Estimation of Sample Reactivity Worth with Differential Operator Sampling Method

Transcription

1 Progress in NUCLEAR SCIENCE and TECHNOLOGY, Vol. 2, pp (2011) ARTICLE Estimation o Sample Reactivity Worth with Dierential Operator Sampling Method Yasunobu NAGAYA and Takamasa MORI Japan Atomic Energy Agency, Tokai-mura, Naka-gun, Ibaraki-ken, , Japan Applicability o the Taylor series approach with the arbitrary-order dierential operator sampling (DOS) method is examined or the calculation o sample reactivity worth. The DOS method is extended to obtain the dierential coeicients o the eective multiplication actor and the perturbed source eect up to the arbitrary order. The methodology is implemented into a continuous-energy Monte Carlo code MVP to perorm benchmark calculations. It is ound that the second-order Taylor series approach gives an enough accurate result or simple ast systems o Godiva and Jezebel. A discrepancy o 10% is, on the other hand, observed or the Np-237 sample worth calculation or the tank-type critical assembly TCA even with the ith-order Taylor series approach. The perturbed source eect has a signiicant contribution or the calculation o sample reactivity worth; it must be estimated or all the cases. Alternative approaches such as the Padé approximation, the midpoint and interpoint methods are also examined or the TCA sample worth problem. They give similar results as the ordinary higher-order Taylor series approach. KEYWORDS: Monte Carlo, perturbation, sample reactivity worth, dierential operator sampling, perturbed source eect I. Introduction Nuclear reactor analysis requires calculations o reactivity worth such as control rod worth, void reactivity worth, sample reactivity worth, etc. It is, however, diicult to perorm such calculations with the Monte Carlo method i the worth is small. In such cases, it might be appropriate to use Monte Carlo perturbation methods: correlated sampling and dierential operator sampling (DOS) methods. Applicability o these methods to reactivity changes has been extended by taking the perturbed source eect (PSE) into account. 1, 2) One, however, must note that the methods are applicable only or small perturbation such as small ractional density change; the variance diverges or large perturbation in the correlated sampling method and the higher-order eect is neglected in the DOS method. In the present work, applicability o the Taylor series approach with the DOS method is examined or the calculation o sample reactivity worth. The DOS method is extended to obtain the dierential coeicients o the eective multiplication actor up to an arbitrary order. The extension has been already proposed by Morillon 3) but we present a ormulation to calculate the contributions rom the PSE explicitly. The normalization condition is also derived or the higher-order dierential coeicient o the eective multiplication actor. The extended DOS method is implemented into the MVP code 4) to perorm benchmark calculations. The benchmark systems are Godiva, Jezebel and TCA. 5) The irst two are simple ast systems and the last is a complicated thermal system. We do not ocus on the comparison between the calculated Corresponding author, nagaya.yasunobu@jaea.go.jp and measured values though measurements o sample reactivity worth were perormed at Jezebel and TCA. The comparison is perormed with the reerence values obtained by two independent Monte Carlo calculations to investigate the applicability o the Taylor series approach. II. Methodology The change in the eective multiplication actor k i o the i-th generation can be expressed with the Taylor series expansion as k i = k i a n! 2 k i 2 ( a)2 n k i n ( a)n, (1) where a is a perturbation parameter such as a material density, etc. Each dierential coeicients at a given state can be estimated with the DOS method. 1. First-Order Dierential Coeicient According to the DOS ormulation by Nagaya and Mori, 1) the irst-order dierential coeicient o k i can be written as ollows: k i = 1 dp S,i (P ) (P m,, P 1, r 0 )F (P m,, P 1, r 0 ), (2) c 2011 Atomic Energy Society o Japan, All Rights Reserved. 842

2 Estimation o Sample Reactivity Worth with Dierential Operator Sampling Method 843 F (P m,, P 1, r 0 ) = νσ (P m ) Σ t (P m ) K(P m; P m 1 ) K(P 2 ; P 1 ) T (P 1 ; r 0 )S,i (r 0, E 1, Ω 1 ), (3) (P m,, P 1, r 0 ) = Σ ( ) t(p l ) νσ (P l ) νσ (P l ) K(P l ; P l 1 ) K(P l; P l 1 ) K(P 2 ; P 1 ) K(P 2; P 1 ) T (P 1 ; r 0 ) T (P 1; r 0 ) S,i (r 0, E 1, Ω 1 ) S,i(r 0, E 1, Ω 1 ), (4) where P is the six-dimensional vector (r, E, Ω) which represents spatial position r, energy E, angle Ω; S,i (P ) is the ission source; Σ t and νσ are the total and production cross sections, respectively; K is the transition kernel which consists o the transport kernel T multiplied by the collision kernel C. The transition, transport and collision kernels are deined as ollows: K(P l ; P l 1 ) = T (P l ; r l 1 ) C(r l 1, E l, Ω l ; E l 1, Ω l 1 ), (5) T (r, E, Ω; r ) = Σ t (r, E) [ r r exp Σ t (r s r r r r )ds 0 δ(ω r r r r 1) r r 2, (6) C(r, E, Ω; E, Ω ) = Σ s(r; E, Ω E, Ω ) Σ t (r, E. (7) ) is the weight actor to estimate the irst-order dierential coeicient. To express the tally score or the irst-order dierential coeicient o k i explicitly, Eq. (2) is transormed into the ollowing orm: k i = 1 dp S,i (P ) [ m (P l,, P 1, r 0 ) νσ (P l ) H(P m,, P 1, r 0 ), (8) H(P m,, P 1, r 0 ) = α(p m )K(P m ; P m 1 ) K(P 2 ; P 1 ) T (P 1 ; r 0 )S,i (r 0, E 1, Ω 1 ), (9) where α is the absorption probability. H denotes a neutron history; it means the existence probability o the neutron. It is unity or analog Monte Carlo and is a particle weight w or non-analog Monte Carlo. Scoring νσ /Σ t or wνσ /Σ t estimates the irst-order dierential coeicient o k i or analog or non-analog Monte Carlo, respectively. Scored values or depend on the perturbation parameter; Reerences 6 and 7 can be consulted or concrete examples. Furthermore, Eq. (8) can be split into the perturbed source and non-perturbed source terms: k i = k NPS,i k PS,i, (10) k NPS,i 1 = dp S,i (P ) [ m νσ (P l ),NPS H, (11) k PS,i 1 = dp S,i (P ) [ m νσ (P l ),PS H, (12) where,nps = Σ ( ) t(p l ) νσ (P l ) νσ (P l ) K(P l ; P l 1 ) K(P l; P l 1 ) (13) K(P 2 ; P 1 ) K(P 2; P 1 ) T (P 1 ; r 0 ) T (P 1; r 0 ),PS = S,i (r 0, E 1, Ω 1 ) S,i(r 0, E 1, Ω 1 ). (14) The use o Eq. (10) enables us to estimate the PSE and its statistical uncertainty quantitatively. In a Monte Carlo power iteration, the ission source is normalized as ollows: S,i (P ) = 1 dp K F (P ; P )S,i 1 (P ), (15) k i 1 dp K F (P ; P )S,i 1 (P ) = de m dω m m=0 dp m 1 dp 0 χ(e, Ω)F (r, E m, Ω m, P m 1,, P 0 ) (16) Thus the normalization condition is also imposed or the dierential coeicient o the ission source. The condition is derived rom Eq. (15) as S,i(P ) = 1 { k i 1 k i 1 k i 1 dp [K F S,i 1 (P ) } dp K F S,i 1 (P ). (17) This condition must be considered or Eq. (14). In the present work, the irst-order dierential coeicient is calculated rom the collision estimates in each batch. The inal value is obtained by averaging over active batches as ollows: E [ k = 1 N N skip N i=n skip 1 k i, (18) VOL. 2, OCTOBER 2011

3 844 Yasunobu NAGAYA et al. where E[ k/ is the estimated k/ value; N and N skip are the total and skipped (inactive) batches, respectively. The variance is simply estimated as ollows: [ k σ 2 1 = N N skip 1 1 N N N skip i=n skip 1 ( ki ) 2 [ k E 2, (19) where σ[ k/ is the standard deviation or the estimated k/ value. The correlation between batches (generations) is not taken into account. 2. Higher-Order Dierential Coeicient The ormula or the second-order dierential coeicient o k i can be derived by dierentiating Eq. (2). W (2) 2 k i 2 = 1 dp S,i (P ) W (2) F, (20) ( ) 2. (21) (1) W (2) W = is the weight actor to estimate the second-order dierential coeicient. Likewise, we can obtain the ormula or the higher-order dierential coeicient o k i as ollows: n k i n = 1 dp S,i (P ) F, (22) (n 1) W = W (n 1). (23) can be calculated rom and its derivatives by using Eq. (23) recursively. The ollowing relation is useul or the recursive calculation: t t = t s=0 t! s!(t s)! s W (n 1) s t s (1) W t s. (24) Equation (22) can be transormed into the tally-score orm: n k i n = 1 dp S,i (P ) [ m νσ (P l ) H. (25) The higher-order dierential coeicients o k i can be estimated by scoring νσ /Σ t at every collision site. The above procedure has been already proposed by Morillon; 3) the detailed algorithm is described in his paper. To estimate the perturbed source eect explicitly, we split Eq. (25) into two terms: n k i k NPS,i n = n n n k PS,i n, (26) n k NPS,i 1 n = dp S,i (P ) [ m νσ (P l ),NPS H, (27) n k PS,i 1 n = dp S,i (P ) [ m νσ (P l ),PS H, (28) where,nps,ps (n 1),NPS = W W (n 1),NPS,NPS (29) (n 1),PS = W [ W (n 1),NPS W (n 1),PS,NPS W (n 1),PS,PS. (30) Equations (29) and (30) can be also calculated with Morillon s procedure. The calculation o Eq.(30) requires the higher-order dierential coeicients o the ission source. They are calculated with the ollowing normalization condition in the power iteration: n n S,i(P ) = 1 k i 1 n k i 1 k i 1 n n 1 s=1 { dp n n [K F S,i 1 (P ) dp K F S,i 1 (P ) } n! s s!(n s)! s S,i 1(P ) n s k i 1 n s. (31) Similarly to the irst-order dierential coeicient, the higher-order dierential coeicients are calculated as ollows: E [ n k n = 1 N N skip N i=n skip 1 n k i n, (32) where E[ n k/ n is the estimated n k/ n value. The variance is estimated as ollows: [ σ 2 n k 1 n = N N skip 1 1 N ( n ) 2 [ k i N N skip n E 2 n k n, (33) i=n skip 1 where σ[ n k/ n is the standard deviation or the estimated n k/ n value. The correlation between batches (generations) is not taken into account. In addition, all the correlations between the dierential coeicients are neglected in estimating the reactivity worth; they are caused by using the same random walk to estimate the dierential coeicients. PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY

4 Estimation o Sample Reactivity Worth with Dierential Operator Sampling Method 845 Table 1 Comparison o the reactivity worth calculated with the DOS method and its reerence values or the Godiva assembly (perectly voided case) Reerence (0.042) Taylor series Without PSE With PSE up to 1st-order (0.020) (0.054) up to 2nd-order (0.023) (0.068) up to 3rd-order (0.023) (0.078) up to 4th-order (0.023) (0.085) The value in the parenthesis stands or one standard deviation o Reerence 1st-order DOS without PSE 2nd-order DOS without PSE 1st-order DOS 1st-order PSE 2nd-order DOS 2nd-order PSE III. Benchmark Calculation Benchmark calculations are perormed or Godiva, Jezebel and TCA assemblies. The irst two assemblies are a ast uranium system and a ast plutonium system, respectively. The TCA assembly is a thermal uranium system. JENDL-3.3 8) has been used or all calculations. 1. Godiva The benchmark model speciied in HEU-MET-FAST-001 o the ICSBEP handbook 9) is employed; the geometry o the assembly is a bare uranium sphere o a radius o cm and the composition is wt% U-235, 5.27 wt% U-238 and 1.02 wt% U-234. The reactivity worth is calculated or the case where the central region o radius less than 1 cm is voided. This is a calculation benchmark since the perturbation is ictitiously introduced: such an experiment has not been perormed. The reerence values are obtained with two independent Monte Carlo runs perormed or 40,000 active and 100 inactive batches with 10,000 histories per batch. A single eigenvalue calculation is, on the other hand, perormed or 4,000 active and 100 inactive batches with 10,000 histories per batch to obtain the reactivity worth with the DOS method. Table 1 shows the comparison o the change in reactivity worth calculated with the DOS method and its reerence values or the Godiva assembly. The signiicant discrepancy can be observed unless the PSE is not taken into account. Comparing the calculated values including the PSE, the irst-order approximation yields a slight overestimation o the negative reactivity worth; the values agree within two standard deviations though they do not agree within one standard deviation. The second-order perturbed source eect improves the calculated result; the values agree within one standard deviation. The higher-order approximation also gives a very good agreement with the reerence value; the dierence rom the reerence value is less than 1%. To examine the applicability o the irst- and second-order approximation against the void raction, we plot the change in reactivity worth as a unction o the ractional decrease in density. Figure 1 shows the comparison o the change in reactivity worth or the irst- and second-order approximations. The Fig. 1 Comparison o the change in reactivity worth calculated with the dierential operator sampling method and its reerence values or the Godiva assembly irst-order approximation is applicable up to about 50% decrease in density but the discrepancy becomes larger or more than 50% decrease. The second-order approximation is, on the other hand, applicable or the perectly voided case (100% decrease in density). 2. Jezebel We perorm a benchmark calculation or a Pu-238 replacement measurement using Jezebel. This experiment has been reported as SPEC-MET-FAST-002 in the ICSBEP handbook. The benchmark geometry o the assembly is a bare plutonium sphere o a radius o cm and the composition is 76.4 at% Pu-239, 20.1 at% Pu-240, 3.1 at% Pu-241 and 0.4 at% Pu The Pu-238 sample o a radius o cm is placed at the center o the Jezebel assembly. The reactivity worth is calculated as the change in reactivity between the Pu-238 sample and a void. The reerence values are obtained with two independent Monte Carlo runs perormed or 160,000 active and 100 inactive batches with 10,000 histories per batch. A single eigenvalue calculation is, on the other hand, perormed or 10,000 active and 100 inactive batches with 10,000 histories per batch to obtain the reactivity worth with the DOS method. Table 2 shows the comparison o the change in reactivity worth calculated with the DOS method and its reerence values or the Jezebel assembly. Almost the same trend as or Godiva is observed or the Jezebel case. The PSE must be calculated and the second- or higher order approximation gives a very good result; the values agree within one standard deviation and the dierence is less than 1%. According to the ICSBEP handbook, the experimental value is 1.461(±0.109) 10 3 k/kk. The β e value o ± is used or the unit conversion. All the calculated data agree with the experimental value within the experimental uncertainty. We do not perorm urther investigation or validation o nuclear data in the present work. However, the reactivity worth calculation with the DOS method can be an applicable tool or it. VOL. 2, OCTOBER 2011

5 846 Yasunobu NAGAYA et al. Table 2 Comparison o the reactivity worth calculated with the DOS method and its reerence values or the Jezebel assembly (perectly voided case) Reerence (0.021) Taylor series Without PSE With PSE up to 1st-order (0.007) (0.020) up to 2nd-order (0.007) (0.022) up to 3rd-order (0.007) (0.022) up to 4th-order (0.007) (0.022) The value in the parenthesis stands or one standard deviation o Reerence 1st-order DOS without PSE 2nd-order DOS without PSE 1st-order DOS 1st-order PSE 2nd-order DOS 2nd-order PSE Fig. 2 Comparison o the change in reactivity worth calculated with the dierential operator sampling method and its reerence values or the Jezebel assembly Figure 2 shows the comparison o the change in reactivity worth or the irst- and second-order approximations. The second-order approximation is applicable or the perectly voided case (100% decrease in density) but the irst-order approximation tends to overestimate the reerence values o the negative reactivity worth slightly as the decrease in density becomes larger. 3. TCA Measurements o Np-237 sample reactivity worth 10) have been perormed with the Tank-Type Critical Assembly (TCA), which is a light-water moderated and relected 2.6 wt% enriched UO 2 lattice. 5) The sample worth was measured or cores with various water-to-uel volume ratios. We perorm benchmark calculations or a core o the water-to-uel volume ratio o 3.0 because the measured reactivity worth is more than 20 cent (> k/kk ). Figure 3 shows the calculation geometry or the benchmark. The experimental channel and the sample are explicitly modeled in the geometry. Reerences 5 and 10 can be consulted or detailed inormation such as dimensions, composition, etc. about TCA and the sample, respectively. Table 3 Comparison o the Np-237 sample reactivity worth calculated with the DOS method and its reerence values or TCA (perectly voided case) Reerence 1.754(0.014) Taylor series Without PSE With PSE up to 1st-order 0.511(0.004) 0.793(0.013) up to 2nd-order 0.754(0.005) 1.194(0.017) up to 3rd-order 0.872(0.006) 1.385(0.021) up to 4th-order 0.932(0.007) 1.494(0.025) up to 5th-order 0.961(0.008) 1.552(0.029) up to 6th-order 0.975(0.009) 1.563(0.033) up to 7th-order 0.980(0.009) 1.579(0.038) up to 8th-order 0.982(0.010) 1.562(0.042) The value in the parenthesis stands or one standard deviation o The reerence values are obtained with two independent Monte Carlo runs perormed or 100,000 active and 100 inactive batches with 64,000 histories per batch. A single eigenvalue calculation is, on the other hand, perormed or 8,000 active and 100 inactive batches with 64,000 histories per batch to obtain the reactivity worth with the DOS method. Table 3 shows the comparison o the change in reactivity worth calculated with the DOS method and its reerence values or the TCA assembly. The Taylor series terms are calculated up to eighth order or this problem. The PSE terms are also calculated up to eighth order. However, the sixth and higher-order PSE terms have not been conidently estimated because the estimates o the dierential coeicients luctuate signiicantly as the number o the propagation batches increases and the relative statistical errors o the dierential coeicients are larger than 100%. The second-order approximation underestimates the reerence result by 30% even i the PSE is taken into account although the approximation gives very good results or the Godiva and Jezebel cases. Consideration o the higher-order PSE improves the calculated result but the ith-order approximation still underestimates the reerence result by 11%. Thereore, the higher-order Taylor series must be taken into account and the sixth- and higherorder PSE terms must be estimated accurately or the TCA case. To investigate the applicability o the DOS method, the change in reactivity worth is plotted as a unction o the ractional decrease in density. Figure 4 shows the comparison between Taylor approximations and the reerence solution or the case where the PSE is not taken into account. Though the reactivity change can be conidently estimated up to the higher order in this case, the calculated results are signiicantly underestimated in all the range o the ractional decrease. Figure 5 shows, on the other hand, or the case where the PSE is taken into account. The third- or higher-order approximation gives a very good result until the density decrease o 50%. However, the discrepancy becomes large even or the ith- PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY

6 Estimation o Sample Reactivity Worth with Dierential Operator Sampling Method 847 Vacuum Air Vacuum Water NpO 2 sample Stainless steel sleeve UO 2 uel rod Air (Void) Guide tube Al plug Vacuum Fig. 3 Calculation geometry or the TCA assembly Reerence 1st-order 2nd-order 3nd-order 4th-order 8th-order Reerence 1st-order 2nd-order 3nd-order 4th-order 5th-order Fig. 4 Comparison o the change in reactivity worth calculated with the dierential operator sampling method and its reerence values or the TCA assembly (no PSE included) Fig. 5 Comparison o the change in reactivity worth calculated with the dierential operator sampling method and its reerence values or the TCA assembly (PSE included) order approximation as the density o the sample decreases. These results suggest that more higher-order terms must be considered. IV. Alternative Approaches In the previous section, it has been ound that the accurate estimation o the sample reactivity worth with the ordinary Taylor series approach is diicult or the TCA case. To mitigate this diiculty, three alternative approaches are examined: the Padé approximation, 11) midpoint method and the interpoint method. The Padé approximation is expected to improve the estimation o higher-order eects. The midpoint and interpoint methods are expected to improve the accuracy o the calculated sample reactivity worth by reducing its magnitude and the density change. 1. Padé approximation Suppose that the eective multiplication actor k is a unction o the variation o a perturbation parameter x = a. Then the Taylor series o k(x) at x = 0 can be expressed as ollows: k(x) = k(0) k (0)x 1 2 k (0)x 2 1 n! k(n) x n, (34) VOL. 2, OCTOBER 2011

7 848 Yasunobu NAGAYA et al Reerence 2nd-order Taylor [1/1 Pade Reerence 4th-order Taylor [2/2 Pade Fig. 6 Comparison o the change in reactivity worth calculated with the second-order Taylor series and the [1/1 Padé approximant or the TCA assembly Fig. 7 Comparison o the change in reactivity worth calculated with the ourth-order Taylor series and the [2/2 Padé approximant or the TCA assembly With the Padé approximation k(x) can be written as a rational unction: R[L/M(x) = p 0 p 1 x p 2 x 2 p L x L 1 q 1 x q 2 x 2 q M x M. (35) To transorm Eq. (34) into the rational orm, the ollowing conditions must be satisied: R[L/M(0) = k(0), (36) d k dx k R[L/M(x) = dk x=0 dx k k(x) (37) x=0 or k = 1, 2,, L M. The second-order Taylor series can be written with the [1/1 Padé approximant: k(0) k (0)x 1 2 k (0)x 2 p 0 p 1 x 1 q 1 x, (38) where the coeicients p 0, p 1 and q 1 can be obtained rom Eqs. (36) and (37). Thus the change in k is expressed as ollows: k = R[1/1(x) R[1/1(0) = (p 1 p 0 q 1 )x 1 q 1 x = k (0) a. (39) 1 k (0) 2k (0) a Likewise, the ourth-order Taylor series can be written with the [2/2 Padé approximant: k(0) k (0)x 1 2 k (0)x 2 1 3! k (0)x 3 1 4! k(4) (0)x 4 p 0 p 1 x p 2 x 2 1 q 1 x q 2 x 2. (40) Table 4 Comparison o the Np-237 sample reactivity worth calculated with the Padé approximation and its reerence value or TCA (perectly voided case) Reerence 1.754(0.014) 2nd-order Taylor series 1.194(0.017) 4th-order Taylor series 1.494(0.025) [1/1 Padé 1.603(0.047) [2/2 Padé 1.587(0.032) The value in the parenthesis stands or one standard deviation o All the calculated results include the PSE. The change in k is expressed as ollows: k = R[2/2(x) R[2/2(0) = c 1 a c 2 ( a) 2 1 q 1 a q 2 ( a) 2, (41) c 1 = p 1 p 0 q 1 = k (0), (42) c 2 = p 2 p 0 q 2 = q 1 k (0) 1 2 k (0), (43) q 1 = 2k (0)k (0) 6k (4) (0)k (0) 4k (0)k (0) 6(k (0)) 2, (44) q 2 = 4(k (0)) 2 3k (4) (0)k (0) 36(k (0)) 2 24k (0)k (0). (45) Figures 6 and 7 show the comparison o the change in the reactivity worth calculated with the second-order Taylor series and the [1/1 Padé approximant, and with the ourthorder Taylor series and the [2/2 Padé approximant, respectively. The Padé approximation gives better results than the Taylor series approximations. In particular, the [1/1 Padé approximation outperorms the second-order Taylor series approximation or the large ractional decrease in density even though the same dierential coeicients up to the second or- PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY

8 Estimation o Sample Reactivity Worth with Dierential Operator Sampling Method 849 Table 5 Comparison o the Np-237 sample reactivity worth calculated by using the midpoint and interpoint methods with the Taylor series approximation and its reerence value or TCA (perectly voided case) Reerence Midpoint 1.754(0.014) Interpoint up to 1st-order 1.415(0.013) 1.793(0.018) up to 2nd-order 1.312(0.026) up to 3rd-order 1.577(0.015) 1.771(0.036) up to 4th-order 1.531(0.050) up to 5th-order 1.611(0.018) 1.707(0.071) up to 6th-order 1.593(0.107) up to 7th-order 1.640(0.021) 1.548(0.156) up to 8th-order 1.423(0.174) The value in the parenthesis stands or one standard deviation o All the calculated results include the PSE. der are used. In addition, the low-order Padé approximation gives similar results as the higher-order Taylor series approximation. However, the discrepancies o 10% can be still observed or the perectly voided case calculated with the Padé approximation as shown in Table Midpoint and Interpoint Methods The interpoint method calculates the sample reactivity worth at the interpoint between the cases where the sample is inserted and is perectly withdrawn. Namely, the positiveand negative-side reactivity worths rom the interpoint (reerence point) are calculated at the same time. The worths can be expressed as ollows: k = k (0) a 1 2 k (0)( a ) 2 1 3! k (0)( a ) 3, (46) k = k (0)( a ) 1 2 k (0)( a ) 2 1 3! k (0)( a ) 3. (47) The total reactivity worth is then calculated as k = k k. (48) I we select such that a = a = a, Eq. (48) is simpliied as ollows: k = 2k (0) a 2 3! k (0)( a) 3. (49) This is the midpoint method. It has an advantage that the midpoint method with the odd-order Taylor series approximation yields the same result as that with one more higher-order Taylor series. For the interpoint method, the reerence point (interpoint) is set to the system with the Np-237 sample voided by 66.7% Table 6 Positive- and negative-side reactivity worths calculated by using the interpoint method with the Taylor series approximation and their reerence values Positive side Negative side 200% increase -100% decrease Reerence (0.014) 0.911(0.014) Interpoint method with Taylor series up to 1st-order (0.016) 0.598(0.008) up to 2nd-order (0.024) 0.758(0.009) up to 3rd-order (0.034) 0.809(0.010) up to 4th-order (0.049) 0.825(0.010) up to 5th-order (0.070) 0.830(0.010) up to 6th-order (0.106) 0.832(0.010) up to 7th-order (0.156) 0.832(0.010) up to 8th-order (0.174) 0.831(0.010) The value in the parenthesis stands or one standard deviation o All the calculated results include the PSE. so that the positive- and negative reactivity worths are roughly the same. Then the positive reactivity worth is calculated or the 200% increase in the sample density and the negative one is or the 100% decrease in the density. Table 5 shows the comparison o the Np-237 sample reactivity worth calculated by using the midpoint and interpoint methods with the Taylor series approximation. The results or the midpoint method becomes better as the order o the Taylor series becomes higher. However, there still exists the discrepancy o 7% even or the seventh-order result. The interpoint method, on the other hand, yields the luctuated results with the order. This luctuation is caused by the positive-side reactivity worth as seen in Table 6. Since the positive-side reactivity worth is calculated or more than 100% change in the density, the higher-order terms o the Taylor series becomes large. To circumvent the luctuation o the Taylor series approximation, the Padé approximation is applied or the interpoint method. Table 7 shows the Np-237 sample reactivity worth calculated by using the interpoint method with the Padé approximation. The Padé approximation yields the relatively stable results or the positive-side reactivity worth. However, the interpoint method with Padé approximation still underestimates the reerence result or the total reactivity worth; it gives similar results as the ordinary Taylor series approximation. V. Concluding Remarks The applicability o the Taylor series approach with the DOS method has been examined or the calculation o sample reactivity worth. The alternative approaches such as the Padé approximation, the midpoint and interpoint methods are also examined. The higher-order DOS method including the PSE has been ormulated. The methodology has been implemented into the MVP code to perorm benchmark calculations. The ollowing conclusions have been obtained. VOL. 2, OCTOBER 2011

9 850 Yasunobu NAGAYA et al. Table 7 Positive- and negative-side reactivity worths calculated by using the interpoint method with the Padé approximation and their reerence value Positive Negative Total (200%) (-100%) Reerence (0.014) 0.911(0.014) 1.754(0.014) [1/1 Padé (0.008) 0.821(0.009) 1.602(0.012) [2/2 Padé (0.037) 0.837(0.011) 1.644(0.038) The value in the parenthesis stands or one standard deviation o All the calculated results include the PSE. The second-order Taylor approximation gives very good results or the sample reactivity worth calculation o the Godiva and Jezebel assemblies; the calculated values agree within one standard deviation. The higher-order Taylor terms must be calculated or the Np-237 sample reactivity worth calculation o the TCA assembly. It is, however, diicult to estimate the higher-order PSE conidently because o the large statistical luctuation. The PSE has a signiicant contribution or all the cases: 53%, 49% and 44% or Godiva, Jezebel and TCA, respectively. It must be thus estimated or the accurate calculation o the reactivity worth. The Padé approximation with the low-order dierential coeicients yields similar reactivity worths as the higherorder Taylor series approximation. However, there still exists a discrepancy between the calculated result with the Padé approximation and the reerence result or the Np-237 sample reactivity worth at TCA. Both the midpoint method with the Taylor series approximation and interpoint methods with the Padé approximation give similar results as the ordinary Taylor series approximation or Np-237 sample reactivity worth at TCA; no improvement can be observed. The Padé approximation gives better results than the Taylor series approximation or a change larger than 100%. In the present work, a case has been shown where it is diicult to estimate the sample reactivity worth with the Taylor series approach and the DOS method. The estimation o the higherorder eect gives a better approximate result even or such a case. Further investigation is, however, required or the accurate estimation o sample reactivity worth. Further research or the area o applicability is also required. Acknowledgments The authors are grateul to Proessor Ken Nakajima o Kyoto University or his support on the present study and to Dr. Takeshi Sakurai o Japan Atomic Energy Agency or providing useul inormation on the sample worth measurements at TCA. Reerences 1) Y. Nagaya, T. Mori, Impact o Perturbed Fission Source on the Eective Multiplication Factor in Monte Carlo Perturbation Calculations, J. Nucl. Sci. Technol., 42[5, (2005). 2) K. F. Raskash, An Improvement o the Monte Carlo Generalized Dierential Operator Method by Taking into Account First- and Second-Order Perturbations o Fission Source, Nucl. Sci. Eng., 162, (2009). 3) B. Morillon, On the use o Monte Carlo perturbation in neutron transport problems, Ann. Nucl. Energy, 25, (1998). 4) Y. Nagaya, K. Okumura, T. Mori, M. Nakagawa, MVP/GMVP II: General Purpose Monte Carlo Codes or Neutron and Photon Transport Calculations Based on Continuous Energy and Multigroup Methods, JAERI 1348, Japan Atomic Energy Research Institute (JAERI) (2005). 5) H. Tsuruta et al., Critical Sizes o Light-Water Moderated UO 2 and PuO 2-UO 2 Lattices, JAERI 1254, Japan Atomic Energy Research Institute (JAERI) (1978). 6) H. Rie, Generalized Monte Carlo perturbation algorithms or correlated sampling and a second-order Taylor series approach, Ann. Nucl. Energy, 11, (1984). 7) D. E. Peplow, K. Verghese Dierential Sampling or the Monte Carlo Practitioner, Prog. Nucl. Energy, 36, (2000). 8) K. Shibata et al., Japanese Evaluated Nuclear Data Library Version 3 Revision-3: JENDL-3.3, J. Nucl. Sci. Technol., 39[11, (2002). 9) OECD/NEA Nuclear Science Committee, International Handbook o Evaluated Criticality Saety Benchmark Experiments, NEA/NSC/DOC(95)03, September 2008 Edition (2008), [CD- ROM. 10) T. Sakurai et al., Measurement and Analysis o Reactivity Worth o 237 Np Sample in Cores o TCA and FCA, J. Nucl. Sci. Technol., 46[6, (2009). 11) G. A. Baker, Jr., Essentials o Padé Approximants, Academic Press, New York (1975). PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY