1 status of calendf-2005 j-ch. sublet and p. ribon cea cadarache, den/der/sprc, 13108 saint paul lez...
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Status of CALENDF-2005
J-Ch. Sublet and P. Ribon
CEA Cadarache, DEN/DER/SPRC,13108 Saint Paul Lez Durance,
France
JEFFDOC-1159
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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
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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)
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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]
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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
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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
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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
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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%
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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
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InelasticInelastic
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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
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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