MONTE CARLO ALGORITHM FOR CLASPING SEARCH AND NEUTRON LEAKAGE

Transcription

1 Sep. 5. Vol. 7. No. Inernaonal Journal o Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved MONTE CARLO ALGORITHM FOR CLASPING SEARCH AND NEUTRON LEAKAGE PEYMAN MAJNOUN, FARID VAKILI-TAHAMI, ARASH MOHAMMAD ALIZADEH-FARD Assocae Proessor, Faculy o Mechancal Engneerng, Unversy o Tabrz-IRAN, MSc & PhD sudens, Faculy o Mechancal Engneerng, Unversy o Tabrz-IRAN Correspondng Emal: p.manoon@gmal.com ABSTRACT A Mone Carlo algorhm has been developed or akng no accoun neuron leakage eec by a claspng. The algorhm allows one o perorm claspng search mode proper value calculaons where he claspng s reaed as a proper value. For nroducng neuron leakage eec, he spaal dependence o neuron nsably s come near a sngle Fourer mode. The parcle wegh s a complex number.the magnary par, however, dsappears and only he real par needs o be reaed symmery exss wh respec o he drecon represened by he claspng and he dsrbung and splng neuron emsson are soropous. The algorhm has been vered by comparng buck lngs acqure by claspng search mode proper value calculaons wh hose by he duson approxmaon and he B mehod.when a horzonal calculave eld has wo orhogonal symmery planes and he dsrbung and splng neuron emsson are soropous he magnary pars o complex weghs dsappear. I does no, complex weghs need o be reaed. The mehodology or reang complex weghs has no ye been esablshed and would be a uure work. The newly developed algorhm can also be appled o leakage-correced Mone Carlo calculaons, generang leakage correced neuron specra and nsably dsrbuons. Alhough hs paper has presened ha he algorhm can conrbue o he generaon o leakage-correced group connuous s based on Mone Carlo calculaon echnques. Keywords: Mone Carlo algorhm, Claspng, Neuron leakage INTRODUCTION Three-dmensonal o core calculaons by Mone Carlo echnques are gradually becomng a calculaon ool or reacor core desgns. Meanwhle, connuous energy Mone Carlo calculaons are now beleved an ecen ool or calculang some neuronc parameers such as duson coecen, whch wll be used or deermnsc core desgn codes. Yoshoka and Ando [], and Leppänen [], have been perormed or generang deermnsc group connuous by usng Mone Carlo echnques. Calculaons are usually perormed or un uel pn cells or uel assembles havng nne lengh n he vercal drecon. To oban accurae group connuous, he eec o neuron leakage o he radal and axal drecons has o be aken no accoun n he Mone Carlo calculaons. Neuron leakage correcons are ncorporaed no deermnsc uel pn cell or uel assembly calculaon codes by usng some leakage correcon echnques, e.g., he B mehod (Dudersad and Hamlon, []. In hs mehod, he neuron leakage eec s aken no accoun by specyng he claspng B o a reacor core. In he eld o Mone Carlo, however, some echnques have been developed o presen he neuron leakage eec no uel pn cell or uel assembly calculaons (Yoshoka and Ando []; Frdman and Leppänen [4]; Shm e al. [5]. Leppänen [], presen an addonal cross secon-lke parameer or ncorporang he neuron leakage eec. Ths echnque s smlar o he calculaons o he proper value mode. A erm DgBg s arcally added as smulaed absorpon where Dg s a duson coecen n he gh energy group and Bg s a geomerc claspng. Ths mehodology s based on he assumpon ha he duson holds whn reasonable accuracy. A dculy n esmang he duson coecens or connuous energy Mone Carlo calculaons sll remans unresolved. The mehods are beleved unsubsanaed and no recommended o be used or group connuous 6

2 Sep. 5. Vol. 7. No. Inernaonal Journal o Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved generaon. Frdman and Leppänen [4] and Shm e al. [5] presen a mehod based on he B mehod or ew-group connuous generaon. A Mone Carlo calculaon s used o produce ne group cross secons used or solvng he B equaons. Then, he B equaons are solved o oban he crcal neuron specrum. Ths mehod s sem-deermnsc, and leakage-correced eec s no explcly ncluded n he Mone Carlo calculaon. The change o ke caused by neuron leakage s calculaed by he perurbaon heory. Ths echnque requres he ad on source dsrbuon ha s usually dcul o be esmaed by a connuous energy Mone Carlo calculaon. Anoher relaed work usng Mone Carlo echnque was perormed by Maorov [6]. To ake no accoun he eec o neuron leakage, parcle weghs are gven by a ormula dependng on a parcle s random walks. Recenly, a leakagecorreced Mone Carlo calculaon echnque was proposed by Yun and Cho [7]. Ths mehod presens he eec o neuron leakage by specyng on he ouer suraces o a calculave eld nsead o nroducng a geomerc claspng. Durng he course o a Mone Carlo crcaly calculaon, an albedo s updaed a each generaon n such a way ha ke s equal o uny. The presen paper ocuses on deermnaon o he geomerc claspng or spaal decompose connuous c n he vercal drecon o a uel pn cell or a uel assembly. I he geomerc claspng n he horzonal drecon s larger han he maeral claspng, he neuron nsably decomposes n he vercal drecon. The spaal decompose connuous s nmaely relaed o he sub crcaly o he uel pn cell or uel assembly and can be used as an ndcaor o nuclear crcaly saey. In such a case, a proper value mode neuron ranspor equaon can be dened. Yamamoo and Myosh [8] proposed a mehod o solve he proper value mode equaon by Mone Carlo. The mehod o he proper value mode calculaon has been mplemened no a connuous energy Mone Carlo code MCNP4C (Bresmeser, [9]. On he oher hand, he geomerc claspng n he horzonal drecon s smaller han he maeral claspng, he vercal neuron nsably dsrbuon s expressed by a cosne uncon where Bz s he geomerc claspng Where s a drecon cosne wh he z-axs, and oher noaons are sandard whn he nuclear engneerng communy. Ths equaon s a proper value equaon whose proper value s he spaal decompose connuous. Ths equaon s o be solved by he repeon mehod as crcaly calculaons. To nclude he las erm n Eq. ( n he z-drecon. The vercal claspng Bz s dened by where H s he acve hegh o a core and s he exrapolaed lengh. Some deermnsc reacor calculaon codes (Fowler e al. [] have a uncon o ndng a crcal geomerc claspng. The presen paper s o develop a new algorhm o Mone Carlo calculaon mehod or ndng a crcal geomerc claspng. I ceran condons dscussed below are me, he algorhm s also avalable or uel pn cell or uel assembly calculaons n whch he eec o neuron leakage speced by a claspng s aken no accoun n he same manner as he B mehod. The B mehod adops ha he leakage curren s a uncon o neuron energy only and ha he leakage occurs whou dreconal dependence. On he oher hand, he mehod o hs paper adops ha he leakage curren depends only on drecon. Crcal claspng's acqure by he presen mehod are o be compared wh he B mehod n one-energy group n a laer secon.. Revs o spaal decompose connuous search by Mone Carlo mehod.. Theory o proper value mode calculaon by Mone Carlo mehod When he horzonal claspng s smaller han he maeral claspng, spaal decompose connuous search mode calculaons by Mone Carlo (Yamamoo and Myosh, [8] are revsed. Suppose ha he horzonal claspng n he x and y-drecons s larger han he maeral claspng and ha an exernal neuron source s locaed n he z-drecon. I he subcrcal sysem s homogeneous n he z- drecon, he asympoc neuron nsably dsrbuon n he z-drecon s gven by where s a spaal decompose connuous n he z-drecon. The hree-dmensonal neuron ranspor equaon s rewren o a wo-dmensonal orm as:. ( x, y,, E ( x, y,, E ( x, y,, E d de ( x, y,, E ( x, y,, E E ( S ( E d de ( x, y,, E (,, (,,, x y E x y E 4 durng he random processes n a Mone Carlo calculaon, he derenal o he wegh o a neuron parcle ha les over an nnesmal dsance ds s gven by (Yamamoo, []: dw Wds ( 7

3 Sep. 5. Vol. 7. No. Inernaonal Journal o Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved Thus, aer ha S n he h pah wh a drecon cosne, he nal wegh W changes o: W W exp( s ( a hs sage s an unknown proper value and s gven rom he prevous generaon. Unless he wegh changes durng s pah, he produc o he rack lengh lengh S and s wegh W s gven by W S. The wegh, however, changes durng s pah. Aer he h pah wh he dsance S, he, he produc o he rack lengh and he parcle s wegh s gven by: exp( s TL W s ds W (4 s exp( TL n Eq. (4 s he rack lengh esmaor o neuron nsably when neuron wegh s a connuously changng uncon across he pah. Ths s no only o he calculaon o k e bu o all alles as well. Maor modcaons o a Mone Carlo crcaly calculaon code needed or proper value mode calculaons are mplemenaon o Eqs. ( and (4. All oher procedures or crcaly calculaons can be ulzed or proper value mode calculaons whou any modcaons. Aer all random processes whn one generaon are compleed, he proper value s esmaed as: N. W W v. TL TL. where s summarze over all pahs whn one generaon, N s he number o source per generaon, D W = W (exp( zgs - and W s s he wegh o S (5 parcles ha sar rom splng source ses. The sarng wegh W s s gven by M/N where M s a nomnal source sze. N pons come rom he splng source pons o he prevous generaon. Thus, he oal source wegh per generaon s a connuous M. An ndcaor o, k, s dened as: v. TL k (6 Ths ndcaor has he same denon as k e and s supposed o be uny converged. The s used or he nex generaon calculaon s deermned n such a way ha k approaches uny as: c( k (7 where he sands or he generaon number and s an unreasonable posve value. The are averaged over all generaons aer generaons o deermne a nal. The mode calculaon s exends o nny lengh along he z-axs... Conrmaon o mode calculaon by Mone Carlo mehod The algorhm o he mode calculaon by Mone Carlo mehod s nsalled no he connuous energy Mone Carlo code MCNP4C. The mode calculaon was numercally vered by comparng decompose connuous calculaed by he mode calculaons wh hose by xed source calculaons or uranyl nrae aqueous uel soluons (Yamamoo and Myosh, [8]. In he xed source calculaons, neuron nsably dsrbuons n he z-drecon were calculaed and he decompose connuous were acqure by ng he nsably dsrbuons. Ths paper presens a numercal conrmaon usng a homogeneous, one-energy group problem wh soropous dsrbung n he laboraory sysem snce n hs suaon an accurae reerence soluon can be acqure by he duson heory. A es Mone Carlo code has been prepared or he numercal es. Suppose ha a recangular parallelepped wh he sde lengh o 4. cm s composed o a homogeneous mulplyng medum. The one-group connuous s shown n Table. The spaal decompose connuous based on he duson heory s gven by: B B B where, X y m X y (8 B B ( / ( H, (H s sde lengh, B ( / D, m a D / (,.74 The spaal decay consan s based on he duson n he horzonal drecon as well as n he vercal drecon. To mprove he accuracy o s as he reerence soluon, anoher spaal decompose connuous s dened by: 8

4 Sep. 5. Vol. 7. No. Inernaonal Journal o Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved B B h m (9 d Spaal decay consan by c-egenvalue mode calculaon. where B ( k D, K n s neuron h n a mulplcaon acor o he recangular parallelepped wh nne lengh n he vercal drecon. K n was calculaed by a Mone Carlo crcaly calculaon. Thus, he spaal decompose connuous s more accurae han snce he horzonal drecon s reaed by he neuron ranspor heory. Anoher spaal decompose connuous s acqure by nsably dsrbuon n he vercal exp( z drecon o. The nsably dsrbuon was calculaed by a xed source calculaon or he recangular parallelepped wh nne lengh n he vercal drecon. In he calculaon or he nsably dsrbuon, an exernal neuron source was placed a z =. The calculaed vercal nsably dsrbuon s shown n Fg.. The nsably s ed n he regon ar enough away rom he source o exclude he eec o hgher harmoncs. Table - Group consans and calculaon resuls or exponenal lux dsrbuon Parameers Values (cm -.8 (cm -.9 (cm -.4 k n ±. (cm - a.9878 (cm - b.9 ±. (cm - c.945 ±. e (cm - d.9489 ±.5 k.999 ±.6 a Spaal decay consan dened by Eq. (8. b Spaal decay consan dened by Eq. (9. c Spaal decay consan ed o exponenal uncon. Vercal lux (arbrary The e was acqure by he mode calculaon wh 55, neurons per generaon, skppng 5 generaons and runnng 85, acve generaons. A he same me, he ndcaor k dened by Eq. (6 was calculaed. The calculaon resuls o he spaal decompose connuous are shown n Table. Four values n Table relavely agree wh each oher. and slghly der rom and e due o he naccuracy nvolved n usng he duson approxmaon. The shows excellen agreemen e wh he decompose connuous ha s rgorously based on he neuron ranspor heory. Thereore, one may conclude ha he algorhm shown n Secon. provdes an accurae Eq. ( and he algorhm s vered numercally.. Claspng search mode calculaon by Mone Carlo mehod.. Theory o claspng search mode calculaon Ths secon deals wh cases where he horzonal claspng s smaller han he maeral claspng. Accordng o ha he energy-dependen horzonal nsably and he energy-ndependen vercal nsably are separable and he vercal nsably dsrbuon s characerzed by a claspng mode as: 9

5 Sep. 5. Vol. 7. No. Inernaonal Journal o Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved (x,y,z, WE, = ( x, y, WE, exp(b z ( where s he magnary un, B s geomerc claspng n he z-drecon. Here, one o he spaal dependence n he z-drecon by a sngle Fourer mode, exp( z. I s assumed ha he vercal nsably s ndependen o neuron energy. Ths assumpon s no correc bu s orced o be presened or represenng he leakage eec by he sngle acor B as n he wdely-used B mehod. Subsue Eq. ( no he hree-dmensonal neuron ranspor equaon. Then, aer a algebrac manpulaon, one obans a wo-dmensonal ranspor equaon ncludng he vercal claspng B:. ( x, y,, E ( x, y, E ( x, y,, E d de ( x, y,, E ( x, y,, E E ( S ( E d de ( x, y,, E (,, (,,, x y E B x y E 4 Where s a drecon cosne wh he z-axs. Eq. ( s very smlar o Eq. (.To ncorporae he las erm n Eq. ( no random processes o a Mone Carlo calculaon, we ollow he same procedure as n Secon.. The derenal o he wegh o a neuron parcle caused by lyng over nnesmal dsance ds s gven by: dw BWds ( Thus, aer a parcle les over a dsance S n he h pah wh a drecon cosne, he nal wegh W changes o: W W exp( B S W (cos( B S sn( B S ( The produc o he rack lengh and he parcle s wegh aer he h pah wh a dsance S s gven by: S TL W exp( B S ds exp( B S W B sn( B S cos( B S W B B (4 Whle he nal wegh o a parcle sarng rom a splng source s a real number, he subsequen weghs can be complex numbers. I he splng neuron emsson s soropous n he laboraory sysem, he magnary par o W + n Eq. (, W sn( Bz s, appears wh he same requency as W sn( Bz s. Furhermore, he dsrbung s soropous, he magnary pars o weghs dsappear or he same reason. Snce he calculave eld s homogeneous n he z-drecon, he weghs are symmerc wh respec o he z-drecon. In he same way, he magnary par o TL n Eq. (4, (cos( Bx S - B x, whch s an odd uncon o x, s supposed o dsappear. Thereore by magnary pars n Eqs. ( and (4 can be omed only he splng neuron emsson and dsrbung are soropous. Then, under such condons, Eqs. ( and (4 are rewren as ollows: W W cos( B S (5 TL sn( B S W (6 B W and TL can be negave when B z > p. When B s large (.e., large neuron leakage n he z- drecon and a mean ree pah s long, a negave wegh ends o occur n Eq. (5. Even aer a wegh urns negave, he parcle wh a negave wegh keeps beng ollowed. The negave parcle wegh s reaed he same way as were posve. Dscardng he negave weghs s equvalen o shorenng he pah lengh ha B z < p. The proposed mehod, however, adops an nne dmenson n he z-drecon. The wegh gven by Eq. (5 s a perodc uncon o he pah lengh, and he negave wegh becomes posve agan. I he pah lengh s very long n such a way ha B z > p, he wegh may become larger han s wha was beore he pah sared. Ths problem may be emphaszed n reacor sysems wh gas-lled regons. Such heerogeneous sysems ncludng vod regons are problemac and should be reaed as a uure work. I a parcle colldes, (n splng source ses are sored or use as splng sources n he nex generaon where: n n = In W + R ( (7 In(... s he neger par, R s unorm smulaed random number n [,. I W <, a negave wegh s s

6 Sep. 5. Vol. 7. No. Inernaonal Journal o Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved s a sgned o he n splng sources. Aer all random processes whn one generaon are compleed, he proper value s esmaed as: N W. S + DW - n. TL B = A Where s summarze over all pahs whn one generaon, N s he number o source per generaon, D W = W (cos( Bz S -, and: é S ù A = Re - x W exp( - B xs ds = ê ë úû W = ( cos( Bx S - B (8 (9 Here, Re[...] s a real par, and N = N p - N m where N p and N m are he numbers o splng source ses wh posve and negave weghs, respecvely. W s s gven by W s = M/N where M s a nomnal source sze. k e o Eq. ( s supposed o be uny and s gven by: n. TL k = ( e M The claspng B used or he nex generaon calculaon s deermned ha k e approaches uny as: = + l B ( - l + Bl d K e ( Where l sands or he generaon number and d s an unreasonable posve value. Splng sources wh negave weghs mus be cancelled a he end o each generaon. Ths presen paper uses he procedure or he wegh cancellaon (Yamamoo, []. A whole regon s dvded no a large number o small regons. Splng sources wh posve and negave weghs are accumulaed n he bns. The number o splng source parcles n he h bn s gven by n = n p - n m where n p and n m are he number o splng sources wh posve and negave weghs, respecvely. n splng sources ha are used or he nex generaon s sarers are dsrbued unormly whn he h bn... Conrmaon o claspng search mode calculaon As shown n Secon., homogeneous one-energy group problems were used or numercal conrmaon o he claspng search mode calculaon. The one-energy group does no lm he generaly o hs developed mehod. I s sraghorward o exend hs mehod o general connuous energy problems. Frs, one-energy group problems ha have nne dmenson n he horzonal drecon were beleved. In hs case, he neuron nsably n he neuron ranspor equaon, Eq. (, becomes a uncon o drecon only as ollows: ( W - ( s + n 4p ( W dw + Bx ( W = 4p ( Ths problem s suable or conrmaon o he claspng search mode calculaon model snce no neuron ranspor n he horzonal drecon s presened. The proper values B s o Eq. ( were acqure by search mode calculaons wh 55, neurons per generaon, skppng 5 generaons and runnng 8, acve generaons. Snce hs problem adops ha he splng neuron emsson and dsrbung are soropous, he magnary pars o wegh cancel ou and he weghs are expressed by usng Eqs. (5 and (6. The calculaons were perormed or several cases. The group connuous and calculaon resuls are shown n Table. B s he claspng based on he duson and s gven by,b m e n Table s he claspng acqure by he claspng search calculaon. In cases,, and n Table, we have a negave wegh n each pah s less han -7 even when n = or - Thus, he eec o negave weghs s almos neglgble. In case 4, he probably s % when n = or - and he racon o splng sources wh negave weghs s.%. Thus, he eec o negave weghs s sgncan n case 4. In cases,, and, he resuls o he claspng search calculaons agree wh hose by he duson whn.%. On he oher hand, n case 4 a noable derence beween B and B e s ound. Fg. shows he dependences o neuron nsably or cases and 4 ha were acqure durng he claspng search mode calculaons. The dependence or case s almos la and, he slgh dependence can be accuraely come near by a lnear uncon o he drecon cosne, whch s one o he requses o he duson approxmaon. Ths ac means ha he duson gves a que accurae claspng. For case 4, however, he dependence clearly

7 Sep. 5. Vol. 7. No. Inernaonal Journal o Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved devaes rom a lnear uncon o he drecon cosne. Thus, B n case 4 s beleved o be based, and he derence beween B and B e may be due o he naccuracy o B. To oban a more accurae claspng, an nsably dsrbuon n he z-drecon was calculaed or a horzonally nne bu vercally ne slab ha s nearly crcal (k e =.8 ±.. The hckness o he crcal slab was 5.75 cm. The vercal nsably dsrbuon was ed o a cosne uncon, he vercal claspng. The vercal nsably dsrbuon, however, s no exacly cosne shape due o neuron ranspor ransen eecs near he ouer boundares. The nsably dsrbuon only near he cener o he slab was used or he ng. The nsably dsrbuon and ed cosne curve are shown n Fg.. The area used or he ng was.5 cm. The claspng acqure rom he nsably dsrbuon, B c, s gven n Table. The claspng B c agrees well wh B e, whch may show conrmaon o he claspng search mode calculaon wh large neuron leakage. Anoher conrmaon can be done by comparng wh he B mehod. The B equaons are gven n (Dudersad and Hamlon, [] as: Table-Group consans and calculaon resuls or he bucklng search mode calculaons wh nne dmenson n he horzonal drecon. Case Case Case Case 4 (cm a (cm (cm B (cm - a B e (cm - b.68 ± ±..74 ±..59e ±. B c (cm - c ±.5 B (cm - d k e. ±.. ±.. ±.. ±.5 a Bucklng by duson approxmaon. b Bucklng by bucklng search mode calculaon. c Bucklng by ng o cosne uncon. d Bucklng by he B mehod. e I he negave weghs are dscarded, Bc =.678 ±. cm-. B J ( u + ( u ( u = = du ( u u ( u + k e u S c ( u du n ( u ( u S B ( u + g( B, u ( u J ( u = u = du ( u u J ( u æ B ö æ ö - B an ç ( ( è ç u ø è u ø g( B, u = æ B - æ ö an B - ç ç ( ç ( è è u u ø ( (4 (5

8 Sep. 5. Vol. 7. No. Inernaonal Journal o Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved B = - - B B B (7 m x y Where u s he neuron lehargy, J(u s neuron curren, s soropous componen o he dsrbung cross secon, and s he lnearly an soropous dsrbung componen o he dsrbung cross secon. The equaons are moded rom he orgnal ones n he leraure. Assumng ha he dsrbung s soropous (.e,. = and he neuron s monoenergec, Eqs. ( and (4 are smpled o: B ( - S g( B + = n g( B (6 k e where he varable u and he subscrps or he dsrbung orders are omed. Usng Eq.(6, he crcal buck lngs or k e = can be easly acqure by erave calculaons. The crcal buck lngs acqure by Eq.(6 are lsed n Table. The crcal bucks lngs acqure by he B mehod agree very well wh he Mone Carlo calculaons or all cases. I he negave eghs are mmedaely dscarded, he calculaed crcal claspng s.688. cm whch s denely wrong. The proposed algorhm or he claspng search calculaon by Mone Carlo mehod can reproduce very accuraely he B mehod. For numercal conrmaon o he mehod or solvng Eq. (, a recangular parallelepped wh a ne sde lengh n he horzonal drecon was beleved or each o cases 4. The group connuous, dmensons, and calculaon resuls are shown n Table.B n Table s a vercal claspng and s dened by: Anoher vercal claspng B s gven n Table, whch s dened by: B = - B B (8 m h where B = ( n k - / D, k = k o h n a m e he recangular parallelepped wh nne lengh n he vercal drecon. k n was calculaed by a Mone Carlo crcaly calculaon. B s more accurae snce he horzonal drecon s reaed by he neuron ranspor heory. Excellen agreemen s ound beween B and B e excep or case 4 where ansoropy o neuron nsably s sgncan due o he large vercal claspng and long mean ree pah. 4.Leakage-correced calculaon by Mone Carlo mehod Suppose ha a claspng n he vercal drecon s known a pror and he horzonal drecon s reaed, ncludng he horzonal neuron leakage, by he neuron ranspor heory. The neuron ranspor equaon or hs case s gven by: WÑ. ( x, y, W, E + ( x, y, E ( x, y, W, E = dw de ( x, y, W, E ( x, y, W W, E E (9 c ( E + dw de ( x, y, W, E n ( x, y, E k 4p e - Bx ( x, y, W, E Where B s no an proper value bu a user-speced claspng n he z-drecon. The calculaon algorhm S

9 Sep. 5. Vol. 7. No. Inernaonal Journal o Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved or solvng Eq.(9 s almos he same as convenonal Mone Carlo crcaly calculaons or seekng k e. The excepons are ha a parcle wegh and he produc o a rack lengh and parcle wegh are gven by Eqs. (5 and 6 ( respecvely. k e appears explcly as an proper value o Eq.(9. The claspng B s no updaed and s xed a he userspeced value hroughou he calculaon. The calculaon n whch neuron leakage n he vercal drecon s correced by a user-speced vercal claspng s easer han he claspng search mode calculaon snce he claspng need no be updaed. By solvng Eq. (9, one can oban a leakagecorreced neuron nsably dsrbuon and neuron specrum, whch wll subsequenly be used or generang group connuous. Nex, consder a consuen un o a whole core conguraon such as a un uel, a sngle uel assembly, or mul-uel assembles. The relecve condons are requre on he boundary suraces o he consuen un. Alhough he neuron ranspor phenomenon s reaed by Mone Carlo mehod whn he range o he un, he eecs o vercal and horzonal neuron leakage are presened by vercal and horzonal buck lngs, respecvely. Agan, he un s homogeneous and nne n he z-drecon. Assume ha he hreedmensonal nsably dsrbuon n an x y z Caresan coordnae sysem s characerzed by a smple claspng mode as: ( x, y, z, W, E = ( ( x, y, W, Eexp( ( B x + B y + B z x y z where B x, B y and B z are geomerc drecon buck lngs n x, y, and z-drecons, respecvely. Subsung Eq. ( no he hree dmensonal neuron ranspor equaon, one obans a wodmensonal ranspor equaon ha s smlar o Eq. (. However, he las erm s deren and s gven by: - ( B m+ B h + B x ( x, y, W, E ( x y z where u, n are drecon cosnes wh he x- and y- axes, respecvely. Eqs. ( and (4 are, respecvely, rewren as: W = W exp( - ( B m + B + h + B x S = x y z = W (cos(( B m + B h + B x S x y z - sn(( B m + B h + B x S S x y z TL = W exp( - ( B m + B h + B x S ds x y z æsn(( xm yh zx S ö B + B + B + xm yh zx B + B + B = W cos(( B xm + B yh + B zx S - ç xm yh zx çè B + B + B ø ( ( Consder ha a horzonal calculave eld has wo orhogonal symmery planes and he dsrbung and splng neuron emsson are soropous. For such a case, Gelbard and Lell [] menoned a claspng canno exs anywhere n he lace one canno a cosne very well o any segmen o hs curve. However, suppose ha one connues o presume, roughly, a cosne shape even or an asymmerc lace cell and he horzonal bucklng B x and B y can be dened whn a ceran range o accuracy. Then, one opon s o neglec he magnary pars o Eqs. ( and ( (whch however may presen errors ha are dcul o be quanavely esmaed. Anoher opon s o rea complex weghs dened by Eq. (. A deermnsc core calculaon code equpped wh he B mehod could be used or conrmaon o he new algorhm ha handles complex weghs. When one s neresed n a crcal claspng search calculaon n a hree-dmensonal conguraon, hree drecon bucklng, B x, B y and B z, need o be deermned. However, he combnaon o hree drecon bucklng s unreasonable. Thus, some resran condons are requred on he combnaon o B x, B y and B z. For nsance, B x and B y are xed a connuous and hen only B z s searched. Or, B x s equal o B y and he rao o he horzonal claspng o he vercal claspng s xed and so on. 5. CONCLUSIONS Ths paper has used a Mone Carlo calculaon algorhm or claspng search mode calculaons where he horzonal drecon s reaed by neuron ranspor calculaon and neuron leakage eec n he remanng vercal drecon whch s aken no accoun by he vercal claspng. When he horzonal claspng s larger han he maeral claspng, he 4

10 Sep. 5. Vol. 7. No. Inernaonal Journal o Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved vercal neuron nsably dsrbuon akes he orm o an exponenal uncon. The spaal decompose connuous o he vercal neuron nsably dsrbuon can be acqure as a proper value by solvng a claspng search mode proper value equaon. The proper value can be vered by comparng wh he spaal decompose connuous calculaed by a xed source problem. On he oher hand, when he horzonal claspng s smaller han he maeral claspng, a proper value equaon s a claspng n he vercal drecon can be dened by he vercal drecon by a sngle Fourer mode, exp(bz. The proper value equaon ncludes a complex erm, whch produces complex parcle weghs durng he Mone Carlo random processes. However, he calculave eld s unorm n he vercal drecon and he dsrbung and splng neuron emsson are soropous, he weghs urn ou o be symmerc wh respec o he vercal drecon, whch elmnaes he magnary pars rom he complex weghs. Thus, only he real pars reman n he random processes. When a claspng s large and a neuron s mean ree pah s long, a negave wegh may appear durng he random processes. The negave wegh s reaed he same way as he posve wegh. When appearance o negave weghs s sgncan, he duson becomes worse due o noable ansoropy o neuron nsably. In such a case, conrmaon by comparng wh he duson s naccurae. The claspng acqure by he claspng search mode proper value calculaon was compared wh he one acqure by he nsably dsrbuon or by he B mehod. Boh bucklng agree well, and he claspng search mode calculaon has been vered or large neuron leakage sysems. Ths Mone Carlo algorhm or nroducng neuron leakage eec by a claspng can also be appled o leakage-correced Mone Carlo calculaons. The bucklng n he x, y, and z-drecons are xed a user-speced values, and hen Ke, neuron specrum, splng source dsrbuon and so on are calculaed or generang group connuous used or a subsequen reacor core desgn. When a horzonal calculave eld has wo orhogonal symmery planes and he dsrbung and splng neuron emsson are soropous he magnary pars o complex weghs dsappear. I does no, complex weghs need o be reaed. The mehodology or reang complex weghs has no ye been esablshed and would be a uure work. The newly developed algorhm or solvng he claspng search mode o proper value equaon was vered hrough comparson wh he claspng does acqure by he duson and he B mehod. As long as he duson s reasonable, boh claspng's acqure by he claspng search mode o proper value and he duson agree well, whch exhbs conrmaon o he new algorhm. REFERENCES. Yoshoka, K., Ando, Y.,. Mulgroup scaerng marx generaon mehod usng wegh-o-lux rao based on a connuous energy Mone Carlo echnque. Journal o Nuclear Scence and Technology 47, Leppänen, J., 7. Developmen o a New Mone Carlo Reacor Physcs Code. VTT Publcaons 64.. Dudersad, J.J., Hamlon, L.J., 976. Nuclear Reacor Analyss. John Wley & Sons, New York. 4. Frdman, E., Leppänen, J.,. On he use o he Serpen Mone Carlo code or ew group cross secon generaon. Annals o Nuclear Energy 8, Shm, H.J., Han, B.S., Jung, J.S., Park, H.J., Km, C.H.,. McCARD: Mone Carlo code or advanced reacor desgn and analyss. Nuclear Engneerng and Technology 44, Maorov, L.V., 985. Calculang neuronlux unconals by he Mone Carlo mehod n breeder sysems wh leakage speced by a geomerc parameer. Translaed rom Aomnaya Énergya 58, Yun, S., Cho, N.Z.,. Mone Carlo depleon under leakage-correced crcal specrum va albedo search. Nuclear Engneerng and Technology 4, Yamamoo, T., Myosh, Y.,. An algorhm o α and c-mode egenvalue calculaons by Mone Carlo mehod. In: Proc. 7h In. Con. on Nuclear 9. Bresmeser, J.F., (Ed.. MCNP A General Mone Carlo N-parcle ranspor code, Verson 4C. LA-79-M.. Fowler, T.B., Vondy, D.R., Cunnngham, G.W., 969. Nuclear Reacor Core Analyss Code: CITATION, ORNL-TM-496, Rev... Yamamoo, T., a. Hgher order a mode egenvalue calculaon by Mone Carlo power eraon. Progress n Nuclear Scence and Technology,

11 Sep. 5. Vol. 7. No. Inernaonal Journal o Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved Yamamoo, T., b. Non-regon wse wegh cancellaon or Mone Carlo hgher order crcaly calculaons usng kernel densy esmaor. Annals o Nuclear Energy 8, Gelbard, E.M., Lell, R., 977. Mone Carlo reamen o undamenal-mode neuron leakage n he presence o vods. Nuclear Scence and Engneerng 6, 9. 6