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Annals of Nuclear Energy 33 (2006) 1141–1146

annals of

NUCLEAR ENERGY

Improvement of moment-based probability table for resonanceself-shielding calculation

Go Chiba a,*, Hironobu Unesaki b

a Japan Atomic Energy Agency, 4002, Narita-cho, O-arai-machi, Ibaraki 311-1311, Japanb Kyoto University Research Reactor Institute, Kumatori-cho, Sennan-gun, Osaka 590-0494, Japan

Received 10 April 2005; accepted 7 May 2006Available online 22 September 2006

Abstract

In the present paper, an improved method has been proposed to produce a probability table needed for the resonance self-shieldingcalculations with the sub-group method. The proposed method is based on a relation between the effective cross section and the crosssection moment, which is obtained from a numerical analysis. Using the proposed method, more accurate probability tables can beobtained with less number of the tabulated steps than the conventional method. This enables us to reduce computation time and com-puter memory storage for the sub-group calculations.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

A calculation for the resonance self-shielding effect hasbeen one of the most important matters in reactor neutron-ics analyses. Recently, continuous-energy Monte-Carlo cal-culations and deterministic calculations with ultra-fineenergy group cross sections have been realized. However,they are time-consuming and not appropriate for designcalculations in which many calculation cases are necessary.The resonance self-shielding calculation therefore remainsan important matter for reactor analyses based on multi-group calculations.

The sub-group method (Cullen, 1974; Levitt, 1972;Nikolaev et al., 1970) is one of the methods for the reso-nance self-shielding calculations. The method has beenapplied both to the fast reactor analyses (Rimpault, 2002)and to the thermal reactor analyses (Coste and Mengelle,1996). In the sub-group method the cross section fluctua-tions are described by sets of Dirac d functions withweights, which are the basic parameters to evaluate theself-shielding effect. The set of the parameters is called a‘probability table’. While there are several methods to pro-

0306-4549/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.anucene.2006.05.013

* Corresponding author. Tel.: +81 29 267 4141; fax: +81 29 267 1676.E-mail address: chiba.go@jaea.go.jp (G. Chiba).

duce a probability table, a method proposed by Ribon(1986) has been often used because of its mathematicalbasis and numerical stability. In Ribon’s method, a proba-bility table is produced in order to preserve the cross sec-tion moments. In this paper, it is mentioned that thesub-group method with the moment-based probabilitytable is merely a Gauss quadrature.

Discretization errors of probability tables generallydepend on the number of tabulated ‘steps’ in each energygroup. Calculation time and computer memory storageneeded for the sub-group calculation are directly affectedby the number of steps of a probability table. Hence, it isdesired to produce an accurate probability table with assmall number of steps as possible.

In the present paper, a relation between the effectivecross section and the cross section moment is obtainedfrom a numerical analysis. A probability table generatedby the newly proposed procedure based on this relationshows a great improvement in discretization errors com-pared to those based on the conventional procedure. Anoverview of Ribon’s method will be described in Section2 and the improved procedure will be proposed in Section3. In Section 4, numerical results and the effectiveness ofthe proposed procedure will be shown and the study willbe concluded in Section 5.

1142 G. Chiba, H. Unesaki / Annals of Nuclear Energy 33 (2006) 1141–1146

2. Method proposed by Ribon

First, we describe an overview of the method proposedby Ribon.

The nth order moment of total cross section in a neutronenergy group g is calculated as

Mn ¼1

DE

ZE2gfrtðEÞgn dE;

where E is a neutron energy. On the other hand, the totalcross section moment based on a probability table is givenas

mn ¼XN

i¼1

pirni :

Parameters of a probability table, pi and ri, are defined soas to make equal the two total cross section moments, i.e.

Mn ¼ mn: ð1ÞNow, the number of unknowns are 2N (N for ri and N forpi), then we are going to use Eq. (1) where n are set to beI 6 n 6 (I + 2N � 1).

For example, a case I = 0 is considered. We suppose afollowing series function of z where mn are used as itscoefficients:

F ðzÞ ¼ m0 þ m1zþ � � � þ m2N�1z2N�1 þ rðz2N Þ

¼XN

i¼1ð1þ rizþ � � � þ r2N�1

i z2N�1Þpi þ rðz2NÞ

¼XN

i¼1

pi

1� rizþ r0ðz2N Þ: ð2Þ

F(z) can be also obtained from Mn as

F ðzÞ ¼ M0 þM1zþ � � � þM2N�1z2N�1 þ Rðz2N Þ: ð3ÞEq. (3) can be transformed utilizing the Pade approxima-tion as

F ðzÞ ¼ a0 þ a1zþ � � � þ aN�1zN�1

1þ b1zþ � � � þ bN zNþ R0ðz2N Þ

� P N�1ðzÞQNðzÞ

þ R0ðz2N Þ; ð4Þ

where

a0 ¼ M0;

a1 ¼ M1 þM0b1;

..

.

aN�1 ¼ MN�1 þMN�2b1 þ � � � þM0bN�1

and

0 ¼ MN þMN�1b1 þ � � � þM0bN ;

0 ¼ MNþ1 þMN b1 þ � � � þM1bN ;

..

.

0 ¼ M2N�1 þM2N�2b1 þ � � � þMN�1bN :

Eq. (4) can be transformed into

F ðzÞ ¼ a0 þ a1zþ � � � þ aN�1zN�1

QNi¼1

1� zzi

� � þ R0ðz2NÞ

¼XN

i¼1

xi

1� zzi

þ R0ðz2N Þ:

Parameters pi and ri are determined as

pi ¼ xi; ri ¼1

zi:

In the present paper, ri is determined by the described pro-cedure and pi is obtained as a solution of the following lin-ear system:

Mn ¼XN

i¼1

pirni :

After obtaining pi and ri, tabulated partial cross sectionsrx,i are obtained from

1

DE

ZE2g

rxðEÞrtðEÞn dE ¼XN

i¼1

pirx;irni :

As described above, N equations involving 2N momentsare needed to obtain ri while N moments and equationsare needed for pi and every rx,i. The 0th moment of totalcross section should be preserved because the summationof pi must be 1. In addition, it is better to preserve the1st moment of total cross section and 0th moment of par-tial cross section because the infinite dilution cross sectionsshould be preserved.

An integration of an arbitrary function f(rt(E)) on anenergy group g can be carried out with a probability table asZ

E2gf ðrtðEÞÞdE ¼

Z rMax:

rMin:

pðrtÞf ðrtÞdrt �X

i

pif ðriÞ;

where rMin. and rMax. are minimum and maximum totalcross sections in the energy group g. In Ribon’s paper, itis mentioned that since QN(rt) in Eq. (4) is an orthogonalfunction relative to p(rt) and ri is determined from rootsof QN(rt) = 0, the sub-group method with the moment-based probability table is merely a Gauss quadrature.

As well known, the Gauss quadrature is accurate when afunction f(rt) can be approximated by polynomials. How-ever, it was described by Yamamoto (2004) that a neutronspectrum in a homogeneous system can be expanded bypolynomials only under limited conditions.

Let us consider a homogeneous medium composed ofone resonant nuclide and non-resonant nuclides. The neu-tron spectrum in the medium /(E) can be expressed usingthe narrow resonance approximation as

/ðEÞ ¼ /Asym:ðEÞrtðEÞ þ r0

;

where /Asym.(E) is an asymptotic neutron flux, rt(E) the to-tal cross section of the resonant nuclide and r0 the dilution

G. Chiba, H. Unesaki / Annals of Nuclear Energy 33 (2006) 1141–1146 1143

cross section. Here, for simplicity, it is assumed that theasymptotic neutron flux is flat within each energy group.

An integration of the neutron spectrum on energy is car-ried out with a probability table asZ

E2g/ðEÞdE ¼

ZE2g

1

rtðEÞ þ r0

dE

¼Z rMax:

rMin:

pðrtÞ1

rt þ r0

drt �X

i

pi

1

ri þ r0

:

However, a function 1/(rt + r0) can be expanded by poly-nomials in limited conditions as follows:

1

rt þ r0

¼1rt

1þ r0

rt

¼ 1

rt1þ � r0

rt

� �þ � r0

rt

� �2

þ � � �" #

ðrt > r0Þ

1

rt þ r0

¼1r0

1þ rtr0

¼ 1

r0

1þ � rt

r0

� �þ � rt

r0

� �2

þ � � �" #

ðrt < r0Þ:

Therefore it appears that probability tables produced bythe conventional procedure warrant accurate integrationover the neutron spectrum only for r0 < rMin. orr0 > rMax..

0

5

10

15

20

25

-1 -0.8 -0.6 -0.4 -0.2 0

0.1 1 10 100 1000 10000 100000 1e+06

Effe

ctiv

e cr

oss

sect

ion

or M

xn /Mtn

Order of cross section moment

Dilution cross section [barn]

(Moment)(Dilution cross section)

Fig. 1. Relation between effective cross section and cross section moment.

3. Relation between effective cross section and cross section

moment

In the present chapter, we will show a relation betweenthe effective cross sections and the cross section moments.Based on the relation, we will propose an improved proce-dure to obtain more accurate probability tables.

An energy-averaged cross section (effective cross sec-tion) for reaction x in an energy group g, rx,eff,g, in a homo-geneous medium can be calculated as

rx;eff;g ¼R

E2grxðEÞ

rtðEÞþr0dER

E2g1

rtðEÞþr0dE

:

When the dilution cross section is infinite, the effectivecross section can be expressed using cross section momentsas

rx;eff;gðr0 ¼ 1Þ ¼R

E2g rxðEÞrtðEÞ0 dERE2g rtðEÞ0 dE

¼ M0;gx

M0;gt

;

where M0;gx and M0;g

t are the 0th moments of the partialcross section for reaction x and the total cross section.

On the other hand, when the dilution cross section iszero, the effective cross section can be also expressed withthe cross section moments as

rx;eff;gðr0 ¼ 0Þ ¼R

E2grxðEÞrtðEÞ dER

E2g1

rtðEÞ dE¼ M�1;g

x

M�1;gt

:

Here, let us consider the quantity Mn;gx

Mn;gt

, which is a function ofn, and observe numerically its behavior in comparison with

the effective cross section, which depends on the dilution

cross section. In Fig. 1,Mn;g

capture

Mn;gt

of uranium-238 in JENDL-

3.2 (Nakagawa, 1995) in an energy group (167-214eV)are plotted in a range �1 6 n 6 0 with the effective cross

sections. It can be observed that bothMn;g

capture

Mn;gt

and the effec-

tive cross section increase as n or dilution cross section in-creases and they are in the range between the infinitedilution cross section and the totally self-shielded cross sec-tion (rx,eff,g(r0 = 0)). From these results, it can be said thatfor every r0, it exists {n; �1 6 n 6 0} such that

rcapture;eff ;gðr0Þ ¼Mn;g

capture

Mn;gt

. Numerical investigations show that

this relation can be observed in other reaction types, inother energy groups and also in other nuclides such as iron.From these numerical results, we can draw a conjecturethat any effective cross section can be expressed using thecross section moments as

rx;eff ;gðr0Þ ¼Mn;g

x

Mn;gt

ð�1 6 n 6 0Þ:

Here the above relation between n and r0 is described as

n ¼ f ðr0Þ:The effective cross section and the ratio of cross section

moments can be also obtained from a probability table as

rPTx;eff;gðr0Þ ¼

PNi¼1

pirx;i

riþr0

PNi¼1

piriþr0

mn;gx

mn;gt¼

PNi¼1

pirx;irni

PNi¼1

pirni

:

This effective cross section also has a relation to the ratio ofthe cross section moments as

0.001

0.01

0.1

1

10

30 35 40 45 50 55

Max

imum

dis

cret

izat

ion

erro

r[%

]

Neutron energy group

’Case_A’’Case_B’’Case_C’

Fig. 2. Maximum discretization error in the ratio of cross sectionmoments.

1144 G. Chiba, H. Unesaki / Annals of Nuclear Energy 33 (2006) 1141–1146

rPTx;eff ;gðr0Þ ¼

mn;gx

mn;gt

ð�1 6 n 6 0Þ:

The above relation between n and r0 is described as

n ¼ ~f ðr0Þ:From the above discussions, we can obtain a following

relation between rx,eff,g and rPTx;eff ;g:

rx;eff ;gðr0Þ ¼Mn;g

x

Mn;gt

� mn;gx

mn;gt¼ rPT

x;eff ;gð ~r0Þ � rPTx;eff ;gðr0Þ: ð5Þ

In the conventional procedure proposed by Ribon, onlythe integer-order moments of cross section are preserved toobtain probability tables. However, we conclude from theabove discussions that preserving some values of the crosssection moments for �1 < n < 0 should improve the accu-racy of the probability table description.

4. Numerical tests and results

4.1. Production of probability table

With the described procedure, probability tables wereproduced for uranium-238 in JENDL-3.2 in each energygroup from 5.04 eV to 912 keV in which a lethargy widthis 0.25. The energy boundaries are shown in Table 1.

We produced three sets of the 5-step probability table inwhich the preserved order of cross section moments are dif-ferent from each other. The detail is shown in Table 2.

Table 1Upper energy boundaries of group structure

Energy group Upper energy (eV) Energygroup

Upper energy (eV)

29 9.1188E+03a 44 2.1445E+0230 7.1017E+03 45 1.6702E+0231 5.5308E+03 46 1.3007E+0232 4.3074E+03 47 1.0130E+0233 3.3546E+03 48 7.8893E+0134 2.6126E+03 49 6.1442E+0135 2.0347E+03 50 4.7851E+0136 1.5846E+03 51 3.7267E+0137 1.2341E+03 52 2.9023E+0138 9.6112E+02 53 2.2603E+0139 7.4852E+02 54 1.7604E+0140 5.8295E+02 55 1.3710E+0141 4.5400E+02 56 1.0677E+0142 3.5358E+02 57 8.3153E+0043 2.7536E+02 58 6.4760E+00

(Lowerboundary)

5.0435E+00

a Read as 9.1188 · 103.

Table 2Preserved moment to produce probability tables

Case Steps For ri

A 5 �5, �4, . . ., 3, 4B 5 �2, �3/2, . . ., 2, 5/2C 5 �1, �3/4,. . ., 1, 5/4

‘Case A’ was produced by the conventional procedurewhich preserves only the integer-order of cross sectionmoments. ‘Case B’ and ‘Case C’ preserve cross sectionmoments at more points from the �1st to the 0th orderthan ‘Case A’.

First, the discretization errors of mn;gx

mn;gt

were estimated. The

errors were calculated as differences between mn;gx

mn;gt

and Mn;gx

Mn;gt

at

101 values of moment n in the range, �1 6 n 6 0. Theabsolute values of the maximum discretization errors ineach energy group are shown in Fig. 2. The errors of ‘CaseB’ and ‘Case C’ are negligible in almost all energy groupswhile the errors in ‘Case A’ are larger than 1% in someenergy groups. The discretization errors for energy groups44 (167–214 eV) and 51 (29.0–37.2 eV) are shown in Figs. 3and 4. The errors in ‘Case C’ are the smallest among threein both energy groups.

Next, we estimated the errors in the effective cross sec-tions calculated with the probability table, rPT

eff . The errorsin energy groups 44 and 51 are shown in Figs. 5 and 6.Maximum and minimum total cross sections, rt,Max. andrt,Min., in group 44 are 5230 barn and 1.3 barn, and thosein group 51 are 13,500 barn and 4.6 barn. In both energygroups, the errors of ‘Case A’ are small in the ranger0 > rt,Max. and r0 < rt,Min. as described in the previouschapter. However, in the range rt,Min. < r0 < rt, Max., thereare significant peaks of errors in both energy groups for‘Case A’. On the other hand, the magnitudes of the errorsare significantly reduced in ‘Case B’ and ‘Case C’ while theerrors become a little larger than ‘Case A’ in r0 < rt,Min..These results show that the proposed procedure can gener-ate more accurate probability table with less number of

For pi For rx,i

�2, �1, 0, 1, 2 �2, �1, 0, 1,2�1, �1/2, 0, 1/2, 1 �1, �1/2, 0, 1/2, 1�1, �3/4, �2/4, �1/4, 0 �1, �3/4, �1/2, �1/4, 0

1e-04

0.001

0.01

0.1

1

10

100

-1 -0.8 -0.6 -0.4 -0.2 0

Err

or[%

]

order of cross section moment

Case ACase BCase C

Fig. 3. Discretization error in the ratio of cross section moments in energygroup 44.

1e-04

0.001

0.01

0.1

1

10

100

-1 -0.8 -0.6 -0.4 -0.2 0

Err

or[%

]

order of cross section moment

Case ACase BCase C

Fig. 4. Discretization error in the ratio of cross section moments in energygroup 51.

1e-04

0.001

0.01

0.1

1

10

100

0.1 1 10 100 1000 10000 100000

Err

or[%

]

Dilution cross section [barn]

Case ACase BCase C

Fig. 5. Error in effective cross section in energy group 44.

0.001

0.01

0.1

1

10

100

0.1 1 10 100 1000 10000 100000

Err

or[%

]

Dilution cross section[barn]

Case ACase BCase C

Fig. 6. Error in effective cross section in energy group 51.

1

10

100

1000

10000

0 0.2 0.4 0.6 0.8 1

Tab

ulat

ed to

tal c

ross

sec

tion

Probability

Case ACase C

Fig. 7. Values of probability table in energy group 44.

1

10

100

1000

10000

100000

0 0.2 0.4 0.6 0.8 1

Tab

ulat

ed to

tal c

ross

sec

tion

Probability

Case ACase C

Fig. 8. Values of probability table in energy group 51.

G. Chiba, H. Unesaki / Annals of Nuclear Energy 33 (2006) 1141–1146 1145

1146 G. Chiba, H. Unesaki / Annals of Nuclear Energy 33 (2006) 1141–1146

tabulated steps than the conventional procedure in animportant range of r0.

Here, we observe a difference in the probability tablesobtained by the conventional and proposed procedures.In Figs. 7 and 8, tabulated total cross sections for ‘CaseA’ and ‘Case C’ are shown with their probabilities inenergy groups 44 and 51. In ‘Case A’, larger probabilitiesare given to the high cross section range and the lowcross section range than ‘Case C’ since ‘Case A’ has topreserve the larger and smaller order of the cross sectionmoments.

5. Conclusions

In the present paper, an improved method has been pro-posed to produce a probability table. The method is basedon a relation between the effective cross section and thecross section moment, which is obtained from a numericalanalysis. Using the proposed method, more accurate prob-ability tables can be obtained with less number of tabulatedsteps than the conventional method. This enables us toreduce computation time and computer memory storagefor the sub-group calculations.

Acknowledgements

The authors would like to express their thanks to Dr. P.Ribon and Dr. M. Coste-Delclaux of CEA/Saclay with

whom they had fruitful discussions. One of the author(G.C.) would like to thank Mr. M. Ishikawa, Mr. A. Sho-no, Mr. T. Hazama and Mr. K. Yokoyama of JAEA fortheir supports to his stay in CEA/Saclay.

References

Coste, M., Mengelle, S., 1996. Implementation of a sub-group method forself-shielding calculations in APOLLO2 code. In: Proceeding of theInternational Conference on the Physics of Reactors (PHYSOR96),September 16–20, 1996, Mito, Japan.

Cullen, D.E., 1974. Application of the probability table method tomultigroup calculations of neutron transport. Nucl. Sci. Eng. 55, 387.

Levitt, B.L., 1972. The probability table method for treatment unresolvedneutron resonances in Monte Carlo calculations. Nucl. Sci. Eng. 49,450.

Nakagawa, T. et al., 1995. Japanese evaluated nuclear data libraryversion 3 revision 2: JENDL-3.2. J. Nucl. Sci. Technol. 32 (12),1259.

Nikolaev, M.N., Ignatov, A.A., Isaev, N.V., Kokhlov, V.F., 1970. Themethod of subgroups for considering the resonance structure of thecross sections in neutron calculations. Sov. At. Energy 29, 689.

Ribon, P., 1986. Probability tables and gauss quadrature application toneutron cross-sections in the unresolved energy rangeProceeding of theTopocal Meeting on Advances in Reactor Physics and Safety,September 17–19, 1986, vol. 1. Saratoga Springs, USA, p. 280.

Rimpault, G.,et al., 2002. The ERANOS code and data system for fastreactor neutronic analyses. In: Proceeding of the InternationalConference on the Physics of Reactors (PHYSOR2002), October 7–10, 2002, Seoul, Korea.

Yamamoto, T., 2004. A generalized approach to optimize subgroupparameters. J. Nucl. Sci. Technol. 41 (4), 425.