Higher Order Mode Eigenvalue Calculation by Monte Carlo Power Iteration

Transcription

1 Proress n NUCLEAR SCIENCE and TECHNOLOGY, Vol., pp (0) ARTICLE Hher Order Mode Eenvalue Calculaton by Monte Carlo Power Iteraton Toshhro YAMAMOTO * Kyoto Unversty, -00 Asashronsh, Kuator-cho, Sennan-un, Osaka, , Japan A recently proposed ethod pleentn a pseudo absorpton ter n the neutron transport equaton for ode eenvalue calculatons n the Monte Carlo technque has been dscussed. Ths ethod s known to overcoe the dffculty n ode eenvalue calculatons for fssonable systes wth lare subcrtcalty. Ths paper has deonstrated that ths technque certanly can provde stable ode eenvalue calculatons for probles that the ornal MCNP 4C fals to solve and that the fure of ert of the calculated value s larely proved. Howeve cauton should be taken that the varance of calculated values s underestated due to the nter-cycle correlaton of the values. A ethod that provdes eenfunctons wth hher order crtcalty eenvalues has been appled to the second order ode eenvalue calculatons. The ethod parttons a whole space nto two reons. The estate of the ultplcaton factor n each reon s forced to be equal to each other. Ths ethod s found to be applcable to calculatons for the second ode eenfuncton. The condtons on converence of the second ode eenfuncton are dscussed. Hotelln s ethod, whch explctly subtracts lower order eenfunctons fro the fsson source dstrbuton, has been appled to a test proble. The thrd and fourth order forward eenfunctons are successfully obtaned wth the use of Hotelln s ethod. KEYWORDS: Monte Carlo, ode, hher order ode, crtcalty calculaton, power teraton, eenvalue, pulsed neutron I. Introducton A propt neutron te decay constant (hereafter called ) s one of portant paraeters to acqure the neutronc property of a fssonable syste. In an experent that uses a fssonable ateral, the constant s one of values that can be ectly easured by the pulsed neutron ethod, Ross- ethod, Feynan- ethod and so on. On the other hand, an effectve neutron ultplcaton factor k eff cannot be exactly easured except when the syste s crtcal. Thus, one of the ethods of valdatn crtcalty calculaton codes s the coparson of the calculated and easured. When valdatn Monte Carlo crtcalty calculaton codes wth value coparson, the capablty of calculatn values s requred n Monte Carlo crtcalty calculaton codes. For exaple, Monte Carlo sulaton for pulsed neutron ethod can provde an value, whch can be avalable for the code valdaton. Another ethod for obtann an value by the Monte Carlo crtcalty calculaton ethod s an eenvalue ode calculaton. ) The functon of eenvalue ode calculaton had been nstalled n a contnuous enery Monte Carlo code MCNP verson 4C. ) The value obtaned by the eenvalue ode calculaton corresponds to a fundaental ode. Lke the crtcalty calculaton for a fundaental ode neutron ultplcaton factor k eff, the eenvalue ode calculaton s relatvely easy as copared to radaton sheldn calculatons that requre elaborate varance reduc- *Correspondn autho E-al: pbrasko@yahoo.co.jp c 0 Atoc Enery Socety of Japan, All Rhts Reserved. 86 ton technques. The eenvalue and the fsson source dstrbuton convere toward the fundaental ode wthout any specal treatents. On the other hand, n the easureent of an value, the contanaton of the easured value by hher order odes s unavodable. The effect of the hher order odes depends on the locaton of a neutron source and neutron detectors. Endo et al. derved eneralzed but coplcated theoretcal forulae for the well-known Feynan- ethod (.e., the second order neutron correlaton technque) and the thrd order neutron correlaton technque that takes nto consderaton hher order ode eenvalues and eenfunctons. 3) A desrable technque of the easureent of a fundaental ode value s to elnate the hher order ode effects as uch as possble. If ths elnaton s successful, sple forulae for the thrd order neutron correlaton can be appled to easurn a fundaental ode value. If the contanaton of hher order odes s nevtable, eneralzed but coplcated forulae have to be used. Therefore, the knowlede on the hher order ode eenvalues and eenfunctons s requred. Fro these vew ponts, the dentfcaton of the ode hher order ode eenfunctons and eenvalues s portant for the easureent of an accurate fundaental ode value as well as the code valdaton usn values. Soe attepts to calculate eenfunctons wth hher order crtcalty eenvalues have been done by Booth 4,5) and Yaaoto. 6) For exaple, to obtan the second eenfuncton, the ethod parttons a whole space nto two reons. In

2 Hher Order α Mode Eenvalue Calculaton by Monte Carlo Power Iteraton 87 the course of the power teraton of a Monte Carlo crtcalty calculaton, the apltude of the eenfuncton n each reon s adjusted such that the estate of the eenvalue n each reon s forced to be equal to each other. Ths technque has an advantae n requrn no nforaton on the lower order eenfunctons and s sutable for Monte Carlo crtcalty calculatons. Therefore, t s expected to be a atter of course that ths technque ay be applcable to enerate ode hher order eenfunctons by Monte Carlo technques. Ths paper ay be the frst attept to obtan hher order ode eenvalues and eenfunctons by the Monte Carlo technque. In ths pape the Monte Carlo calculaton technque of an ode eenvalue s revsted. Recently, a ethod for provn the relablty of the ode eenvalue calculaton wth Monte Carlo technques has been proposed. 7) The effect of ths proveent s dscussed. Then, the ethod for hher order crtcalty eenfunctons wth Monte Carlo technques s appled to eneratn the second order ode eenvalue. The condtons on the converence of the second order eenfuncton are treated. Fnally, the prospect of further hher order ode eenvalue s dscussed. II. Revew of Mode Eenvalue Calculaton wth Monte Carlo Technque. Alorth of Mode Eenvalue Calculaton n Monte Carlo Technque Before dscussn the hher order ode calculaton, the fundaental ode calculaton for an eenvalue wth Monte Carlo technque s dscussed. A neutron transport equaton n the propt neutron decay constant eenvalue ode s wrtten as (, t ( (, d' de' ( ', E') s ( ', E' () p ( d' de' ( ', E') p f ( E') 4 (, / v(, r s a three-densonal poston and v( s a neutron velocty wth enery E. The subscrpt p stands for the value of propt neutrons. Other notatons are standard n neutroncs. In ths transport equaton for a subcrtcal syste, the fundaental ode neutron flux chanes exponentally wth the fundaental ode eenvalue (.e., te-decay constant) (>0) after hher order odes de out as (, E, t) e t. () If a fssonable syste s supercrtcal wthout delayed neutrons, the neutron flux rows exponentally wth the te-decay constant (<0). In a subcrtcal fssonable syste, the last ter n Eq. () plays a role as a source ter to ake the eenvalue ode equaton hold stable. Yaaoto et al. adopted a follown dfferent approach for pleentn the last ter n Eq. () durn the rando walk of Monte Carlo calculatons. 8) The exstence of the last ter n Eq. () shows that neutron weht ncrease or decrease when a neutron fles throuh a edu. If the neutron flux n the last ter n Eq. () s replaced by the neutron weht, the ter represents the weht chane per unt partcle flht dstance. Thus, the weht chane dw of a neutron partcle that fles by an nfntesal lenth of ds s ven by dw / v( W ds. (3) Here, suppose that the value s known fro the calculaton n the prevous cycle of a power teraton. After the flht of a lenth s, an ntal weht W 0 s chaned to W as W= W exp( s / v( )). (4) 0 E T, the product of the track lenth and the weht n the -th flht wth a lenth s, s defned by s T W0 exp( s '/ v ( ) ds ' 0 (5) W0 (exp( s / v ( )) ) / ( / v ( ). v ( s the neutron velocty of the -th flht. The nuber of fsson neutrons n used for the startn source neutrons n the next cycle s deterned by ) n INT ( W p f t (6) s a unfor pseudorando nuber between 0 and, and INT stands for the nteer part of the value wthn the parentheses. Note that the second ter n the parentheses n Eq. (6) s not dvded by k eff unlke crtcalty calculatons. The nteraton of two ters on the left hand sde n Eq. () over all phase space throuhout one cycle calculaton s equal to total loss of neutrons due to absorpton and escape. Thus, the nteraton s ven by the su of the total weht of startn source partcles and ncrease n weht due to Eq. (3). The scattern ter n Eq. () does not contrbute to the weht chane. By nteratn the last two ters n Eq. () over all phase space throuhout one cycle and solvn for, an eenvalue s estated as N W p f T T / v ( (7) s sued over all track lenths n a cycle, N = nuber of source hstores per cycle, W W0 (exp( s / v ( ) ). (8) Usn Eqs. (5) and (8), Eq. (7) s rewrtten as N p f T. (9) W Then, the next cycle calculaton uses for the value n Eqs. (4) and (5). A fnal soluton of eenvalue s ven by sply averan the values of all actve cycles after dscardn soe ntal cycles that ay be contanated by not fully convered fsson source dstrbuton. VOL., OCTOBER 0

3 88 Toshhro YAMAMOTO For a fssonable syste wth a lare subcrtcalty, the last ter n Eq. () (.e., source ter) becoes ore donant than the second last ter n Eq. () (.e., fsson ter) snce the lare subcrtcal syste has a lare value. In such a case, the neutron weht becoes too lare to keep sustanable Monte Carlo power teraton. 9) Abnoral ternatons n ode eenvalue calculatons were reported n the ornal MCNP verson 4C. 7) To ake ode eenvalue calculatons ore stable, Yen et al. ntroduced a pseudo neutron absorpton ter n the eenvalue ode neutron transport equaton as * (, t ( (, d ' de' ( ', E') s ( ', E' (0) p ( d ' de' ( ', E') p f ( E') 4 ( ) (, / v(, * t ( t ( / v(. () The ter / v( n Eq. () ncreases the acroscopc total cross sectons, leadn to ncrease n neutron absorpton. A paraeter s approprately adjusted so that the eenvalue ode calculaton can be stable. The pseudo absorpton ter can be pleented by replacn by (+) n Eqs. (4), (5) and (8). Equaton (7) s stll avalable for obtann the for the next cycle calculaton even when the pseudo absorpton ter s ntroduced. The last ter n the nuerator n Eq. (9) s alost equal to N, the nuber of source hstores per cycle, reardless of after the power teraton enters a stable cycle. On the other hand, the denonator n Eq. (9) ncreases wth ncreasn. Therefore, as seen n Eq. (9), the chane n the value (.e., ) n each teraton s suppressed by ntroducn the pseudo absorpton ter. The effect of the pseudo absorpton ter s shown below.. Exaples of Mode Eenvalue Calculatons n Monte Carlo Technque As nuercal tests for the alorth of ode eenvalue calculatons explaned above, ode eenvalue calculatons were perfored for a soluton fuel of enrched uranyl ntrate that was used n the Statc Crtcalty Experental Faclty (STACY) at the Japan Atoc Enery Aency. 0) Althouh the experents n Ref. 0) were perfored for 9.97-wt.%-enrched uranyl ntrate soluton, the 35 U enrchent for the test calculatons was approxately 6 wt.% that was used for the experent capan of the STACY n 004. The uranu concentraton was approxately 378 /. The experental core confuraton for the nuercal tests was an unreflected cylncal for wth the daeter of 79 c and the heht of approxately 55 c. The soluton fuel was contaned n a cylncal core tank ade of stanless steel. The k eff of the fuel soluton was approxately α (/sec) λ= 0 λ=5 averae Cycle F. Calculated value transtons wth and wthout pseudo absorpton ter The alorth explaned above was pleented nto MCNP 4C. The ode eenvalue calculatons for the cylncal fuel soluton syste were perfored wth 5 nactve cycles and,000 actve cycles at 0,000 neutrons per cycle. The calculated values, ther standard devatons and the relatve fure of ert (FOM) for several adjustent paraeters are lsted n Table. The k eff s, whch would be exactly unty n the ode eenvalue calculatons, are also lsted n Table. The FOM are defned as /T, T s the CPU te and s the standard devaton. Note that these values and ther standard devatons are the results of sple statstcal processn over,000 actve cycles. Table shows that the standard devaton decreases as the paraeter ncreases. Fure shows the transtons of the calculated values n each cycle for several values. As shown n F., the fluctuaton of calculated values becoes suppressed as the paraeter ncreases. Furtherore, an apparent correlaton of values between nehborn cycles s seen for larer. In other words, the value calculated by Eq. (7) s nfluenced by the value that s used n Eqs. (4) and (5). In the alorth of ths pape the value of Eqs. (4) and (5) n the next cycle uses the value obtaned n the prevous cycle. Under such a stuaton wth the nter-cycle correlaton, sple statstcal processn of values underestates the standard devaton. Ths s a coon proble arsn n crtcalty calculatons that always ental the nter-cycle correlaton of fsson source dstrbuton. How to estate an value used for the next cycle and how to estate the true standard devaton would be a research ssue n the future. Howeve t s certan that ntroducn the pseudo absorpton ter n the eenvalue ode neutron transport equaton akes ode eenvalue calculatons stable and relable. To verfy the values calculated by the eenvalue ode, an equvalent value was obtaned by a dfferent calculaton technque usn the sae Monte Carlo code (MCNP 4C) and the sae cross secton lbrary (JENDL-3.3). As one of the technques for the verfcaton, the sulaton of the pulsed neutron experent was perfored. Pulsed fsson neutrons were enerated unforly wthn the whole fuel reon, and then the subsequent neutron counts were accuulated n te bns. The decay constant, whch s PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY

4 Hher Order α Mode Eenvalue Calculaton by Monte Carlo Power Iteraton 89 ode eenvalue calculaton Pulsed neutron ethod sulaton Table Calculated values of 6-wt.%-enrched uranyl ntrate soluton for several 's (s - Relatve ) FOM of W ± ± ± ± ± ± ± ± ± ± 0. k eff ode eenvalue calculaton Pulsed neutron ethod sulaton Table Calculated values of 4 wt.% enrched uranu soluton for several 's (s - ) Relatve FOM of W k eff ± ± ± ± ± ± ± ± ± ± 0.8 deeed to be equvalent to the ode eenvalue, was obtaned by fttn the neutron te-decay n the duraton when the hher order ode fluxes apparently decay. The value obtaned by the pulsed neutron sulaton s also lsted n Table. The values obtaned by the dfferent values and the pulsed neutron experent sulaton ndcate areeent wth each other wthn the statstcal uncertantes. Ths test proble s a oderate subcrtcal syste that MCNP 4C can successfully yeld a convered eenvalue. In another test proble, the 35 U enrchent and the uranu concentraton were reduced to 4 wt.% and 00 /, respectvely. The heht of the cylncal fuel soluton was set to 35 c wth other condtons ben dentcal to the prevous test proble. The k eff of ths fuel soluton was approxately An ode eenvalue calculaton by the ornal MCNP 4C abnorally halted before all cycles to be done were copleted. The odfed MCNP 4C wth the adjustent paraeter = 0 also faled to keep a sustanable power teraton of ode eenvalue. The paraeter was adjusted so that the ode eenvalue calculatons were norally ternated. Table also shows the results wth several values for the lare subcrtcal syste. In ths syste, the calculatons were found to be successfully fnshed when the paraeter was larer than.0. Aan, F. shows the transtons of the calculated values n each cycle for several 's. Even for the sallest, the values oscllate around the averae value wth a loner perod as opposed to the case wth = 0 n F.. Thus, a certan level of nter-cycle correlaton of an value s nevtable for a lare subcrtcal fssonable syste. The sulaton of the pulsed neutron experent was also perfored for ths lare subcrtcal syste. In the sae anner as the prevous test proble, the te-decay constant was obtaned and s lsted n Table. The values n Table aree well wth each other reardless of the paraeter and the calculaton ethods. α (/sec) λ=.0 λ=5.0 averae Cycle F. Calculated value transtons for = and 5 III. Mode Hher Order Eenvalue Calculaton n Monte Carlo Technque. Converence of Fundaental Mode Eenvalue In ths secton, hher order ode eenvalue calculatons usn the power teraton ethod are dscussed. Hher order ode calculatons usually requre subtracton of lower order eenfunctons fro the fsson source dstrbuton as represented by Hotelln s ethod.,) Monte Carlo ethods, howeve have dffcultes to ake subtracton snce Monte Carlo ethods defne the neutron flux dstrbuton by any partcles ben randoly dstrbuted throuhout a fssonable syste. Soe attepts to perfor hher order crtcalty eenvalues wthout subtracton technque have been done by Booth 4,5) and Yaaoto. 6) Ths technque s sutable for Monte Carlo calculatons. Ths paper attepts to apply the technque to hher order ode eenvalue calculatons. In the power teraton ethod for a fundaental ode eenvalue calculaton, the ntal fsson source dstrbuton, VOL., OCTOBER 0

5 830 Toshhro YAMAMOTO whch ay be far fro the fundaental ode, s ven at the bennn of the calculaton. The rato of a hher order eenfuncton s apltude to the fundaental ode s one attenuates at each power teraton, then the fsson source dstrbuton converes to the fundaental ode. Let the fsson source dstrbuton and the -th ode eenfuncton be P ( and (, =,,, respectvely. The fsson source dstrbuton s ven by P( f ( (, de d, () 4 0 f ( = acroscopc fsson cross secton of neutron enery E at poston r. Here, k s defned as the -th eenvalue of -ode equaton of Eq. () for a ven value, and s ordered k >k k 3,, >0 the eenvalues are assued to be real and postve. If the value s equal to, the fundaental ode eenvalue of Eq. (), then k =. If >, then k >. An operator A, whch corresponds to a snle power teraton, s ntroduced, then A ( k (. (3) Because of copleteness, an arbtrary fsson source dstrbuton P ( can be expanded as a lnear cobnaton of the eenfunctons P( c (, (4) c s an apltude of the -th eenfuncton. P ( s noralzed at each teraton throuhout the Monte Carlo calculaton. For exaple, N P(, (5) N s the nuber of partcles per teraton. Applyn the operator A to P ( stands for obtann the fsson source dstrbuton n the next teraton: AP( ck (. (6) Snce k > k k 3,, > 0, applyn the operator A any tes wth a constant value has the lt as: n l A P( c (, (7) n k n only the eenfuncton wth the larest eenvalue survves. Actually, the value s updated at each teraton by Eq. (7). If > > 0 and P ( s assued to be approxately proportonal to the fundaental ode eenfuncton ( ) after soe teratons wth the constant value, then r p f T k N N. (8) due to k >.Thus, as can be understood fro Eq. (9), <, whch says that the value does not dvere fro the fundaental ode eenvalue correspondn to k =.. Converence of Second Mode Eenvalue Here, a ethod for obtann hher order crtcalty eenvalues, whch was proposed by Booth, 4) s appled to seekn the second ode eenvalue. Ths ethod parttons the space nto two reons R I and R II. Two k s n the n-th teraton are defned as: and I I ( n) I k P P, (9) II II ( n) II k P P, (0) ( n) ( n) P I P ( RI ( n) ( n) P II P ( RII I ( n), (), () P AP (, (3) II RI ( n) P AP (, (4) ( ) P n RII and ( s the fsson source dstrbuton n the (n-)th teraton. How the space s parttoned nto two reons ht be arbtrary. Howeve t s preferable to partton the space accordn to the sn of P (. 6) For exaple, P ( >0 n R I and P ( <0 n R II. I II The ethod utlzes the property that k = k I II = k when P( (. When k k, ths fact eans that the coponent n the reon R I s rown faster than n the reon R II and vce versa. Modfyn the fsson source dstrbuton to keep two coponents rown at the sae rate leads to suppressn the fundaental ode eenfuncton relatve to the second one. Ths was proved by Booth 4) and Yaaoto 6) for crtcalty eenvalue probles. Such a odfcaton of the fsson source dstrbuton after each power teraton would be expected eventually to lead to converence toward the second ode eenfuncton. Booth proposes that the fsson source dstrbuton for the ( ) next teraton P n ( be odfed as or P ( P ( II I ( n) k k AP ( I II ( n) k k AP ( for for r RI, (5) r RII, (6) >. Then, the odfed fsson source dstrbuton ( ) P n ( ets renoralzed as n Eq. (5). The odfed dstrbuton s used as P ( n the next cycle. In the case of a crtcalty eenvalue proble, the rane of that allows a PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY

6 Hher Order α Mode Eenvalue Calculaton by Monte Carlo Power Iteraton 83 Vacuu 0 F. 3 5 c 5 c Reon I Materal Materal Materal 5 c converence toward the second eenfuncton s ven by 4,6) ( k k) ( k k). (7) In the case of an ode eenvalue proble, such a rane of for converence cannot be strahtforwardly derved snce the value s updated each teraton. The eenfunctons chane wth the value whle the eenfunctons of a crtcalty ode eenvalue proble are fxed throuhout the calculaton. The rane of for converence s to be studed eprcally n the follown secton. IV. Nuercal Applcatons Reon II Vacuu Calculaton odel for test proble. Second Mode Eenvalue The technque for the second ode eenvalue that uses Eq. (5) or (6) was appled to a sple test proble n two neutron enery roups. A syste for the test proble was coposed of two fssonable nfnte slabs decoupled by an ntervenn non-fssonable ateral. The eoetry s syetrc as shown n F. 3. The two-enery roup constants for the test proble are shown n Table 3. The scattern was sotropc n the laboratory syste. The notatons n Table 3 are standard. Ths syste s subcrtcal, and the k eff wthout delayed neutrons s For hher order ode calculatons n the Monte Carlo technque, postve and neatve wehts need to be cancelled durn the course of the calculaton. The cancellaton should be conducted n a pontwse anner. Monte Carlo ethods, howeve have a dffculty to ake such a pontwse cancellaton of postve and neatve wehts. Booth 4) used a pont-detector lke technque for the pont cancellaton. Ths technque, howeve rases a proble how to decde the pont postons and the nuber of ponts. Furtherore, due to the pont-detector lke property, t takes lon coputaton te. The coputn burden becoes larer n proporton to the square of the nuber of ponts used for the pontwse cancellaton. Recently, Booth et al. developed an exact ethod that cancels postve and neatve wehts over a reon nstead of at ponts or n sall dscretzed bns and has the potental of ben snfcantly ore effcent than the other two. 3) Here, for the cancellaton of postve and neatve wehts, the eoetry of ths proble was dvded nto 00 zones n the fssonable slabs and 40 zones n the ntervenn slab. The fsson reacton rates calculated wth the track lenth x Table 3 Two-roup constants for test proble Materal Materal * t (c - ) t (c - ) a (c - ) a (c - ) f (c - ) f (c - ) s (c - ) s (c - ) s (c - ) s (c - ) p.5 0 v (c/s) v (c/s) * The superscrpts eans neutron enery roup. estators were accuulated n the zones. The postve and neatve wehts of the sned partcles were cancelled n the dvded zones. At the end of each cycle, the relatve fsson source ntensty n each zone was deterned n proporton to the fsson reacton rate n the zone. The fsson source stes for the next cycle were postoned unforly wthn each zone. Ths cancellaton technque s not exact as copared to the pontwse cancellaton usn the pont detector-lke technque. The source noralzaton schee and cancellaton technque that were used n ths test proble would not snfcantly affect calculaton results f a zone dvson s fne enouh. Monte Carlo calculatons were perfored for,070 cycles wth 0,000 neutrons per cycle wth the frst 70 cycles skpped. The adjustent paraeter n Eq. (0) was set to = 0 throuhout the calculatons of the test proble. The frst ode eenfuncton and eenvalue for the forward and adjont probles were obtaned by the technque n Secton II. The adjont eenfuncton was calculated because t s used later for hher order ode calculatons. The calculated values are lsted n Table 4 alon wth the value calculated by the ONEDANT deternstc dscrete ordnates code 4) wth the sae roup constants. The ONEDANT calculaton was perfored wth S 4 quaature and esh boundares at 0.5 c ntervals. Three values ndcate areeent wthn the statstcal uncertantes. The whole space was parttoned syetrcally nto two reons. The left sde fssonable ateral (0 x 5 c) VOL., OCTOBER 0

7 83 Toshhro YAMAMOTO Table 4 Frst ode eenvalue for test proble (s - ) k eff value by Monte Carlo ± 55.4 ± 6. calculaton (forward) value by Monte Carlo ± 5.6 ± 6.5 calculaton (adjont) value by ONEDANT 57.8 was assned to the reon I, and the rht sde (30 x 55 c) to the reon II as shown n F. 3. The ntal startn partcles of postve weht + were randoly assned n the reon 0 x 5 c and 30 x 38 c. On the other hand, the ntal startn partcles of neatve weht were randoly assned n the reon 38 x 55 c. The asyetrc dstrbuton of the postve and neatve wehts was ntentonally ven for testn the converence of the second eenfuncton. Follown Eqs. (9) throuh (4), the second eenvalue k n the -th dvded reon were calculated by: N k t j j p f w j wk, = I or II, (8) k j s sued over all trajectores n the -th reon, and w j = partcle weht n the j-th trajectory (postve or neatve), t j = track lenth n the j-th trajectory, N = the nuber of partcles startn fro the -th reon, w k = partcle weht of the k-th startn partcle (postve or neatve). The ters n Eq. (7) for obtann an value over the whole syste were calculated as follows: Ns N w, (9) j p f T W W j, (30) t j j p f w j, (3) T / v ( t j j w j, (3) was sued over the all dscretzaton zones, and j was sued over all trajectores n the -th dscretzaton zone, and N s = nuber of all startn neutrons n one cycle, W j = weht chane n the j-th trajectory n the -th dscretzaton zone (postve or neatve), t j = track lenth n the j-th trajectory n the -th dscretzaton zone, w j = partcle weht n the j-th trajectory n the -th dscretzaton zone (postve or neatve). A ethod s ntroduced to assess the converence of the fsson source dstrbuton toward the second eenfuncton. 6) Auxlary eenvectors f and are defned that are assocated wth the frst eenfuncton and second eenfuncton as follows, respectvely: f t t f,, (33),, (34) t ples transpose, and f ( (, (35) RII RI ( (. (36) RII RI ( ) P n It s assued that ( conssts of the frst eenfuncton ( and second eenfuncton ( and that the eenfunctons c ( wth > already becoe nsnfcant after soe teratons. That s, P ( c ( c (. (37) The apltudes of ( and ( n the n-th teraton, whch are denoted by c and c, respectvely, are ven by: c c P U I, (38) PII U f f f. (39) / A paraeter c c becoes saller as the fsson source dstrbuton converes to the second eenfuncton. Thus, the paraeter c c s used as a easure for / assessn how the second eenfuncton s donant over the frst one and how the converence to the second eenfuncton s attaned. Snce the test proble s syetrcal, t follows that f = and =. The transtons of the paraeter c / c durn the ntal cycles were calculated for several values (defned n Eqs. (5) or (6)), and they are shown n F. 4. When s close to unty, e.., =., c c becoes zero slowly. As becoes lare / / c c becoes zero earler. Howeve snfcant undershoots of c / c are seen for =.8 and 5.0. If s larer than around 8.0, the calculaton s abnorally ternated due to the lare undershoot. Fure 4 llustrates that the second ode eenvalue calculatons see to be perfored successfully by selectn a value approprately. PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY

8 Hher Order α Mode Eenvalue Calculaton by Monte Carlo Power Iteraton 833 The calculated second eenvalues for the forward and adjont probles are lsted n Table 5. Snce no exstn calculaton ethod s avalable for verfcaton of the second eenvalues, the sulaton of a pulsed neutron ethod was perfored. Pulsed neutrons n the frst enery roup were njected at x = 8.0 c, and then the subsequent neutron counts were accuulated n te bns wth the nterval of 0 second. The pont the source neutrons were enerated was deterned so as to excte the second and thrd eenfunctons. The te-dependent anular flux dstrbutons ( Ω, t) n the -th enery roup can be expanded as a lnear cobnaton of the eenfunctons, ( Ω, t) c ( t) (, (40) c (t) s a te-dependent apltude of the -th eenfuncton, and ( s the -th eenfuncton of the forward anular flux n the -th enery roup. The forward anular eenfuncton and adjont anular eenfuncton * ( satsfy the orthoonal condtons, G V and G c / c V 4 v 4 v * ( n ( 0 * ( n ( 0 for n, (4) for = n, (4) v s the neutron velocty n the -th enery roup. Usn Eq. (40) and ths orthoonalty, the te-dependent apltude of the -th eenfuncton s ven by c ( t) G V G V v 4 v 4 * * γ=. γ=.0 γ=.8 γ= Cycle F. 4 c / c transtons for several 's. ( ( Ω, t) (. (43) ( The te-dependent apltude of the -th eenfuncton asyptotcally decays wth te as follows: t c ( t) e (44) s the -th ode eenvalue. The relatve apltudes of the frst and second eenfunctons are shown n F. 5. The second ode eenvalue that s lsted n the last row n Table 5 was obtaned by fttn the te-decay of the apltude of the second eenfuncton. The second ode eenvalues calculated by the ode eenvalue calculatons ndcate ood areeent wth the one by the pulsed neutron sulaton. Ths areeent consttutes verfcaton of the eenvalue ode calculaton for the second ode.. Thrd and Hher Mode Eenvalue Booth successfully obtaned the thrd and hher crtcalty eenvalues by the ethod that parttons the whole space nto several reons wthout known lower order eenfunctons. 4) The author attepted n van to obtan the thrd ode eenvalue. Then, Hotelln s ethod, whch explctly subtracts the frst and second eenfuncton durn the power teraton, was adopted for the thrd ode eenvalue ( n) calculaton. Suppose that the anular flux ( s obtaned after the n-th power teraton. The anular flux ( n) ( used for a starter n the (n+)th power teraton s obtaned by subtractn the frst and second ode eenfunctons as Table 5 Second ode eenvalue for test proble value by Monte Carlo calculaton (forward) value by Monte Carlo calculaton (adjont) Pulsed neutron ethod sulaton ( n) ( ( n) ( c ( c ( G V 4 v c G V 4 v (s - ) ± ± 6.3 * ( * ( ( n) ( (, (45) =,. (46) Ths subtracton procedure s also perfored for the calculaton of the adjont anular flux. As far as the test proble n ths paper s concerned, converence of the ode eenvalue calculatons was acheved up to the forth ode eenvalue calculaton. The calculated thrd eenvalues for the forward and adjont probles are lsted n Table 6. The relatve apltude of the thrd eenfuncton k eff.000 ± ± ± 0.8 VOL., OCTOBER 0

9 834 Toshhro YAMAMOTO Apltude of eenfuncton F. 5.E+0.E+00.E-0.E-0.E-03.E-04 Table 6 Thrd ode eenvalue for test proble value by Monte Carlo calculaton (forward) value by Monte Carlo calculaton (adjont) Pulsed neutron ethod sulaton calculated by Eq. (46) s also shown n F. 5. The thrd ode eenvalue obtaned by fttn the te-decay of the apltude of the thrd eenfuncton s lsted n Table 6. As te elapses after the pulse eneraton, the te-dependent flux dstrbutons ( Ω, t) et closer to the frst eenfuncton due to the faster decays of hher order eenfunctons. The apltude of the thrd eenfuncton c 3 (t) approaches zero rapdly, whch nevtably akes the calculaton of c 3 (t) naccurate. Howeve the areeent between the thrd ode eenvalue by the pulsed neutron ethod and the one by the ode eenvalue calculaton ay be satsfactory by consdern the standard devatons. The calculated fourth ode eenvalue was 5,478 ± 68 (s ). It was unable to perfor the fourth order adjont calculaton and calculatons hher than the fourth order wth Hotelln s ethod. Relatve forward eenfuncton dstrbutons of the frst (.e., fundaental ode), second, thrd, and fourth orders are shown n F. 6. V. Conclusons Frst eenfuncton Second eenfuncton Thrd eenfuncton.e-05 0.E+00.E-04.E-04 3.E-04 4.E-04 5.E-04 6.E-04 Te after pulse eneraton (sec) Te decay of apltudes of -st, -nd and 3-rd order (s - ) 6845 ± 6908 ± k eff ± ± ± Relatve forward eenfuncton F. 6 Frst Second Thrd Fourth x (c) Eenfuncton dstrbutons of test proble It has been reported n any lteratures that ode eenvalue calculatons n the Monte Carlo technque are dffcult for fssonable systes wth lare subcrtcalty. Ipleentn a pseudo absorpton ter n the neutron transport equaton, whch was proposed by Yen et al., overcoes the dffculty. The author deonstrated that ths technque certanly can provde stable ode eenvalue calculatons for probles that the ornal MCNP 4C fals to solve. In addton, the fure of ert of the calculated value s larely proved by ntroducn the pseudo absorpton ter. Howeve the value calculated n a cycle s nfluenced by the value used for the calculaton when ntroducn the pseudo absorpton ter. Thus, f the value obtaned n the prevous cycle s used n the next cycle calculaton, the varance of the calculated values would be underestated. How to estate an value used for the next cycle and how to estate the true standard devaton would be a research ssue n the future. The values calculated by the ode eenvalue calculatons were verfed by coparn the values obtaned by the pulsed neutron sulaton calculatons. The ethod that provdes eenfunctons wth hher order crtcalty eenvalues was appled to hher order ode eenvalue calculatons. To obtan the second eenfuncton, the ethod parttons the whole space nto two reons. The estate of the ultplcaton factor n each reon s forced to be equal to each other. The ethod was found to be applcable to calculatons for the second ode eenfuncton. Ths paper ay be the frst attept to obtan hher order ode eenvalues and eenfunctons by the Monte Carlo technque. The factor (>) n Eqs. (5) or (6) needs to be specfed n ths ethod. Howeve the theoretcal bass of the approprate rane of the factor n Eqs. (5) or (6) has not been ven n ths paper. As t now stands, the factor has to be deterned throuh a tral and error process. The nuercal exaples n ths work ndcates that the second eenvalue calculated by the proposed ethod was verfed by the sulaton of the pulsed neutron ethod. For ode eenvalue proble hher than the second orde Hotelln s ethod was appled. Ths ethod explctly subtracts lower order eenfunctons fro the fsson source dstrbuton. The thrd and fourth order forward eenfunctons of the test proble were successfully obtaned wth the use of Hotelln s ethod. Lke hher order crtcalty calculatons, hher order ode eenvalue calculatons n the Monte Carlo ethod requres the cancellaton of postve and neatve wehts of partcles. Ths work adopted the space dscretzaton technque for the cancellaton of the sned partcles as an PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY

10 Hher Order α Mode Eenvalue Calculaton by Monte Carlo Power Iteraton 835 alternatve. Recently, an exact ethod for canceln postve and neatve partcle wehts has been developed by Booth et al. The adopton of the new ethod would be prosn for hher order ode calculatons by the Monte Carlo ethod. References ) D. Brockway, P. Soran, P. Whalen, Monte Carlo calculaton, LA-UR-85-4, Los Alaos Natonal Laboratory (LANL) (985). ) J. F. Breseste (Ed.), MCNP A eneral Monte Carlo N-partcle transport code, verson 4C, LA-3709-M, Los Alaos Natonal Laboratory (LANL) (000). 3) T. Endo, Y. Yaane, A. Yaaoto, Space and enery dependent theoretcal forula for the thrd order neutron correlaton technque, Ann. Nucl. Enery, 33, (006). 4) T. E. Booth, Coputn the hher k-eenfunctons by Monte Carlo power teraton: a conjecture, Nucl. Sc. En., 43, (003). 5) T. E. Booth, Power teraton ethod for the several larest eenvalues and eenfunctons, Nucl. Sc. En., 54, 48 6 (006). 6) T. Yaaoto, Converence of the second eenfuncton n Monte Carlo power teraton, Ann. Nucl. Enery, 36, 7 4 (009). 7) T. Yen, C. Chen, W. Sun, B. Zhan, D. Tan, Propt te constants of a reflected reacto T. Fukahor (Ed.), Proc. 006 Syposu on Nuclear Data, Toka-ura, Ibarak-ken, Japan, Jan. 5 6, 007, Nuclear Data Dvson, Atoc Enery Socety of Japan (007), [CD-ROM]. 8) T. Yaaoto, Y. Myosh, An alorth of - and -ode eenvalue calculatons by Monte Carlo ethod, Proc. 7th Int. Conf. on Nuclear Crtcalty Safety, ICNC 003, Toka-ura, Ibarak, Japan, Oct. 0 4, 003, JAERI-Conf , Japan Atoc Enery Research Insttute (JAERI) (003). 9) D. E. Kornrech, D. K. Parsons, Te-eenvalue calculatons n ult-reon Cartesan eoetry usn Green s functon, Ann. Nucl. Enery, 3, (005). 0) T. Yaaoto, Y. Myosh, T. Kyosu, Valdatn JENDL-3.3 for water-reflected low-enrched uranu soluton systes usn STACY ICSBEP benchark odels, Nucl. Sc. En., 45, 3 44 (003). ) H. Yoshkawa, J. Wakabayash, An approxate calculaton ethod of the, p, d eenvalue proble of the roup dffuson equaton, J. Nucl. Sc. Technol., 7[7], (970). ) K. Hashoto, K. Nshna, Calculatons of spatal haroncs n two-densonal ultplcatve systes, J. Atoc Enery Soc. Japan, 33[9], (99), [n Japanese]. 3) T. E. Booth, J. E. Gubernats, Exact reonal Monte Carlo weht cancellaton for second eenfuncton calculatons, Nucl. Sc. En., 65, 83 9 (00). 4) R. E. Alcouffe, R. S. Bake F. W. Brnkley, D. R. Mar R. D. O Dell, W. F. Walters, DANTSYS: A dffuson accelerated neutral partcle transport code syste, LA-969-M, Los Alaos Natonal Laboratory (LANL) (995). VOL., OCTOBER 0