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### Status of CALENDF-2005J-Ch. Sublet and P. RibonCEA Cadarache, DEN/DER/SPRC,13108 Saint Paul Lez Durance, France

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

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)

Gauss Quadrature and PT-Mt

XS distribution in a

group G

Cross section over energy

PT discretisation

G= [Einf, Esup]

Padé Approximant and Gauss Quadrature

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

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.

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

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

Formalism interpretation- approximation

Coded MLBW leads to the worst results

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%

Ratio of subsequent energies points

Reconstruction accuracy

0.1%

1.6Kev – 0.99eV

points

IP= 1 32256

IP =2 44848

IP =3 63054

IP =4 90231

IP =5 130920

IP =6 186678 ref.

x1.4 steps

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

ECCO group library scheme

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

Codes

Cross-sections

Angular distributions

Emitted spectra

Interfaces

NJOY

(99-125)

Data

GENDF

GENDF*

ENDF-6

MERGE

(3.8)

GECCO

(1.5)

+ updates: Dimensions, …

NJOY-99 I/O

Fission matrix

mt = 5, mf = 6

Thermal scattering

(inel, coh.-incoh. el.)

CALENDF

(2005 Build 69)

Cross-sections

Probability Tables

Temperatures

293.2 573.2

973.2 1473.2

2973.2 5673.2

GENDF*

MF 1 Header

MF 3 Cross sections

MF 5 Fission spectra

MF 6 Scatter matrices

MF 50 Sub group data

ECCOLIB-JEFF-3.11968 groups with Probability Table

- 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…)

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:

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

CALENDF-2005-Resonance energies sampling (600 1100)

-Improved resonance grid

-Improved Gaussian quadrature table computation

-Total = partials sum # MT=1

-Probability tables order reduction

CALENDF-2005 input data

CALENDF

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

Energy range

Group structure

Weighting spectrum

Temperature

Dilution

Mat. and ENDF file

Output stream name - unit

Calculational accuracy indice

Output dumps or prints on unit 6 indices

CALENDF-2005 input data

REGROUTP

NFTP 10 './pu239.tp'

NFTPR 17 './pu239.tpr'

NIMP 0 80

REGROUSF

NFSF 12 './pu239.sf'

NFSFR 13 './pu239.sfdr'

NIMP 0 80

REGROUSF

NFSF 11 './pu239.sft'

NFSFR 14 './pu239.sftr'

NIMP 0 80

COMPSF

NFSF1 13 './pu239.sfdr'

NFSF2 14 './pu239.sftr'

NFSFDR 20 './pu239.err'

NFSFDA 21 './pu239.era'

NIMP 0 80

END

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

Pointwise cross section comparison: total

A Cubic interpolation

requires less points

than a linear one

But many more points

exists in the CALENDF

pointwise file in

the URR, tenths of

thousand …

CALENDF 115156 pts

NJOY 72194 pts

Groupwise cross section: total

ECCO 1968 Gprs

Groupwise cross section: fission

ECCO 1968 Gprs

CALENDF-2005 TPR

NOR = table order

NPAR = partials

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

15

N,xN

I = first negatif moment

1

Total

2

Elastic

101

Absorption

18

Fission

4

Inelastic

Probability

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

Total

Elastic

Absorption

Fission

Inelastic

N,xN

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

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

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’), ….
- ……..

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

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