1

Status of CALENDF-2005

J-Ch. Sublet and P. Ribon

CEA Cadarache, DEN/DER/SPRC,13108 Saint Paul Lez Durance,

France

JEFFDOC-1159

2

CALENDF-2005

Probability tables means a natural discretisation of the cross section data to describe an entire energy range

Circa 1970, Nikolaev described a sub-group method and Levitt a probability table method for Monte Carlo

The probability table (PT) approach has been introduced, exploited in both resolved (RRR) and unresolved (URR) resonance ranges

The Ribon CALENDF approach is based on Gauss quadrature as a probability table definition

This approach introduces mathematical rigorousness, procuring a better accuracy and some treatments that would be prohibited under other table definition such as group condensation and interpolation, isotopic smearing

3

Gauss Quadrature and PT-Mt

A probability distribution is exactly defined by its infinite moment sequence

A PT-Mt is formed of N doublets (pi,σi, i=1,N) exactly describing a

sever sequence of 2N moments of the σt(E) distribution

Such a probability table is a Gauss quadrature and as such will benefit from their entire mathematical settings

The only degree of freedom is in the choice of moments for which a standard is proposed in CALENDF, dependant on the table order, and associated to the required accuracy

pi, σt,i [σx,i , x = elastic, inelastic, fission, absorption, n,xn]

with i=1 to N (steps)

4

Gauss Quadrature and PT-Mt

Ni

i

nitit

ntt

E

E

nt pdpdEE

G 1,

in

max

min

sup

f

)()(.1

Cross section over energyXS distribution in a

group GPT discretisation

G= [Einf, Esup]

5

Padé Approximant and Gauss Quadrature

N

N

xp

i

ii

NN

iNi

NN

NN

Q

NN

NN

NN

N

NN

zRzx

pzR

zx

zbzbb

zRzazaza

zbzbzbb

zRzMzMzMMdxzx

xpzI

ii

NN

22

le tabquadrature ,

22

1

1110

22

nt approxima Pade

221

11

2210

22

moments 2

1212

2210

1

1

.....

.....1

.....

.....1

)()(

1,

Moments, othogonal polynomials, Padé approximants and Gauss quadrature are

closely related and allow to establish a quadrature table

The second line is the Padé approximant that introduces an

approximate description of higher order momenta

6

Partials cross sections

Partials cross sections steps follow this equation

The consistency between total and partials is obtained, ascertained by a suitable choice of the indices n

In the absence of mathematical background there is no reason why partial cross section steps cannot be slightly negative, and sometimes this is the case.

However, the effective cross section reconstructed from the sum of the steps values is always positive.

N to1n with ,,

)()(.1

1

ni

ixG

ntn ix

niti

pdEEEG

P

7

PT-Moments

The moments taken into account are not only from 0 to 2N-1 for the total, but negative moments are also introduced in order to obtain a better numerical description of the excitation function deeps (opposed to peaks) of the cross-section

CALENDF standard choice ranges from 1-N to N for total cross section, and -N/2 to (N-1)/2 for the partials

It is also possible to bias the PT by a different choice of moment (reduortp code word), this feature allows a better accuracy to be reached according to the specific use of a table of reduced order

For examples for deep penetration simulation or small dilution positive moments are not of great importance

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Unresolved Resonance Range

Generation of random ladders of resonance: the “statistical Hypothesis”

For each group, or several in case of fine structure, an energy range is defined taking into account both the nuclei properties and the neutronic requirement (accuracy and grid)

A stratified algorithm, improved by an antithetic method creates the partials widths

The treatment of these ladders is then the same as for the RRR (except, in case of external, far-off resonance)

Formalism: Breit Wigner Multi Niveau (# MLBW) or R-Matrix if necessary

9

Formalism interpretation- approximation

100000 120000 1400000

20

40

t, ba

rns

En, eV

Reich-Moore Multi-Level Breit-Wigner Breit-Wigner Multi-Niveau <RM> = 11.61 barns <ML> = 17.03 barns <MN> = 11.78 barns

42Ca - JEFF-3.1

122k

eV

127k

eV

Coded MLBW leads to the worst results

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Interpolation law

The basic interpolation law is cubic, based over 4 points

y = Pn(x)y = a + bx +cx2 + dx3

applicable to interpolate between

xi and x i+1 taking into account x i-1 and x i+2.

In this example cubic interpolation always gives an accuracy bellow 10-3 for an energy grid spacing up to 40%

10 -5

10 -4

1 1.2

1.4 1.6 1.8 2

E i+1 /E i

Error in

linear Int. cubic Int.

10 -3

10 -2

10 -1

Ratio of subsequent energies points

E1 E)

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Reconstruction accuracy

1.6Kev – 0.99eV

pointsIP= 1 32256IP =2 44848IP =3 63054IP =4 90231IP =5 130920IP =6 186678 ref.

x1.4 steps

0.1%

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CALENDF 2005

CALENDF-2005 is composed of modules, each performing a set of specific tasks

Each module is call specifically by a code word followed by a set of options and/or instructions particular to the task in hand

Input and output streams are module specific

Dimensional options have been made available to the user

Sometimes complex input variables are exemplified in the User Manual, around 30 cases

As always, QA test cases are a good starting point for new user

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ECCO group library scheme

CALENDF PT’s are used by the neutronic codes ERANOS, APOLLO and TRIPOLI

NJOY(99-125)NJOY

(99-125)

CALENDF(2005 Build 69)

CALENDF(2005 Build 69)

MERGE(3.8)

MERGE(3.8)

GECCO(1.5)

GECCO(1.5)

ENDF-6ENDF-6

Cross-sections Probability Tables

Cross-sectionsAngular distributionsEmitted spectra

Fission matrixmt = 5, mf = 6Thermal scattering(inel, coh.-incoh. el.)

Codes

DataInterfaces

NJOY-99 I/O

GENDF GENDF*

+ updates: Dimensions, …

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ECCOLIB-JEFF-3.1

Temperatures

293.2 573.2 973.2 1473.22973.2 5673.2

GENDF*

– MF 1 Header– MF 3 Cross sections– MF 5 Fission spectra– MF 6 Scatter matrices– MF 50 Sub group dataMF 50 Sub group data

Reactions

Total: mt1

Five partial bundles

Elastic 2: mt2

Inelastic 4: mt4 (22,23,28,29,32-36)

(n, n’-n’-n’3-n’p-n’2…)

N,xN 15: mt16,17 (24,25,30) 37 (41,42)

(n, 2n-3n-2n-3n,n,2n2-n,4n-2np-3np)

Fission 18: mt18 (n, f-nf-2nf)

Absorption 101: mt102-109, 111 (116) (n, -p-d-t-He--2-3-2p…)

1968 groups with Probability Table

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CALENDF-2005

CALENDF-2005 Fortran 90/95 SUN, IBM, Linux and

Window XP (both with Lahey) Apple OsX with g95 and ifort

User manual QA Many changes in format, usage

and some in physics:-Resonance energies sampling (600 1100) -Improved resonance grid -Improved Gaussian quadrature table computation-Total = partials sum # MT=1 -Probability tables order reduction

Test cases, ~ 30 Group boundaries hard coded

(Ecco33, Ecco1968, Xmas172, Trip315, Vitj175)

Probability table and effective cross sections comparison

Pointwise cross sections Increased accuracy and

robustness

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CALENDF-2005 input dataCALENDF ENERgies 1.0E-5 20.0E+6 MAILlage READ XMAS172 SPECtre (borne inferieure, ALPHA) 1 zones 0. -1. TEFF 293.6 NDIL 1 1.0E+10 NFEV 9 9437 './jeff31n9437_1.asc' SORTies NFSFRL 0 './pu239.sfr' NFSF 12 './pu239.sf' NFSFTP 11 './pu239.sft' NFTP 10 './pu239.tp' IPRECI 4 NIMP 0 80

Mat. and ENDF file

Energy range

Group structure

Weighting spectrum

TemperatureDilution

Calculational accuracy indice

Output dumps or prints on unit 6 indices

Output stream name - unit

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CALENDF-2005 input data

REGROUTP NFTP 10 './pu239.tp' NFTPR 17 './pu239.tpr' NIMP 0 80REGROUSF NFSF 12 './pu239.sf' NFSFR 13 './pu239.sfdr' NIMP 0 80REGROUSF NFSF 11 './pu239.sft' NFSFR 14 './pu239.sftr' NIMP 0 80COMPSF NFSF1 13 './pu239.sfdr' NFSF2 14 './pu239.sftr' NFSFDR 20 './pu239.err' NFSFDA 21 './pu239.era' NIMP 0 80END

Regroup probability tables computed on

several zones of a singular energy group,

used also for several isotopes

Regroup effective cross section computed on

several zones of a singular energy group

Idem but for the cross section computed

from the probability tables

Compare the effective cross section files

-Relative difference as the Log of the ratio

-Absolute difference as the ratio

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Pointwise cross section comparison: total

CALENDF 115156 ptsNJOY 72194 pts

A Cubic interpolationrequires less pointsthan a linear one

But many more pointsexists in the CALENDF pointwise file inthe URR, tenths ofthousand …

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Pointwise cross section comparison : capture

ReconstructionCriteria:

CALENDF 0.02%NJOY 0.1%

20

Groupwise cross section: total

ECCO 1968 Gprs

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Groupwise cross section: total

ECCO 1968 Gprsin the URR

2.5 to 300 KeV

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Groupwise cross section: fission

ECCO 1968 Gprs

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Groupwise cross section: fission

ECCO 1968 Gprsin the URR

2.5 to 30 KeV

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CALENDF-2005 TPR

ZA= 94239. MAT=9490 TEFF= 293.6 1968 groupes de 1.0000E-5 A 1.9640E+7 IPRECI=4 IG 1 ENG=1.947734E+7 1.964033E+7 NOR= 1 I= 0 NPAR=5 KP= 2 101 18 4 15 1.000000+0 6.115624+0 3.168116+0 1.724428-3 2.239388+0 2.630841-1 4.402475-1------------ IG 1000 ENG=4.962983E+3 5.004514E+3 NOR= 6 I= -5 NPAR=4 KP= 2 101 18 4 0 3.531336-2 1.001996+1 8.673775+0 4.299923-1 8.187744-1 4.878567-2 3.248083-1 1.299016+1 1.116999+1 5.483423-1 1.174122+0 4.879677-2 4.085168-1 1.686617+1 1.278619+1 1.593832+0 2.388719+0 4.880138-2 1.616318-1 2.349794+1 1.635457+1 3.590329+0 3.454910+0 4.884996-2 4.310538-2 3.445546+1 2.438486+1 4.303144+0 5.670905+0 4.876728-2 2.662435-2 4.254442+1 2.965644+1 7.196256+0 5.593669+0 4.874651-2

NOR = table order NPAR = partials

I = first negatif moment

Probability22

ElasticElastic

101101

AbsorptionAbsorption

1818

FissionFission

44

InelasticInelastic

11

TotalTotal

1515

N,xNN,xN

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CALENDF-2005 SFR

ZA= 94239 MAT=9490 TEFF=293.6 1968 gr de 1.0000E-5 a 1.9640E+7 IP=4 NDIL= 1 SDIL= 1.00000E+10 IG 1 ENG=1.947734E+7 1.964033E+7 NK=1 NOR= 1 NPAR=5 KP= 2 101 18 4 15 SMOY= 6.115624+0 3.168116+0 1.724428-3 2.239388+0 2.630841-1 4.402475-1 SEF(0)= 6.115624+0 SEF(1)= 3.168116+0 SEF(2)= 1.724428-3 SEF(3)= 2.239388+0 SEF(4)= 2.630841-1 SEF(5)= 4.402475-1- - -- - -IG 1000 ENG=4.962983E+3 5.004514E+3 NK=1 NOR= 6 NPAR=4 KP= 2 101 18 4 0 SMOY= 1.787921+1 1.364190+1 1.801794+0 2.337908+0 4.880425-2 SEF(0)= 1.787921+1 SEF(1)= 1.364190+1 SEF(2)= 1.801794+0 SEF(3)= 2.337908+0 SEF(4)= 4.880425-2

TotalTotalElasticElastic

AbsorptionAbsorptionFissionFission

InelasticInelasticN,xNN,xN

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Neutronic Applications

The PT are the basis for the sub-group method, proposed in the 70’s, a method that allow to avoid the use of “effective cross section” to account for the surrounding environment. Method largely used in the “fast” ERANOS2 code system

The PT are also the basis behind a the sub-group method implemented in the LWR cell code APOLLO2:

In the URR, with large multigroup (Xmas 172) In all energy range, with fine multigroup (Universal 11276) It allows to account for mixture self-shielding effects(mixture = isotopes of the same element or of different nature)

The PT are also used to replace advantageously the “averaged, smoothed, monotonic, …” pointwise cross section in the URR; method used by the Monte Carlo code TRIPOLI-4.4

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PT’s impact on the ICSBEP benchmarks

Code Tripoli-4.4.1Library JEFF-3.1

Experiment Calculation Whitout PT Δ PT'sICSBEP FastIMF-007 Kef f Unc. Kef f S.D. Kef f S.D.Big Ten deta. 1.0045 70 0.99863 13 0.99415 13

simp. 1.0045 70 0.99790 13 0.99337 12Δ (C-E) -623 -1074 450

t.z.h. 0.9948 130 0.98830 12 0.98435 12Δ (C-E) -650 -1045 395

IMF-012ZPR(16%) c-1 1.0007 270 1.00261 13 0.99959 13

Δ (C-E) 191 -111 302IMF-10ZPR-U9 c-1 0.9954 240 0.99181 12 0.98640 12

Δ (C-E) -359 -900 541IMF-002

c-1 1.0000 300 0.99216 10 0.99223 10Δ (C-E) -784 -777 6

IMF-001Jemima c-2 1.0000 120 0.99837 12 0.99868 13

c-3 1.0000 100 0.99741 12 0.99835 12c-4 1.0000 100 0.99850 12 0.99905 12

Average 0.99809 0.99869Δ (C-E) -191 -131 60

Specifications to ICSBEP NEA/NSC/DOC(95)03September 2005 Handbook Edition

Excellent way totest the influence

of the URR

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Neutronic Applications

Data manipulation processes are efficient and strict : isotopic smearing, condensation, interpolation and table order reduction

“Statistical Hypothesis”, exact at “high energy”, it means for 239 Pu > few hundred eV

In APOLLO2 the PT are used in the reactions rates equivalence in homogeneous media

The level of information in PT are greater than in effective cross section

Integral calculation: speed and accuracy

29

Future work

Introduction of probability table based on half integer moments, as suggested by Go Chiba & Hironobu Unesaki

Fluctuation factors computation using an extrapolation method based on Padé approximant

Increases of the number of partial widths, to account for improvement in evaluation format; i.e. (n,γf), (n,n’), ….

……..

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Conclusions

CALENDF-2002 http://www.nea.fr/abs/html/nea-1278.html

Improved version !! CALENDF-2005; now Full release through the

OECD/NEA and RSICC, this time …

Agenda