adjustment parameter 120578 of zero meansthe conventional 120572-k iterationmethod [1] without the pseudoabsorption adjustment From the table one can see that theMC results from the 120572 iteration method agree well with theanalytic references within 95 confidence intervals while the120572-k iteration method fails when 119896119901 are 03 and 015 due toabnormal terminations

These abnormal terminations of the 120572-k iterationmethodcan be explained by counting the number of time sourceneutrons generated from a source neutron In the 120572-k iter-ation algorithm with the 120578 parameter a time source neutronis generated at each collision site with the probability of 119901120572119892(=(1 + 120578)120572V119892(Σ119905119892 + 120578120572V119892)) while the neutron is absorbedwith the probability of119901119886119892 (=Σ119905119892+120578120572V119892minusΣ119904119892(Σ119905119892+120578120572V119892))Considering that the time source generation is dominated bythe second-group neutron by1199011205722 ≫ 1199011205721 for a small value of 120578let us calculate the expected number of time source neutrons119873time2 during theMCneutron transport simulations startingfrom a second-group source until it is absorbed BecauseΣ11990412 = 0 as given in Table 1119873time2 can be written as

119873time2 = 1199011205722 + (1 minus 1199011198862) 1199011205722 + (1 minus 1199011198862)21199011205722 + sdot sdot sdot

=

1199011205722

1 minus (1 minus 1199011198862)

=

(1 + 120578) 120572

V2 (Σ1199052 minus Σ1199042) + 120578120572

(18)

When 119873time2 is greater than or equal to 1 the totalnumber of time source neutrons generated in a source historybecomes infinity which leads to the abnormal terminationBy inserting the analytic solution of the fundamental modealpha-eigenvalue in Table 2 and the constants given in Table 1into (18)119873time2 is found to be closer to 1 with values of 011034 057 080 and 097 when 120578 = 0 and 028 061 080092 and 099 when 120578 = 2 as the subcriticality is deeper from090 to 015 Because the 120572 estimate from the 120572-k iterationmethod is updated iteratively and fluctuated due to a finite

100000

120000

140000

160000

180000

200000

220000

240000

260000

280000

120572limit

120572-k iteration method with 120578 = 0

120572-k iteration method with 120578 = 2

120572

Iteration index0 1 2 3 4 5 6 7 8 9

Figure 1 Iteration-wise 120572 values estimated by the 120572-k iterationmethod for the two-group infinite homogeneous problem of 119896119901 =03

number of MC history simulations a stability condition canbe expressed from (18) as

120572119894

120572-119896 lt 120572limit = V2 (Σ1199052 minus Σ1199042) (19)

where 120572119894120572-119896 is the fundamental mode alpha-eigenvalue esti-

mated at the 119894th fission source iteration From Table 1120572limit for the two-group infinite homogeneous problems isdetermined to be 231967 Figure 1 shows 120572119894

120572-119896 for the cases of119896119901 of 03 with different 120578 values of 0 and 2 on 10

7 histories periteration starting from an initially guessed 120572 1205720

120572-119896 of 185568From the figure we can observe that 120572119894

120572-119896 becomes close to120572limit of 231967 at cycle 8 in the run with 120578 of 0 and at cycle6 with 120578 of 2 which yield the abnormal terminations at thevery next iteration calculations

The 120572-k iteration method with the pseudo absorptionterm may reduce occurrence of the infinite number oftime source generations by suppressing fluctuations of 120572119894

120572-119896for MC calculations with continuous cross section data asreported in [12] However it is noteworthy that the abnormalterminations in the 120572-k iteration calculations can happen byinevitable statistical fluctuations of 120572 estimates

32 Application to the Th-ADS Experimental BenchmarksThe proposed 120572 iteration method is applied for the exper-imental benchmarks on thorium-loaded accelerator-drivensystem (Th-ADS) [16] The Th-ADS experiments were per-formed using the solid-moderated and solid-reflected type-A core of KUCA for seven core configurations combinedwith a 14MeV pulsed neutron generator or a synchrotrontype proton acceleratorThehighly enriched uranium (HEU)thorium (Th) and natural uranium (NU) fuel was loadedtogether with the reflectors including polyethylene (PE)graphite (Gr) and beryllium (Be) Figure 2 shows a coreconfiguration with three 3He detectors used for Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PE cores and one with two

Science and Technology of Nuclear Installations 5

Table 2 Comparisons of 120572 estimates for the infinite homogeneous problem

119896119901 Ref 120572 120572-119896 iteration with 120578 = 0 (SD) 120572-119896 iteration with 120578 = 2 (SD) 120572 iteration (SD)090 265071 265001 (128) 265456 (148) 265244 (140)070 795234 795746 (338) 794557 (387) 795226 (181)050 1325440 1325180 (708) 1324310 (944) 1325390 (209)030 1855680 Fail Fail 1855540 (266)015 2253380 Fail Fail 2253280 (313)

F

F

F

F

F

F F F

F F F

F F F

F F F

F F F

F

F

F

F

F

Tritium targetDeuteron beams Aluminum sheath

PolyethyleneF Fuel

He detector

3He 1

3He 2 3He 3

(a) Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PE

F

F

F

F

F

F F F

F F F

F F F

F F F

F F F

F

F

F

F

F

Tritium targetDeuteron beams Aluminum sheath

PolyethyleneF Fuel

He detector

3He 1 3He 2

(b) Th-HEU-5PE andTh-HEU-Gr-PE

Figure 2 Th-ADS core configurations with 14MeV neutrons

3He detectors for Th-HEU-5PE and Th-HEU-Gr-PE coreshaving the 14MeV pulsed neutron generator The effectivemultiplication factors 119896eff rsquos of the Th-PE Th-Gr Th-BeTh-HEU-PE NU-PE Th-HEU-5PE and Th-HEU-Gr-PEconfigurations are reported to be 000613 000952 000765058754 050867 085121 and 035473 respectively [16]

The McCARD calculations are performed with contin-uous-energy cross section libraries produced from JENDL-40 for the seven cores with the 14MeV pulsed neutrongenerator The alpha-static calculations by the 120572 iterationalgorithm are conducted with 5000 active iterations on10000 sources per iteration For the comparison McCARDsimulations for the PNS experiments are performed on 20billion of 14MeV neutron sources Figure 3 shows flux tallyresults of the three detectors for the Th-Gr core and the twodetectors for the Th-HEU-5PE core in 1 120583s time bins from00ms to 40ms A time decay constant 120572 can be obtained byfitting the tally results to the exponential function 120601 in thetime range when the higher-mode fluxes sufficiently decay as[2]

120601 (119905) = 119860119890minus120572(119905minus119905

119904)+ 119861 (20)

where 119860 and 119861 are fitting constants and 119905 and 119905119904 are thetime after the neutron burst and the starting time of fittingrespectively Figure 4 shows the convergence of 120572 to a stableone with increasing 119905119904 for the fixed fitting intervals of 800120583sFrom these convergence diagnoses 119905119904 for the Th-PE Th-GrTh-Be Th-HEU-PE NU-PE Th-HEU-5PE and Th-HEU-Gr-PE cores is determined to be 13 16 16 17 17 16 and13ms respectively

Table 3 shows comparisons of 120572rsquos estimated by the 120572

iteration method with the measurements given in [16] andthe 120572 values from the McCARD PNS simulations which arecalculated by averaging fitted values of 3He detectors 2 and3 for the Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PEcores and 3He detectors 1 and 2 for the Th-HEU-5PE andTh-HEU-Gr-PE cores From the comparisons between the 120572values calculated by the120572 iterationmethod and theMcCARDPNS simulations we can see that the absolute values of therelative difference for the Th-PE Th-Be Th-HEU-5PE andTh-HEU-Gr-PE cores are less than 10 while being 1321 and 34 for theTh-GrTh-HEU-PE andNU-PE coresrespectively From the comparisons with measurements wecan see that the 120572 iteration results are quite comparable with

6 Science and Technology of Nuclear Installations

Table 3 Comparisons of 120572 estimates for the Th-ADS experimental benchmark

Core Measurements with 14MeV neutrons (SD) MC PNS simulation (SD) MC 120572 iteration (SD)3He 1 3He 2 3He 3

Th-PE 6642 (11) 6224 (27) 5751 (25) 52454 (49) 52433 (13)Th-Gr 6451 (12) 5945 (15) 5701 (17) 53101 (77) 52397 (13)Th-Be 6515 (8) 6111 (17) 5746 (20) 52242 (72) 52401 (14)Th-HEU-PE 5692 (11) 5275 (7) 5231 (9) 50256 (63) 51312 (28)NU-PE 5748 (11) 6592 (15) 5010 (11) 49951 (64) 51638 (35)Th-HEU-5PE 3110 (11) 3104 (10) mdash 29771 (14) 29541 (12)Th-HEU-Gr-PE 4980 (40) 4939 (50) mdash 47610 (29) 47667 (102)

0 1 2 3 4

He number 1He number 2

He number 3

Neu

tron

flux

Time (ms)

101001

1E minus 31E minus 41E minus 51E minus 61E minus 71E minus 81E minus 91E minus 101E minus 111E minus 121E minus 131E minus 141E minus 151E minus 161E minus 17

(a) Th-Gr core

He number 1He number 2

0 1 2 3 4

Neu

tron

flux

Time (ms)

1

01

001

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

1E minus 8

1E minus 9

1E minus 10

1E minus 11

1E minus 12

1E minus 13

(b) Th-HEU-5PE core

Figure 3 Time-dependent flux for the Th-ADS benchmark prob-lems

5000

5200

5400

5600

5800

6000

Starting time of fitting (ms)02 04 06 08 10 12 14 16 18

He number 1He number 2

He number 3

120572

(a) Th-Gr core

1800

2000

2200

2400

2600

2800

3000

3200

He number 1He number 2

Starting time of fitting (ms)02 04 06 08 10 12 14 16 18

120572

(b) Th-HEU-5PE core

Figure 4 Fitted 120572 values according to the starting time of fitting forthe Th-ADS benchmark problems

Science and Technology of Nuclear Installations 7

the experimental results for core configurations where twoestimates from different detectors show good agreementsthat is the Ph-HEU-PE Th-HEU-5PE and Ph-HEU-Gr-PEcores

4 Conclusion

A new MC alpha-static calculation method is developed tostably estimate the fundamental-mode alpha-eigenvalue ofprompt subcritical systems While the fission source distri-bution is updated in the existing 120572-k iteration method thenew method named as the 120572 iteration algorithm directlyupdates the time source distribution iteration-by-iterationThe calculation stability of the 120572 iteration algorithm isachieved by controlling the number of time source neutronsgenerated at each iteration by the normalization scheme forthe time source distribution

It is demonstrated that the 120572 iteration algorithm doesnot suffer from the instability problem in the two-grouphomogeneous problems with deep subcriticalities where the120572-k iteration method fails to obtain the 120572 value due to thehuge number of neutron productions From the comparisonswithmeasurements in theTh-ADS experimental benchmarks[16] it is observed that the 120572 results calculated by the 120572

iterationmethod are quite comparable with each other for thecases where the experiments provide reliable estimates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Nuclear RampDProgram through theNational Research Foundation of Korea(NRF) funded by MSIP (Ministry of Science ICT and FuturePlanning) (no 2010-0018758)

References

[1] D E Cullen C J Clouse R Procassini and R C Little ldquoStaticand dynamic criticality are they differentrdquo Report UCRL-TR-201506 Lawrence Livermore National Laboratory LivermoreCalif USA 2003

[2] T Suzaki ldquoSubcritically determination of low-enriched UO2lattices in water by exponential experimentrdquo Journal of NuclearScience and Technology vol 28 no 12 pp 1067ndash1077 1991

[3] T Yamamoto K Sakurai T Arakawa and Y Naito ldquoAccu-rate estimation of subcriticality using indirect bias estimationmethod (I) theoryrdquo Journal of Nuclear Science and Technologyvol 34 no 5 pp 454ndash460 1997

[4] D E Cullen ldquoTART 2002 a coupled neutron-photon 3-Dcombinatorial geometry time dependent Monte Carlo trans-port coderdquo Tech Rep UCRL-ID-126455 Rev 4 LawrenceLivermore National Laboratory 2002

[5] D Brockway P Soran and P Whalen ldquoMonte Carlo 120572 calcu-lationrdquo Tech Rep LA-UR-85-1224 Los Alamos National Labo-ratory 1985

[6] T Ye C Chen W Sun B Zhang and D Tian ldquoPrompt timeconstants of a reflected reactorrdquo in Proceedings of the Symposiumon Nuclear Data T Fukahori Ed Ibaraki-ken Japan January2007

[7] S D Nolen T R Adams and J E Sweezy ldquoIntegral criticalityestimators in MCATK(U)rdquo Tech Rep LA-UR-12-25458 LosAlamos National Laboratory Los Alamos NM USA 2012

[8] A Zoia E Brun and F Malvagi ldquoAlpha eigenvalue calculationswith TRIPOLI-4rdquo Annals of Nuclear Energy vol 63 pp 276ndash284 2014

[9] A Zoia E Brun and F Malvagi ldquoA Monte Carlo method forprompt and delayed alpha eigenvalue calculationsrdquo in Proceed-ings of the International Conference on the Role of Reactor PhysicsToward a Sustainable Future (PHYSOR rsquo14) CD-ROM KyotoJapan September-October 2014

[10] J Lieberoth ldquoAMonte Carlo technique to solve the static eigen-value problemof the boltzmann transport equationrdquoNukleonikvol 11 article 213 1968

[11] T Yamamoto and Y Miyoshi ldquoAn algorithm of 120572- and 120574-modeeigenvalue calculations by Monte Carlo methodrdquo in Proceed-ings of the 7th International Conference on Nuclear CriticalitySafety (ICNC rsquo03) JAERI-Conf 2003-019 Japan Atomic EnergyResearch Institute Tokai Japan October 2003

[12] T Yamamoto ldquoHigher order 120572mode eigenvalue calculation byMonte Carlo power iterationrdquo Progress in Nuclear Science andTechnology vol 2 pp 826ndash835 2011

[13] O Petit F-X Hugot Y-K Lee C Jouanne and A MazzoloTRIPOLI-4 Version 4 User Guide CEA-R-6169 CEA Saclay2008

[14] S Nakamura Computational Methods in Engineering and Sci-ence With Applications to Fluid Dynamics and Nuclear SystemsWiley-Interscience 1977

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

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

[17] G I Bell and S Glasstone Nuclear Reactor Theory VanNostrand Reinhold Company New York NY USA 1970

[18] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976

[19] I Lux andLKoblingerMonteCarlo Particle TransportMethodsNeutron and Photon Calculations CRC Press Boca Raton FlaUSA 1991

[20] J F Briesmeister ldquoMCNP A general Monte Carlo N-particletransport code version 4Brdquo Tech Rep LA-13181 Los AlamosNational Laboratory Los Alamos NM USA 1997

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Science and Technology of Nuclear Installations 3

119870119901119895 (r1015840 1198640Ω0 997888rarr r 119864Ω) = int119889r1 int1198891198641

sdot int 119889Ω1 sdot sdot sdot int 119889r119895minus1 int119889119864119895minus1 int119889Ω119895minus1

sdot 119870119901 (r119895minus1 119864119895minus1Ω119895minus1 997888rarr r 119864Ω) sdot sdot sdot 119870119901 (r1015840 1198640Ω0

997888rarr r1 1198641Ω1)

(10)

Multiplying 120572Vminus1Σ119905minus1 on both sides of (9) one can

obtain the 120572-mode eigenvalue equation for the time sourcedistribution as

119878119905 = 120572R119878119905 (11)

R119878119905 =1

V (119864) Σ119905 (r 119864)

infin

sum

119895=0

int119889r1015840 int1198891198640 int119889Ω0

sdot 119870119901119895 (r1015840 1198640Ω0 997888rarr r 119864Ω) int119889r0

sdot 119879 (1198640Ω0 r0 997888rarr r1015840) 119878119905 (r0 1198640Ω0)

(12)

To calculate the fundamental mode alpha-eigenvaluefrom (11) we directly apply the power method [14] to (11)

119878119894

119905= 120572119894R119878119894minus1119905 (13)

120572119894=

1

int 119889rint119889119864int119889ΩR119878119894minus1119905 (14)

where 119894 is the iteration index Equation (13) implies that thefundamental-mode alpha-eigenvalue and 119878119905 are calculatedby iterative updates of the times source distribution untilthey converge It is notable that the transition kernels inoperator R defined by (12) are the same as the ones usedin the fixed source mode MC calculations except that thedelayed fission neutrons are not considered which meansthat all the prompt fission neutrons should be simulatedwithin an iteration Note that an expected number of promptfission neutrons generated within an iteration is less than 1 forprompt subcritical systems of which the prompt criticality 119896119901is less than 1

22 Application to Monte Carlo Calculations The derived 120572iteration method can be implemented by slightly modifyingthe 119896 iteration algorithm [10 20] used in existing MC codesThe main difference between the 120572 iteration and the 119896

iteration methods is that the time source distribution isupdated in the 120572 iteration algorithm while the fission sourcedistribution in the 119896 iteration oneWhen a collision occurs inthe course of the 120572 iteration at iteration 119894 (= 1 2 ) whichis governed by (12) and (13) the time source neutrons for thenext iteration can be sampled at the collision site as many as

119872119894119895119896 = lfloor120572119894minus1

sdot

119908119894119895119896

V (119864119894119895119896) Σ119905 (r119894119895119896 119864119894119895119896)+ 120585rfloor (15)

where 119895 and 119896 are indices of time source neutrons andcollisions respectively lfloor119909rfloor denotes the largest integer not

exceeding 119909 119908119894119895119896 is the neutron weight for the 119896th collisionfrom the 119895th time source neutron at iteration 119894 120585 is a uniformrandom number on the interval of (0 1) 120572119894minus1 is an alpha-eigenvalue estimated at iteration 119894 minus 1 When 119894 = 1 1205720indicates an initial 120572 value guessed by a code user When119872119894119895119896 is greater than or equal to one the location energyand direction of the collision neutron (r119894119895119896 119864119894119895119896Ω119894119895119896) areset to the sampled119872119894119895119896 sourcesrsquo parameters It is meaningfulto compare (15) with a typical fission source site samplingformulation of lfloor1119896

119894minus1

sdot((119908119894119895119896sdot]Σ119891(r119894119895119896 119864119894119895119896))Σ119905(r119894119895119896 119864119894119895119896))+120585rfloor[20] where 119896

119894minus1

denotes a 119896-eigenvalue estimated at cycle 119894minus1Note that the direction of the collision neutron Ω119894119895119896 shouldbe stored in the 120572 iteration algorithm while the directionof a fission source neutron may not be banked in the 119896

iteration algorithm because of an assumption of its isotropicdistributionThemultiplication of 120572119894minus1 in the right hand sideof (15) plays an important role in controlling the total numberof time source neutrons per iteration like the division by 119896

119894minus1

in the 119896 iteration algorithm As noted in the previous sectionthe fission reactions should be sampled in a routine of thecollision type selection from the collision kernel defined by(7) in the 120572 iteration algorithm whereas the fission reactionis not allowed to occur in a common implementation of the119896 iteration algorithm [20]

At the beginning of the next iteration 119894 + 1 the initialweight of the time source neutrons 119908119894+1 is determined fromthe number of the sampled sources at iteration 119894 by

119908119894+1 =119872

sum119895sum119896119872119894119895119896

(16)

where 119872 is a number of time source neutrons per iterationinputted by a code user

To estimate 120572119894 (119894 = 1 2 ) by (14) a collision estimatorfor an inverse of 120572

119894 that is int119889rint119889119864int119889ΩR119878119894minus1119905

sumsup 119908119894119895119896V(119864119894119895119896)Σ119905(r119894119895119896 119864119894119895119896) at each collision Then afterfinishing each iteration for the 119872 time source neutrons120572119894(119894 = 1 2 ) can be calculated from a mean of119872 history

results as

120572119894=

119872

sum119895sum119896 119908119894119895119896V (119864119894119895119896)minus1

Σ119905 (r119894119895119896 119864119894119895119896)minus1 (17)

In the same reason to apply the inactive cycle runs in the119896 iteration method the fundamental-mode alpha-eigenvalueshould be estimated by averaging 120572119894rsquos at active iterations afterproper number of inactive iterations to converge the timesource distribution

3 Numerical Results

31 Two-Group Infinite Homogeneous Problems In order toinvestigate the effectiveness of the proposed method the 120572and the 120572-k iteration algorithms implemented in McCARD[15] are tested for two-group infinite homogeneous mediumproblems Table 1 shows two-group cross sections varying theprompt criticality 119896119901 The differential scattering cross section

4 Science and Technology of Nuclear Installations

Table 1 Two-group cross sections for the infinite homogeneousproblem

Cross section First group (119892 = 1) Second group (119892 = 2)Σ119905119892 050 050Σ119891119892 0025 0175]119901119892 20 20Σ119904119892119892 010 020Σ1199041198921015840119892 (119892 = 119892

1015840) Variable 000120594119901119892 10 001V119892 [seccm] 228626 times 10minus10 129329 times 10minus6

of the first group Σ11990421 is set to 0265714 0197143 01285710060000 or 0008571 which correspond to 119896119901 of 09 07 0503 or 015

The MC 120572 calculations are performed for 1000 activeiterations on 10000 sources per iteration Table 2 showscomparisons of 120572rsquos calculated by the new algorithm andthe 120572-k iteration method applying the pseudo absorptionadjustment [6] with analytic solutions In the table thepseudo absorption adjustment parameter 120578 of zero meansthe conventional 120572-k iterationmethod [1] without the pseudoabsorption adjustment From the table one can see that theMC results from the 120572 iteration method agree well with theanalytic references within 95 confidence intervals while the120572-k iteration method fails when 119896119901 are 03 and 015 due toabnormal terminations

These abnormal terminations of the 120572-k iterationmethodcan be explained by counting the number of time sourceneutrons generated from a source neutron In the 120572-k iter-ation algorithm with the 120578 parameter a time source neutronis generated at each collision site with the probability of 119901120572119892(=(1 + 120578)120572V119892(Σ119905119892 + 120578120572V119892)) while the neutron is absorbedwith the probability of119901119886119892 (=Σ119905119892+120578120572V119892minusΣ119904119892(Σ119905119892+120578120572V119892))Considering that the time source generation is dominated bythe second-group neutron by1199011205722 ≫ 1199011205721 for a small value of 120578let us calculate the expected number of time source neutrons119873time2 during theMCneutron transport simulations startingfrom a second-group source until it is absorbed BecauseΣ11990412 = 0 as given in Table 1119873time2 can be written as

119873time2 = 1199011205722 + (1 minus 1199011198862) 1199011205722 + (1 minus 1199011198862)21199011205722 + sdot sdot sdot

=

1199011205722

1 minus (1 minus 1199011198862)

=

(1 + 120578) 120572

V2 (Σ1199052 minus Σ1199042) + 120578120572

(18)

When 119873time2 is greater than or equal to 1 the totalnumber of time source neutrons generated in a source historybecomes infinity which leads to the abnormal terminationBy inserting the analytic solution of the fundamental modealpha-eigenvalue in Table 2 and the constants given in Table 1into (18)119873time2 is found to be closer to 1 with values of 011034 057 080 and 097 when 120578 = 0 and 028 061 080092 and 099 when 120578 = 2 as the subcriticality is deeper from090 to 015 Because the 120572 estimate from the 120572-k iterationmethod is updated iteratively and fluctuated due to a finite

100000

120000

140000

160000

180000

200000

220000

240000

260000

280000

120572limit

120572-k iteration method with 120578 = 0

120572-k iteration method with 120578 = 2

120572

Iteration index0 1 2 3 4 5 6 7 8 9

Figure 1 Iteration-wise 120572 values estimated by the 120572-k iterationmethod for the two-group infinite homogeneous problem of 119896119901 =03

number of MC history simulations a stability condition canbe expressed from (18) as

120572119894

120572-119896 lt 120572limit = V2 (Σ1199052 minus Σ1199042) (19)

where 120572119894120572-119896 is the fundamental mode alpha-eigenvalue esti-

mated at the 119894th fission source iteration From Table 1120572limit for the two-group infinite homogeneous problems isdetermined to be 231967 Figure 1 shows 120572119894

120572-119896 for the cases of119896119901 of 03 with different 120578 values of 0 and 2 on 10

7 histories periteration starting from an initially guessed 120572 1205720

120572-119896 of 185568From the figure we can observe that 120572119894

120572-119896 becomes close to120572limit of 231967 at cycle 8 in the run with 120578 of 0 and at cycle6 with 120578 of 2 which yield the abnormal terminations at thevery next iteration calculations

The 120572-k iteration method with the pseudo absorptionterm may reduce occurrence of the infinite number oftime source generations by suppressing fluctuations of 120572119894

120572-119896for MC calculations with continuous cross section data asreported in [12] However it is noteworthy that the abnormalterminations in the 120572-k iteration calculations can happen byinevitable statistical fluctuations of 120572 estimates

32 Application to the Th-ADS Experimental BenchmarksThe proposed 120572 iteration method is applied for the exper-imental benchmarks on thorium-loaded accelerator-drivensystem (Th-ADS) [16] The Th-ADS experiments were per-formed using the solid-moderated and solid-reflected type-A core of KUCA for seven core configurations combinedwith a 14MeV pulsed neutron generator or a synchrotrontype proton acceleratorThehighly enriched uranium (HEU)thorium (Th) and natural uranium (NU) fuel was loadedtogether with the reflectors including polyethylene (PE)graphite (Gr) and beryllium (Be) Figure 2 shows a coreconfiguration with three 3He detectors used for Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PE cores and one with two

Science and Technology of Nuclear Installations 5

Table 2 Comparisons of 120572 estimates for the infinite homogeneous problem

119896119901 Ref 120572 120572-119896 iteration with 120578 = 0 (SD) 120572-119896 iteration with 120578 = 2 (SD) 120572 iteration (SD)090 265071 265001 (128) 265456 (148) 265244 (140)070 795234 795746 (338) 794557 (387) 795226 (181)050 1325440 1325180 (708) 1324310 (944) 1325390 (209)030 1855680 Fail Fail 1855540 (266)015 2253380 Fail Fail 2253280 (313)

F

F

F

F

F

F F F

F F F

F F F

F F F

F F F

F

F

F

F

F

Tritium targetDeuteron beams Aluminum sheath

PolyethyleneF Fuel

He detector

3He 1

3He 2 3He 3

(a) Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PE

F

F

F

F

F

F F F

F F F

F F F

F F F

F F F

F

F

F

F

F

Tritium targetDeuteron beams Aluminum sheath

PolyethyleneF Fuel

He detector

3He 1 3He 2

(b) Th-HEU-5PE andTh-HEU-Gr-PE

Figure 2 Th-ADS core configurations with 14MeV neutrons

3He detectors for Th-HEU-5PE and Th-HEU-Gr-PE coreshaving the 14MeV pulsed neutron generator The effectivemultiplication factors 119896eff rsquos of the Th-PE Th-Gr Th-BeTh-HEU-PE NU-PE Th-HEU-5PE and Th-HEU-Gr-PEconfigurations are reported to be 000613 000952 000765058754 050867 085121 and 035473 respectively [16]

The McCARD calculations are performed with contin-uous-energy cross section libraries produced from JENDL-40 for the seven cores with the 14MeV pulsed neutrongenerator The alpha-static calculations by the 120572 iterationalgorithm are conducted with 5000 active iterations on10000 sources per iteration For the comparison McCARDsimulations for the PNS experiments are performed on 20billion of 14MeV neutron sources Figure 3 shows flux tallyresults of the three detectors for the Th-Gr core and the twodetectors for the Th-HEU-5PE core in 1 120583s time bins from00ms to 40ms A time decay constant 120572 can be obtained byfitting the tally results to the exponential function 120601 in thetime range when the higher-mode fluxes sufficiently decay as[2]

120601 (119905) = 119860119890minus120572(119905minus119905

119904)+ 119861 (20)

where 119860 and 119861 are fitting constants and 119905 and 119905119904 are thetime after the neutron burst and the starting time of fittingrespectively Figure 4 shows the convergence of 120572 to a stableone with increasing 119905119904 for the fixed fitting intervals of 800120583sFrom these convergence diagnoses 119905119904 for the Th-PE Th-GrTh-Be Th-HEU-PE NU-PE Th-HEU-5PE and Th-HEU-Gr-PE cores is determined to be 13 16 16 17 17 16 and13ms respectively

Table 3 shows comparisons of 120572rsquos estimated by the 120572

iteration method with the measurements given in [16] andthe 120572 values from the McCARD PNS simulations which arecalculated by averaging fitted values of 3He detectors 2 and3 for the Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PEcores and 3He detectors 1 and 2 for the Th-HEU-5PE andTh-HEU-Gr-PE cores From the comparisons between the 120572values calculated by the120572 iterationmethod and theMcCARDPNS simulations we can see that the absolute values of therelative difference for the Th-PE Th-Be Th-HEU-5PE andTh-HEU-Gr-PE cores are less than 10 while being 1321 and 34 for theTh-GrTh-HEU-PE andNU-PE coresrespectively From the comparisons with measurements wecan see that the 120572 iteration results are quite comparable with

6 Science and Technology of Nuclear Installations

Table 3 Comparisons of 120572 estimates for the Th-ADS experimental benchmark

Core Measurements with 14MeV neutrons (SD) MC PNS simulation (SD) MC 120572 iteration (SD)3He 1 3He 2 3He 3

Th-PE 6642 (11) 6224 (27) 5751 (25) 52454 (49) 52433 (13)Th-Gr 6451 (12) 5945 (15) 5701 (17) 53101 (77) 52397 (13)Th-Be 6515 (8) 6111 (17) 5746 (20) 52242 (72) 52401 (14)Th-HEU-PE 5692 (11) 5275 (7) 5231 (9) 50256 (63) 51312 (28)NU-PE 5748 (11) 6592 (15) 5010 (11) 49951 (64) 51638 (35)Th-HEU-5PE 3110 (11) 3104 (10) mdash 29771 (14) 29541 (12)Th-HEU-Gr-PE 4980 (40) 4939 (50) mdash 47610 (29) 47667 (102)

0 1 2 3 4

He number 1He number 2

He number 3

Neu

tron

flux

Time (ms)

101001

1E minus 31E minus 41E minus 51E minus 61E minus 71E minus 81E minus 91E minus 101E minus 111E minus 121E minus 131E minus 141E minus 151E minus 161E minus 17

(a) Th-Gr core

He number 1He number 2

0 1 2 3 4

Neu

tron

flux

Time (ms)

1

01

001

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

1E minus 8

1E minus 9

1E minus 10

1E minus 11

1E minus 12

1E minus 13

(b) Th-HEU-5PE core

Figure 3 Time-dependent flux for the Th-ADS benchmark prob-lems

5000

5200

5400

5600

5800

6000

Starting time of fitting (ms)02 04 06 08 10 12 14 16 18

He number 1He number 2

He number 3

120572

(a) Th-Gr core

1800

2000

2200

2400

2600

2800

3000

3200

He number 1He number 2

Starting time of fitting (ms)02 04 06 08 10 12 14 16 18

120572

(b) Th-HEU-5PE core

Figure 4 Fitted 120572 values according to the starting time of fitting forthe Th-ADS benchmark problems

Science and Technology of Nuclear Installations 7

the experimental results for core configurations where twoestimates from different detectors show good agreementsthat is the Ph-HEU-PE Th-HEU-5PE and Ph-HEU-Gr-PEcores

4 Conclusion

A new MC alpha-static calculation method is developed tostably estimate the fundamental-mode alpha-eigenvalue ofprompt subcritical systems While the fission source distri-bution is updated in the existing 120572-k iteration method thenew method named as the 120572 iteration algorithm directlyupdates the time source distribution iteration-by-iterationThe calculation stability of the 120572 iteration algorithm isachieved by controlling the number of time source neutronsgenerated at each iteration by the normalization scheme forthe time source distribution

It is demonstrated that the 120572 iteration algorithm doesnot suffer from the instability problem in the two-grouphomogeneous problems with deep subcriticalities where the120572-k iteration method fails to obtain the 120572 value due to thehuge number of neutron productions From the comparisonswithmeasurements in theTh-ADS experimental benchmarks[16] it is observed that the 120572 results calculated by the 120572

iterationmethod are quite comparable with each other for thecases where the experiments provide reliable estimates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Nuclear RampDProgram through theNational Research Foundation of Korea(NRF) funded by MSIP (Ministry of Science ICT and FuturePlanning) (no 2010-0018758)

References

[1] D E Cullen C J Clouse R Procassini and R C Little ldquoStaticand dynamic criticality are they differentrdquo Report UCRL-TR-201506 Lawrence Livermore National Laboratory LivermoreCalif USA 2003

[2] T Suzaki ldquoSubcritically determination of low-enriched UO2lattices in water by exponential experimentrdquo Journal of NuclearScience and Technology vol 28 no 12 pp 1067ndash1077 1991

[3] T Yamamoto K Sakurai T Arakawa and Y Naito ldquoAccu-rate estimation of subcriticality using indirect bias estimationmethod (I) theoryrdquo Journal of Nuclear Science and Technologyvol 34 no 5 pp 454ndash460 1997

[4] D E Cullen ldquoTART 2002 a coupled neutron-photon 3-Dcombinatorial geometry time dependent Monte Carlo trans-port coderdquo Tech Rep UCRL-ID-126455 Rev 4 LawrenceLivermore National Laboratory 2002

[5] D Brockway P Soran and P Whalen ldquoMonte Carlo 120572 calcu-lationrdquo Tech Rep LA-UR-85-1224 Los Alamos National Labo-ratory 1985

[6] T Ye C Chen W Sun B Zhang and D Tian ldquoPrompt timeconstants of a reflected reactorrdquo in Proceedings of the Symposiumon Nuclear Data T Fukahori Ed Ibaraki-ken Japan January2007

[7] S D Nolen T R Adams and J E Sweezy ldquoIntegral criticalityestimators in MCATK(U)rdquo Tech Rep LA-UR-12-25458 LosAlamos National Laboratory Los Alamos NM USA 2012

[8] A Zoia E Brun and F Malvagi ldquoAlpha eigenvalue calculationswith TRIPOLI-4rdquo Annals of Nuclear Energy vol 63 pp 276ndash284 2014

[9] A Zoia E Brun and F Malvagi ldquoA Monte Carlo method forprompt and delayed alpha eigenvalue calculationsrdquo in Proceed-ings of the International Conference on the Role of Reactor PhysicsToward a Sustainable Future (PHYSOR rsquo14) CD-ROM KyotoJapan September-October 2014

[10] J Lieberoth ldquoAMonte Carlo technique to solve the static eigen-value problemof the boltzmann transport equationrdquoNukleonikvol 11 article 213 1968

[11] T Yamamoto and Y Miyoshi ldquoAn algorithm of 120572- and 120574-modeeigenvalue calculations by Monte Carlo methodrdquo in Proceed-ings of the 7th International Conference on Nuclear CriticalitySafety (ICNC rsquo03) JAERI-Conf 2003-019 Japan Atomic EnergyResearch Institute Tokai Japan October 2003

[12] T Yamamoto ldquoHigher order 120572mode eigenvalue calculation byMonte Carlo power iterationrdquo Progress in Nuclear Science andTechnology vol 2 pp 826ndash835 2011

[13] O Petit F-X Hugot Y-K Lee C Jouanne and A MazzoloTRIPOLI-4 Version 4 User Guide CEA-R-6169 CEA Saclay2008

[14] S Nakamura Computational Methods in Engineering and Sci-ence With Applications to Fluid Dynamics and Nuclear SystemsWiley-Interscience 1977

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

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

[17] G I Bell and S Glasstone Nuclear Reactor Theory VanNostrand Reinhold Company New York NY USA 1970

[18] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976

[19] I Lux andLKoblingerMonteCarlo Particle TransportMethodsNeutron and Photon Calculations CRC Press Boca Raton FlaUSA 1991

[20] J F Briesmeister ldquoMCNP A general Monte Carlo N-particletransport code version 4Brdquo Tech Rep LA-13181 Los AlamosNational Laboratory Los Alamos NM USA 1997

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

4 Science and Technology of Nuclear Installations

Table 1 Two-group cross sections for the infinite homogeneousproblem

Cross section First group (119892 = 1) Second group (119892 = 2)Σ119905119892 050 050Σ119891119892 0025 0175]119901119892 20 20Σ119904119892119892 010 020Σ1199041198921015840119892 (119892 = 119892

1015840) Variable 000120594119901119892 10 001V119892 [seccm] 228626 times 10minus10 129329 times 10minus6

of the first group Σ11990421 is set to 0265714 0197143 01285710060000 or 0008571 which correspond to 119896119901 of 09 07 0503 or 015

The MC 120572 calculations are performed for 1000 activeiterations on 10000 sources per iteration Table 2 showscomparisons of 120572rsquos calculated by the new algorithm andthe 120572-k iteration method applying the pseudo absorptionadjustment [6] with analytic solutions In the table thepseudo absorption adjustment parameter 120578 of zero meansthe conventional 120572-k iterationmethod [1] without the pseudoabsorption adjustment From the table one can see that theMC results from the 120572 iteration method agree well with theanalytic references within 95 confidence intervals while the120572-k iteration method fails when 119896119901 are 03 and 015 due toabnormal terminations

These abnormal terminations of the 120572-k iterationmethodcan be explained by counting the number of time sourceneutrons generated from a source neutron In the 120572-k iter-ation algorithm with the 120578 parameter a time source neutronis generated at each collision site with the probability of 119901120572119892(=(1 + 120578)120572V119892(Σ119905119892 + 120578120572V119892)) while the neutron is absorbedwith the probability of119901119886119892 (=Σ119905119892+120578120572V119892minusΣ119904119892(Σ119905119892+120578120572V119892))Considering that the time source generation is dominated bythe second-group neutron by1199011205722 ≫ 1199011205721 for a small value of 120578let us calculate the expected number of time source neutrons119873time2 during theMCneutron transport simulations startingfrom a second-group source until it is absorbed BecauseΣ11990412 = 0 as given in Table 1119873time2 can be written as

119873time2 = 1199011205722 + (1 minus 1199011198862) 1199011205722 + (1 minus 1199011198862)21199011205722 + sdot sdot sdot

=

1199011205722

1 minus (1 minus 1199011198862)

=

(1 + 120578) 120572

V2 (Σ1199052 minus Σ1199042) + 120578120572

(18)

When 119873time2 is greater than or equal to 1 the totalnumber of time source neutrons generated in a source historybecomes infinity which leads to the abnormal terminationBy inserting the analytic solution of the fundamental modealpha-eigenvalue in Table 2 and the constants given in Table 1into (18)119873time2 is found to be closer to 1 with values of 011034 057 080 and 097 when 120578 = 0 and 028 061 080092 and 099 when 120578 = 2 as the subcriticality is deeper from090 to 015 Because the 120572 estimate from the 120572-k iterationmethod is updated iteratively and fluctuated due to a finite

100000

120000

140000

160000

180000

200000

220000

240000

260000

280000

120572limit

120572-k iteration method with 120578 = 0

120572-k iteration method with 120578 = 2

120572

Iteration index0 1 2 3 4 5 6 7 8 9

Figure 1 Iteration-wise 120572 values estimated by the 120572-k iterationmethod for the two-group infinite homogeneous problem of 119896119901 =03

number of MC history simulations a stability condition canbe expressed from (18) as

120572119894

120572-119896 lt 120572limit = V2 (Σ1199052 minus Σ1199042) (19)

where 120572119894120572-119896 is the fundamental mode alpha-eigenvalue esti-

mated at the 119894th fission source iteration From Table 1120572limit for the two-group infinite homogeneous problems isdetermined to be 231967 Figure 1 shows 120572119894

120572-119896 for the cases of119896119901 of 03 with different 120578 values of 0 and 2 on 10

7 histories periteration starting from an initially guessed 120572 1205720

120572-119896 of 185568From the figure we can observe that 120572119894

120572-119896 becomes close to120572limit of 231967 at cycle 8 in the run with 120578 of 0 and at cycle6 with 120578 of 2 which yield the abnormal terminations at thevery next iteration calculations

The 120572-k iteration method with the pseudo absorptionterm may reduce occurrence of the infinite number oftime source generations by suppressing fluctuations of 120572119894

120572-119896for MC calculations with continuous cross section data asreported in [12] However it is noteworthy that the abnormalterminations in the 120572-k iteration calculations can happen byinevitable statistical fluctuations of 120572 estimates

32 Application to the Th-ADS Experimental BenchmarksThe proposed 120572 iteration method is applied for the exper-imental benchmarks on thorium-loaded accelerator-drivensystem (Th-ADS) [16] The Th-ADS experiments were per-formed using the solid-moderated and solid-reflected type-A core of KUCA for seven core configurations combinedwith a 14MeV pulsed neutron generator or a synchrotrontype proton acceleratorThehighly enriched uranium (HEU)thorium (Th) and natural uranium (NU) fuel was loadedtogether with the reflectors including polyethylene (PE)graphite (Gr) and beryllium (Be) Figure 2 shows a coreconfiguration with three 3He detectors used for Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PE cores and one with two

Science and Technology of Nuclear Installations 5

Table 2 Comparisons of 120572 estimates for the infinite homogeneous problem

119896119901 Ref 120572 120572-119896 iteration with 120578 = 0 (SD) 120572-119896 iteration with 120578 = 2 (SD) 120572 iteration (SD)090 265071 265001 (128) 265456 (148) 265244 (140)070 795234 795746 (338) 794557 (387) 795226 (181)050 1325440 1325180 (708) 1324310 (944) 1325390 (209)030 1855680 Fail Fail 1855540 (266)015 2253380 Fail Fail 2253280 (313)

F

F

F

F

F

F F F

F F F

F F F

F F F

F F F

F

F

F

F

F

Tritium targetDeuteron beams Aluminum sheath

PolyethyleneF Fuel

He detector

3He 1

3He 2 3He 3

(a) Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PE

F

F

F

F

F

F F F

F F F

F F F

F F F

F F F

F

F

F

F

F

Tritium targetDeuteron beams Aluminum sheath

PolyethyleneF Fuel

He detector

3He 1 3He 2

(b) Th-HEU-5PE andTh-HEU-Gr-PE

Figure 2 Th-ADS core configurations with 14MeV neutrons

3He detectors for Th-HEU-5PE and Th-HEU-Gr-PE coreshaving the 14MeV pulsed neutron generator The effectivemultiplication factors 119896eff rsquos of the Th-PE Th-Gr Th-BeTh-HEU-PE NU-PE Th-HEU-5PE and Th-HEU-Gr-PEconfigurations are reported to be 000613 000952 000765058754 050867 085121 and 035473 respectively [16]

The McCARD calculations are performed with contin-uous-energy cross section libraries produced from JENDL-40 for the seven cores with the 14MeV pulsed neutrongenerator The alpha-static calculations by the 120572 iterationalgorithm are conducted with 5000 active iterations on10000 sources per iteration For the comparison McCARDsimulations for the PNS experiments are performed on 20billion of 14MeV neutron sources Figure 3 shows flux tallyresults of the three detectors for the Th-Gr core and the twodetectors for the Th-HEU-5PE core in 1 120583s time bins from00ms to 40ms A time decay constant 120572 can be obtained byfitting the tally results to the exponential function 120601 in thetime range when the higher-mode fluxes sufficiently decay as[2]

120601 (119905) = 119860119890minus120572(119905minus119905

119904)+ 119861 (20)

where 119860 and 119861 are fitting constants and 119905 and 119905119904 are thetime after the neutron burst and the starting time of fittingrespectively Figure 4 shows the convergence of 120572 to a stableone with increasing 119905119904 for the fixed fitting intervals of 800120583sFrom these convergence diagnoses 119905119904 for the Th-PE Th-GrTh-Be Th-HEU-PE NU-PE Th-HEU-5PE and Th-HEU-Gr-PE cores is determined to be 13 16 16 17 17 16 and13ms respectively

Table 3 shows comparisons of 120572rsquos estimated by the 120572

iteration method with the measurements given in [16] andthe 120572 values from the McCARD PNS simulations which arecalculated by averaging fitted values of 3He detectors 2 and3 for the Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PEcores and 3He detectors 1 and 2 for the Th-HEU-5PE andTh-HEU-Gr-PE cores From the comparisons between the 120572values calculated by the120572 iterationmethod and theMcCARDPNS simulations we can see that the absolute values of therelative difference for the Th-PE Th-Be Th-HEU-5PE andTh-HEU-Gr-PE cores are less than 10 while being 1321 and 34 for theTh-GrTh-HEU-PE andNU-PE coresrespectively From the comparisons with measurements wecan see that the 120572 iteration results are quite comparable with

6 Science and Technology of Nuclear Installations

Table 3 Comparisons of 120572 estimates for the Th-ADS experimental benchmark

Core Measurements with 14MeV neutrons (SD) MC PNS simulation (SD) MC 120572 iteration (SD)3He 1 3He 2 3He 3

Th-PE 6642 (11) 6224 (27) 5751 (25) 52454 (49) 52433 (13)Th-Gr 6451 (12) 5945 (15) 5701 (17) 53101 (77) 52397 (13)Th-Be 6515 (8) 6111 (17) 5746 (20) 52242 (72) 52401 (14)Th-HEU-PE 5692 (11) 5275 (7) 5231 (9) 50256 (63) 51312 (28)NU-PE 5748 (11) 6592 (15) 5010 (11) 49951 (64) 51638 (35)Th-HEU-5PE 3110 (11) 3104 (10) mdash 29771 (14) 29541 (12)Th-HEU-Gr-PE 4980 (40) 4939 (50) mdash 47610 (29) 47667 (102)

0 1 2 3 4

He number 1He number 2

He number 3

Neu

tron

flux

Time (ms)

101001

1E minus 31E minus 41E minus 51E minus 61E minus 71E minus 81E minus 91E minus 101E minus 111E minus 121E minus 131E minus 141E minus 151E minus 161E minus 17

(a) Th-Gr core

He number 1He number 2

0 1 2 3 4

Neu

tron

flux

Time (ms)

1

01

001

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

1E minus 8

1E minus 9

1E minus 10

1E minus 11

1E minus 12

1E minus 13

(b) Th-HEU-5PE core

Figure 3 Time-dependent flux for the Th-ADS benchmark prob-lems

5000

5200

5400

5600

5800

6000

Starting time of fitting (ms)02 04 06 08 10 12 14 16 18

He number 1He number 2

He number 3

120572

(a) Th-Gr core

1800

2000

2200

2400

2600

2800

3000

3200

He number 1He number 2

Starting time of fitting (ms)02 04 06 08 10 12 14 16 18

120572

(b) Th-HEU-5PE core

Figure 4 Fitted 120572 values according to the starting time of fitting forthe Th-ADS benchmark problems

Science and Technology of Nuclear Installations 7

the experimental results for core configurations where twoestimates from different detectors show good agreementsthat is the Ph-HEU-PE Th-HEU-5PE and Ph-HEU-Gr-PEcores

4 Conclusion

A new MC alpha-static calculation method is developed tostably estimate the fundamental-mode alpha-eigenvalue ofprompt subcritical systems While the fission source distri-bution is updated in the existing 120572-k iteration method thenew method named as the 120572 iteration algorithm directlyupdates the time source distribution iteration-by-iterationThe calculation stability of the 120572 iteration algorithm isachieved by controlling the number of time source neutronsgenerated at each iteration by the normalization scheme forthe time source distribution

It is demonstrated that the 120572 iteration algorithm doesnot suffer from the instability problem in the two-grouphomogeneous problems with deep subcriticalities where the120572-k iteration method fails to obtain the 120572 value due to thehuge number of neutron productions From the comparisonswithmeasurements in theTh-ADS experimental benchmarks[16] it is observed that the 120572 results calculated by the 120572

iterationmethod are quite comparable with each other for thecases where the experiments provide reliable estimates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Nuclear RampDProgram through theNational Research Foundation of Korea(NRF) funded by MSIP (Ministry of Science ICT and FuturePlanning) (no 2010-0018758)

References

[1] D E Cullen C J Clouse R Procassini and R C Little ldquoStaticand dynamic criticality are they differentrdquo Report UCRL-TR-201506 Lawrence Livermore National Laboratory LivermoreCalif USA 2003

[2] T Suzaki ldquoSubcritically determination of low-enriched UO2lattices in water by exponential experimentrdquo Journal of NuclearScience and Technology vol 28 no 12 pp 1067ndash1077 1991

[3] T Yamamoto K Sakurai T Arakawa and Y Naito ldquoAccu-rate estimation of subcriticality using indirect bias estimationmethod (I) theoryrdquo Journal of Nuclear Science and Technologyvol 34 no 5 pp 454ndash460 1997

[4] D E Cullen ldquoTART 2002 a coupled neutron-photon 3-Dcombinatorial geometry time dependent Monte Carlo trans-port coderdquo Tech Rep UCRL-ID-126455 Rev 4 LawrenceLivermore National Laboratory 2002

[5] D Brockway P Soran and P Whalen ldquoMonte Carlo 120572 calcu-lationrdquo Tech Rep LA-UR-85-1224 Los Alamos National Labo-ratory 1985

[6] T Ye C Chen W Sun B Zhang and D Tian ldquoPrompt timeconstants of a reflected reactorrdquo in Proceedings of the Symposiumon Nuclear Data T Fukahori Ed Ibaraki-ken Japan January2007

[7] S D Nolen T R Adams and J E Sweezy ldquoIntegral criticalityestimators in MCATK(U)rdquo Tech Rep LA-UR-12-25458 LosAlamos National Laboratory Los Alamos NM USA 2012

[8] A Zoia E Brun and F Malvagi ldquoAlpha eigenvalue calculationswith TRIPOLI-4rdquo Annals of Nuclear Energy vol 63 pp 276ndash284 2014

[9] A Zoia E Brun and F Malvagi ldquoA Monte Carlo method forprompt and delayed alpha eigenvalue calculationsrdquo in Proceed-ings of the International Conference on the Role of Reactor PhysicsToward a Sustainable Future (PHYSOR rsquo14) CD-ROM KyotoJapan September-October 2014

[10] J Lieberoth ldquoAMonte Carlo technique to solve the static eigen-value problemof the boltzmann transport equationrdquoNukleonikvol 11 article 213 1968

[11] T Yamamoto and Y Miyoshi ldquoAn algorithm of 120572- and 120574-modeeigenvalue calculations by Monte Carlo methodrdquo in Proceed-ings of the 7th International Conference on Nuclear CriticalitySafety (ICNC rsquo03) JAERI-Conf 2003-019 Japan Atomic EnergyResearch Institute Tokai Japan October 2003

[12] T Yamamoto ldquoHigher order 120572mode eigenvalue calculation byMonte Carlo power iterationrdquo Progress in Nuclear Science andTechnology vol 2 pp 826ndash835 2011

[13] O Petit F-X Hugot Y-K Lee C Jouanne and A MazzoloTRIPOLI-4 Version 4 User Guide CEA-R-6169 CEA Saclay2008

[14] S Nakamura Computational Methods in Engineering and Sci-ence With Applications to Fluid Dynamics and Nuclear SystemsWiley-Interscience 1977

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

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

[17] G I Bell and S Glasstone Nuclear Reactor Theory VanNostrand Reinhold Company New York NY USA 1970

[18] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976

[19] I Lux andLKoblingerMonteCarlo Particle TransportMethodsNeutron and Photon Calculations CRC Press Boca Raton FlaUSA 1991

[20] J F Briesmeister ldquoMCNP A general Monte Carlo N-particletransport code version 4Brdquo Tech Rep LA-13181 Los AlamosNational Laboratory Los Alamos NM USA 1997

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Science and Technology of Nuclear Installations 5

Table 2 Comparisons of 120572 estimates for the infinite homogeneous problem

119896119901 Ref 120572 120572-119896 iteration with 120578 = 0 (SD) 120572-119896 iteration with 120578 = 2 (SD) 120572 iteration (SD)090 265071 265001 (128) 265456 (148) 265244 (140)070 795234 795746 (338) 794557 (387) 795226 (181)050 1325440 1325180 (708) 1324310 (944) 1325390 (209)030 1855680 Fail Fail 1855540 (266)015 2253380 Fail Fail 2253280 (313)

F

F

F

F

F

F F F

F F F

F F F

F F F

F F F

F

F

F

F

F

Tritium targetDeuteron beams Aluminum sheath

PolyethyleneF Fuel

He detector

3He 1

3He 2 3He 3

(a) Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PE

F

F

F

F

F

F F F

F F F

F F F

F F F

F F F

F

F

F

F

F

Tritium targetDeuteron beams Aluminum sheath

PolyethyleneF Fuel

He detector

3He 1 3He 2

(b) Th-HEU-5PE andTh-HEU-Gr-PE

Figure 2 Th-ADS core configurations with 14MeV neutrons

3He detectors for Th-HEU-5PE and Th-HEU-Gr-PE coreshaving the 14MeV pulsed neutron generator The effectivemultiplication factors 119896eff rsquos of the Th-PE Th-Gr Th-BeTh-HEU-PE NU-PE Th-HEU-5PE and Th-HEU-Gr-PEconfigurations are reported to be 000613 000952 000765058754 050867 085121 and 035473 respectively [16]

The McCARD calculations are performed with contin-uous-energy cross section libraries produced from JENDL-40 for the seven cores with the 14MeV pulsed neutrongenerator The alpha-static calculations by the 120572 iterationalgorithm are conducted with 5000 active iterations on10000 sources per iteration For the comparison McCARDsimulations for the PNS experiments are performed on 20billion of 14MeV neutron sources Figure 3 shows flux tallyresults of the three detectors for the Th-Gr core and the twodetectors for the Th-HEU-5PE core in 1 120583s time bins from00ms to 40ms A time decay constant 120572 can be obtained byfitting the tally results to the exponential function 120601 in thetime range when the higher-mode fluxes sufficiently decay as[2]

120601 (119905) = 119860119890minus120572(119905minus119905

119904)+ 119861 (20)

where 119860 and 119861 are fitting constants and 119905 and 119905119904 are thetime after the neutron burst and the starting time of fittingrespectively Figure 4 shows the convergence of 120572 to a stableone with increasing 119905119904 for the fixed fitting intervals of 800120583sFrom these convergence diagnoses 119905119904 for the Th-PE Th-GrTh-Be Th-HEU-PE NU-PE Th-HEU-5PE and Th-HEU-Gr-PE cores is determined to be 13 16 16 17 17 16 and13ms respectively

Table 3 shows comparisons of 120572rsquos estimated by the 120572

iteration method with the measurements given in [16] andthe 120572 values from the McCARD PNS simulations which arecalculated by averaging fitted values of 3He detectors 2 and3 for the Th-PE Th-Gr Th-Be Th-HEU-PE and NU-PEcores and 3He detectors 1 and 2 for the Th-HEU-5PE andTh-HEU-Gr-PE cores From the comparisons between the 120572values calculated by the120572 iterationmethod and theMcCARDPNS simulations we can see that the absolute values of therelative difference for the Th-PE Th-Be Th-HEU-5PE andTh-HEU-Gr-PE cores are less than 10 while being 1321 and 34 for theTh-GrTh-HEU-PE andNU-PE coresrespectively From the comparisons with measurements wecan see that the 120572 iteration results are quite comparable with

6 Science and Technology of Nuclear Installations

Table 3 Comparisons of 120572 estimates for the Th-ADS experimental benchmark

Core Measurements with 14MeV neutrons (SD) MC PNS simulation (SD) MC 120572 iteration (SD)3He 1 3He 2 3He 3

Th-PE 6642 (11) 6224 (27) 5751 (25) 52454 (49) 52433 (13)Th-Gr 6451 (12) 5945 (15) 5701 (17) 53101 (77) 52397 (13)Th-Be 6515 (8) 6111 (17) 5746 (20) 52242 (72) 52401 (14)Th-HEU-PE 5692 (11) 5275 (7) 5231 (9) 50256 (63) 51312 (28)NU-PE 5748 (11) 6592 (15) 5010 (11) 49951 (64) 51638 (35)Th-HEU-5PE 3110 (11) 3104 (10) mdash 29771 (14) 29541 (12)Th-HEU-Gr-PE 4980 (40) 4939 (50) mdash 47610 (29) 47667 (102)

0 1 2 3 4

He number 1He number 2

He number 3

Neu

tron

flux

Time (ms)

101001

1E minus 31E minus 41E minus 51E minus 61E minus 71E minus 81E minus 91E minus 101E minus 111E minus 121E minus 131E minus 141E minus 151E minus 161E minus 17

(a) Th-Gr core

He number 1He number 2

0 1 2 3 4

Neu

tron

flux

Time (ms)

1

01

001

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

1E minus 8

1E minus 9

1E minus 10

1E minus 11

1E minus 12

1E minus 13

(b) Th-HEU-5PE core

Figure 3 Time-dependent flux for the Th-ADS benchmark prob-lems

5000

5200

5400

5600

5800

6000

Starting time of fitting (ms)02 04 06 08 10 12 14 16 18

He number 1He number 2

He number 3

120572

(a) Th-Gr core

1800

2000

2200

2400

2600

2800

3000

3200

He number 1He number 2

Starting time of fitting (ms)02 04 06 08 10 12 14 16 18

120572

(b) Th-HEU-5PE core

Figure 4 Fitted 120572 values according to the starting time of fitting forthe Th-ADS benchmark problems

Science and Technology of Nuclear Installations 7

the experimental results for core configurations where twoestimates from different detectors show good agreementsthat is the Ph-HEU-PE Th-HEU-5PE and Ph-HEU-Gr-PEcores

4 Conclusion

A new MC alpha-static calculation method is developed tostably estimate the fundamental-mode alpha-eigenvalue ofprompt subcritical systems While the fission source distri-bution is updated in the existing 120572-k iteration method thenew method named as the 120572 iteration algorithm directlyupdates the time source distribution iteration-by-iterationThe calculation stability of the 120572 iteration algorithm isachieved by controlling the number of time source neutronsgenerated at each iteration by the normalization scheme forthe time source distribution

It is demonstrated that the 120572 iteration algorithm doesnot suffer from the instability problem in the two-grouphomogeneous problems with deep subcriticalities where the120572-k iteration method fails to obtain the 120572 value due to thehuge number of neutron productions From the comparisonswithmeasurements in theTh-ADS experimental benchmarks[16] it is observed that the 120572 results calculated by the 120572

iterationmethod are quite comparable with each other for thecases where the experiments provide reliable estimates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Nuclear RampDProgram through theNational Research Foundation of Korea(NRF) funded by MSIP (Ministry of Science ICT and FuturePlanning) (no 2010-0018758)

References

[1] D E Cullen C J Clouse R Procassini and R C Little ldquoStaticand dynamic criticality are they differentrdquo Report UCRL-TR-201506 Lawrence Livermore National Laboratory LivermoreCalif USA 2003

[2] T Suzaki ldquoSubcritically determination of low-enriched UO2lattices in water by exponential experimentrdquo Journal of NuclearScience and Technology vol 28 no 12 pp 1067ndash1077 1991

[3] T Yamamoto K Sakurai T Arakawa and Y Naito ldquoAccu-rate estimation of subcriticality using indirect bias estimationmethod (I) theoryrdquo Journal of Nuclear Science and Technologyvol 34 no 5 pp 454ndash460 1997

[4] D E Cullen ldquoTART 2002 a coupled neutron-photon 3-Dcombinatorial geometry time dependent Monte Carlo trans-port coderdquo Tech Rep UCRL-ID-126455 Rev 4 LawrenceLivermore National Laboratory 2002

[5] D Brockway P Soran and P Whalen ldquoMonte Carlo 120572 calcu-lationrdquo Tech Rep LA-UR-85-1224 Los Alamos National Labo-ratory 1985

[6] T Ye C Chen W Sun B Zhang and D Tian ldquoPrompt timeconstants of a reflected reactorrdquo in Proceedings of the Symposiumon Nuclear Data T Fukahori Ed Ibaraki-ken Japan January2007

[7] S D Nolen T R Adams and J E Sweezy ldquoIntegral criticalityestimators in MCATK(U)rdquo Tech Rep LA-UR-12-25458 LosAlamos National Laboratory Los Alamos NM USA 2012

[8] A Zoia E Brun and F Malvagi ldquoAlpha eigenvalue calculationswith TRIPOLI-4rdquo Annals of Nuclear Energy vol 63 pp 276ndash284 2014

[9] A Zoia E Brun and F Malvagi ldquoA Monte Carlo method forprompt and delayed alpha eigenvalue calculationsrdquo in Proceed-ings of the International Conference on the Role of Reactor PhysicsToward a Sustainable Future (PHYSOR rsquo14) CD-ROM KyotoJapan September-October 2014

[10] J Lieberoth ldquoAMonte Carlo technique to solve the static eigen-value problemof the boltzmann transport equationrdquoNukleonikvol 11 article 213 1968

[11] T Yamamoto and Y Miyoshi ldquoAn algorithm of 120572- and 120574-modeeigenvalue calculations by Monte Carlo methodrdquo in Proceed-ings of the 7th International Conference on Nuclear CriticalitySafety (ICNC rsquo03) JAERI-Conf 2003-019 Japan Atomic EnergyResearch Institute Tokai Japan October 2003

[12] T Yamamoto ldquoHigher order 120572mode eigenvalue calculation byMonte Carlo power iterationrdquo Progress in Nuclear Science andTechnology vol 2 pp 826ndash835 2011

[13] O Petit F-X Hugot Y-K Lee C Jouanne and A MazzoloTRIPOLI-4 Version 4 User Guide CEA-R-6169 CEA Saclay2008

[14] S Nakamura Computational Methods in Engineering and Sci-ence With Applications to Fluid Dynamics and Nuclear SystemsWiley-Interscience 1977

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

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

[17] G I Bell and S Glasstone Nuclear Reactor Theory VanNostrand Reinhold Company New York NY USA 1970

[18] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976

[19] I Lux andLKoblingerMonteCarlo Particle TransportMethodsNeutron and Photon Calculations CRC Press Boca Raton FlaUSA 1991

[20] J F Briesmeister ldquoMCNP A general Monte Carlo N-particletransport code version 4Brdquo Tech Rep LA-13181 Los AlamosNational Laboratory Los Alamos NM USA 1997

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

6 Science and Technology of Nuclear Installations

Table 3 Comparisons of 120572 estimates for the Th-ADS experimental benchmark

Core Measurements with 14MeV neutrons (SD) MC PNS simulation (SD) MC 120572 iteration (SD)3He 1 3He 2 3He 3

Th-PE 6642 (11) 6224 (27) 5751 (25) 52454 (49) 52433 (13)Th-Gr 6451 (12) 5945 (15) 5701 (17) 53101 (77) 52397 (13)Th-Be 6515 (8) 6111 (17) 5746 (20) 52242 (72) 52401 (14)Th-HEU-PE 5692 (11) 5275 (7) 5231 (9) 50256 (63) 51312 (28)NU-PE 5748 (11) 6592 (15) 5010 (11) 49951 (64) 51638 (35)Th-HEU-5PE 3110 (11) 3104 (10) mdash 29771 (14) 29541 (12)Th-HEU-Gr-PE 4980 (40) 4939 (50) mdash 47610 (29) 47667 (102)

0 1 2 3 4

He number 1He number 2

He number 3

Neu

tron

flux

Time (ms)

101001

1E minus 31E minus 41E minus 51E minus 61E minus 71E minus 81E minus 91E minus 101E minus 111E minus 121E minus 131E minus 141E minus 151E minus 161E minus 17

(a) Th-Gr core

He number 1He number 2

0 1 2 3 4

Neu

tron

flux

Time (ms)

1

01

001

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

1E minus 8

1E minus 9

1E minus 10

1E minus 11

1E minus 12

1E minus 13

(b) Th-HEU-5PE core

Figure 3 Time-dependent flux for the Th-ADS benchmark prob-lems

5000

5200

5400

5600

5800

6000

Starting time of fitting (ms)02 04 06 08 10 12 14 16 18

He number 1He number 2

He number 3

120572

(a) Th-Gr core

1800

2000

2200

2400

2600

2800

3000

3200

He number 1He number 2

Starting time of fitting (ms)02 04 06 08 10 12 14 16 18

120572

(b) Th-HEU-5PE core

Figure 4 Fitted 120572 values according to the starting time of fitting forthe Th-ADS benchmark problems

Science and Technology of Nuclear Installations 7

the experimental results for core configurations where twoestimates from different detectors show good agreementsthat is the Ph-HEU-PE Th-HEU-5PE and Ph-HEU-Gr-PEcores

4 Conclusion

A new MC alpha-static calculation method is developed tostably estimate the fundamental-mode alpha-eigenvalue ofprompt subcritical systems While the fission source distri-bution is updated in the existing 120572-k iteration method thenew method named as the 120572 iteration algorithm directlyupdates the time source distribution iteration-by-iterationThe calculation stability of the 120572 iteration algorithm isachieved by controlling the number of time source neutronsgenerated at each iteration by the normalization scheme forthe time source distribution

It is demonstrated that the 120572 iteration algorithm doesnot suffer from the instability problem in the two-grouphomogeneous problems with deep subcriticalities where the120572-k iteration method fails to obtain the 120572 value due to thehuge number of neutron productions From the comparisonswithmeasurements in theTh-ADS experimental benchmarks[16] it is observed that the 120572 results calculated by the 120572

iterationmethod are quite comparable with each other for thecases where the experiments provide reliable estimates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Nuclear RampDProgram through theNational Research Foundation of Korea(NRF) funded by MSIP (Ministry of Science ICT and FuturePlanning) (no 2010-0018758)

References

[1] D E Cullen C J Clouse R Procassini and R C Little ldquoStaticand dynamic criticality are they differentrdquo Report UCRL-TR-201506 Lawrence Livermore National Laboratory LivermoreCalif USA 2003

[2] T Suzaki ldquoSubcritically determination of low-enriched UO2lattices in water by exponential experimentrdquo Journal of NuclearScience and Technology vol 28 no 12 pp 1067ndash1077 1991

[3] T Yamamoto K Sakurai T Arakawa and Y Naito ldquoAccu-rate estimation of subcriticality using indirect bias estimationmethod (I) theoryrdquo Journal of Nuclear Science and Technologyvol 34 no 5 pp 454ndash460 1997

[4] D E Cullen ldquoTART 2002 a coupled neutron-photon 3-Dcombinatorial geometry time dependent Monte Carlo trans-port coderdquo Tech Rep UCRL-ID-126455 Rev 4 LawrenceLivermore National Laboratory 2002

[5] D Brockway P Soran and P Whalen ldquoMonte Carlo 120572 calcu-lationrdquo Tech Rep LA-UR-85-1224 Los Alamos National Labo-ratory 1985

[6] T Ye C Chen W Sun B Zhang and D Tian ldquoPrompt timeconstants of a reflected reactorrdquo in Proceedings of the Symposiumon Nuclear Data T Fukahori Ed Ibaraki-ken Japan January2007

[7] S D Nolen T R Adams and J E Sweezy ldquoIntegral criticalityestimators in MCATK(U)rdquo Tech Rep LA-UR-12-25458 LosAlamos National Laboratory Los Alamos NM USA 2012

[8] A Zoia E Brun and F Malvagi ldquoAlpha eigenvalue calculationswith TRIPOLI-4rdquo Annals of Nuclear Energy vol 63 pp 276ndash284 2014

[9] A Zoia E Brun and F Malvagi ldquoA Monte Carlo method forprompt and delayed alpha eigenvalue calculationsrdquo in Proceed-ings of the International Conference on the Role of Reactor PhysicsToward a Sustainable Future (PHYSOR rsquo14) CD-ROM KyotoJapan September-October 2014

[10] J Lieberoth ldquoAMonte Carlo technique to solve the static eigen-value problemof the boltzmann transport equationrdquoNukleonikvol 11 article 213 1968

[11] T Yamamoto and Y Miyoshi ldquoAn algorithm of 120572- and 120574-modeeigenvalue calculations by Monte Carlo methodrdquo in Proceed-ings of the 7th International Conference on Nuclear CriticalitySafety (ICNC rsquo03) JAERI-Conf 2003-019 Japan Atomic EnergyResearch Institute Tokai Japan October 2003

[12] T Yamamoto ldquoHigher order 120572mode eigenvalue calculation byMonte Carlo power iterationrdquo Progress in Nuclear Science andTechnology vol 2 pp 826ndash835 2011

[13] O Petit F-X Hugot Y-K Lee C Jouanne and A MazzoloTRIPOLI-4 Version 4 User Guide CEA-R-6169 CEA Saclay2008

[14] S Nakamura Computational Methods in Engineering and Sci-ence With Applications to Fluid Dynamics and Nuclear SystemsWiley-Interscience 1977

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

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

[17] G I Bell and S Glasstone Nuclear Reactor Theory VanNostrand Reinhold Company New York NY USA 1970

[18] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976

[19] I Lux andLKoblingerMonteCarlo Particle TransportMethodsNeutron and Photon Calculations CRC Press Boca Raton FlaUSA 1991

[20] J F Briesmeister ldquoMCNP A general Monte Carlo N-particletransport code version 4Brdquo Tech Rep LA-13181 Los AlamosNational Laboratory Los Alamos NM USA 1997

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Science and Technology of Nuclear Installations 7

the experimental results for core configurations where twoestimates from different detectors show good agreementsthat is the Ph-HEU-PE Th-HEU-5PE and Ph-HEU-Gr-PEcores

4 Conclusion

A new MC alpha-static calculation method is developed tostably estimate the fundamental-mode alpha-eigenvalue ofprompt subcritical systems While the fission source distri-bution is updated in the existing 120572-k iteration method thenew method named as the 120572 iteration algorithm directlyupdates the time source distribution iteration-by-iterationThe calculation stability of the 120572 iteration algorithm isachieved by controlling the number of time source neutronsgenerated at each iteration by the normalization scheme forthe time source distribution

It is demonstrated that the 120572 iteration algorithm doesnot suffer from the instability problem in the two-grouphomogeneous problems with deep subcriticalities where the120572-k iteration method fails to obtain the 120572 value due to thehuge number of neutron productions From the comparisonswithmeasurements in theTh-ADS experimental benchmarks[16] it is observed that the 120572 results calculated by the 120572

iterationmethod are quite comparable with each other for thecases where the experiments provide reliable estimates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Nuclear RampDProgram through theNational Research Foundation of Korea(NRF) funded by MSIP (Ministry of Science ICT and FuturePlanning) (no 2010-0018758)

References

[1] D E Cullen C J Clouse R Procassini and R C Little ldquoStaticand dynamic criticality are they differentrdquo Report UCRL-TR-201506 Lawrence Livermore National Laboratory LivermoreCalif USA 2003

[2] T Suzaki ldquoSubcritically determination of low-enriched UO2lattices in water by exponential experimentrdquo Journal of NuclearScience and Technology vol 28 no 12 pp 1067ndash1077 1991

[3] T Yamamoto K Sakurai T Arakawa and Y Naito ldquoAccu-rate estimation of subcriticality using indirect bias estimationmethod (I) theoryrdquo Journal of Nuclear Science and Technologyvol 34 no 5 pp 454ndash460 1997

[4] D E Cullen ldquoTART 2002 a coupled neutron-photon 3-Dcombinatorial geometry time dependent Monte Carlo trans-port coderdquo Tech Rep UCRL-ID-126455 Rev 4 LawrenceLivermore National Laboratory 2002

[5] D Brockway P Soran and P Whalen ldquoMonte Carlo 120572 calcu-lationrdquo Tech Rep LA-UR-85-1224 Los Alamos National Labo-ratory 1985

[6] T Ye C Chen W Sun B Zhang and D Tian ldquoPrompt timeconstants of a reflected reactorrdquo in Proceedings of the Symposiumon Nuclear Data T Fukahori Ed Ibaraki-ken Japan January2007

[7] S D Nolen T R Adams and J E Sweezy ldquoIntegral criticalityestimators in MCATK(U)rdquo Tech Rep LA-UR-12-25458 LosAlamos National Laboratory Los Alamos NM USA 2012

[8] A Zoia E Brun and F Malvagi ldquoAlpha eigenvalue calculationswith TRIPOLI-4rdquo Annals of Nuclear Energy vol 63 pp 276ndash284 2014

[9] A Zoia E Brun and F Malvagi ldquoA Monte Carlo method forprompt and delayed alpha eigenvalue calculationsrdquo in Proceed-ings of the International Conference on the Role of Reactor PhysicsToward a Sustainable Future (PHYSOR rsquo14) CD-ROM KyotoJapan September-October 2014

[10] J Lieberoth ldquoAMonte Carlo technique to solve the static eigen-value problemof the boltzmann transport equationrdquoNukleonikvol 11 article 213 1968

[11] T Yamamoto and Y Miyoshi ldquoAn algorithm of 120572- and 120574-modeeigenvalue calculations by Monte Carlo methodrdquo in Proceed-ings of the 7th International Conference on Nuclear CriticalitySafety (ICNC rsquo03) JAERI-Conf 2003-019 Japan Atomic EnergyResearch Institute Tokai Japan October 2003

[12] T Yamamoto ldquoHigher order 120572mode eigenvalue calculation byMonte Carlo power iterationrdquo Progress in Nuclear Science andTechnology vol 2 pp 826ndash835 2011

[13] O Petit F-X Hugot Y-K Lee C Jouanne and A MazzoloTRIPOLI-4 Version 4 User Guide CEA-R-6169 CEA Saclay2008

[14] S Nakamura Computational Methods in Engineering and Sci-ence With Applications to Fluid Dynamics and Nuclear SystemsWiley-Interscience 1977

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

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

[17] G I Bell and S Glasstone Nuclear Reactor Theory VanNostrand Reinhold Company New York NY USA 1970

[18] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976

[19] I Lux andLKoblingerMonteCarlo Particle TransportMethodsNeutron and Photon Calculations CRC Press Boca Raton FlaUSA 1991

[20] J F Briesmeister ldquoMCNP A general Monte Carlo N-particletransport code version 4Brdquo Tech Rep LA-13181 Los AlamosNational Laboratory Los Alamos NM USA 1997

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014