1

Overview of FRENDY Ver. 2
Japan Atomic Energy Agency (JAEA)
Kenichi TADA
https://rpg.jaea.go.jp/main/en/program_frendy/index.html

2Outline• Overview of nuclear data processing• Development of FRENDY• Features of new functions in FRENDY Ver. 2• Input format of FRENDY• Future work

3

Overview of nuclear dataprocessing

4

What is nuclear data processing?
[Example of processing]• Nuclear• Not just a converter• It performs many processes.
• Linearization, Resonancereconstruction, Doppler broadening, …

Evaluated Nuclear Data(JENDL, ENDF/B, JEFF, …)Cross Section (b)dataprocessing
generates cross section (XS)
libraries for particle transport
codes.

Nuclear Data Processing
105
Total XS of 235U (293.6 K)
104
103
102
101
100 10-4 10-2 100 102 104 106
Incident Neutron Energy (eV)

Nuclear Calculation• NJOY (LANL) and PREPRO

Cross Section(XS) Library(IAEA) are well-known in theworld.

Transport Codes

5

Status of nuclear data processing systemdevelopment• Development of nuclear data processing system has

been started in many institutes.
• To process their own nuclear data library• To handle new nuclear data format GNDS
• GNDS: Generalized Nuclear Data Structure[Nuclear data processing system development in the world]
CALENDF(CEA)GRUCON(Kurchatov)

Ruler(CIAE)PREPRO (IAEA)GALILEE (CEA),GAIA (IRSN)

NECP-Atlas(Xi’an Jiaotong Uni.)

FRENDY(JAEA)FUDGE(LLNL)NJOY(LANL)AMPX(ORNL)

Existing code
New code
Available code

Ref. D. Brown, “The New Evaluated Nuclear Data File Processing Capabilities,” INDC(NDS)-0695.

Evaluated nuclear 6
data file

XS library generation for
Resonance reconstruction(Linearization)
continuous energy MC codes
Doppler broadening• Many Monte Carlo (MC) codes use

ACE file.

Gas production XS
calculation• MCNP, PHITS, Serpent2, OpenMC,

SuperMC, …
• Current version of FRENDY does not
implement Heating XS calculation
function.

Heating XS calculation(KERMA, DPA)

Probability table
generation• KERMA：kinetic energy release in materials• DPA：displacement per atom

ACE file generation
ACE file

7

XS library generation for
multi-group calculation codes• Many processes are identical to

the XS library for continuous
energy MC codes.
• Processing of thermal scattering law

data is required to consider up
scattering.

Evaluated nuclear
data file
Resonance reconstruction(Linearization)
Doppler broadening
Gas production XS
calculation
Heating XS calculation(KERMA, DPA)
Probability table
generation

Multi-group file generation
Multi-group XS file

Processing of thermal
scattering law data
(Free gas model)

8Development of FRENDY

9Development of nuclear data processing
system FRENDY• JAEA started developing a new nuclear dataprocessing system FRENDY in 2013.
• FRom Evaluated Nuclear Data librarY to any application• To process nuclear data library by JAEA’s nuclear

application code users with simple input file.
• FRENDY Version 2 was released in 2022.
• https://rpg.jaea.go.jp/main/en/program_frendy/
k-eff , flux, …

Nucleardata library

Nuclear dataprocessing
systemFRENDY

Neutronics
calculation codes
MVP, PHITS

10

Features of FRENDY• Utilization of modern programming techniques• C++, BoostTest library, Git• Improvement of quality and reliability• Consideration of maintainability, modularity, andflexibility• Encapsulate all classes• Minimize the function of module• Maintain the independence of each module• Processing methods of FRENDY are similar to

NJOY.
Ref. K. Tada, et. al., “Development of nuclear data processing code FRENDY version 2,” J.
Nucl. Sci. Technol. (2023). (https://doi.org/10.1080/00223131.2023.2278600)

11Structure of FRENDY• Modularity is carefully considered.
• Modules of FRENDY can be used other calculation codes byadding only a few lines.
ENDF-6
formatEndf6Parser/Writer

Endf6
Converter

Resonance
Reconstructor

NuclearData
ObjectDopplerBroader

AceDataGenerator

GasProduction
CrossSection
Calculator
UnresolvedResonance
DataProcessor

AceDataObject

AceDataParser/Writer

ACE format

ThermalScattering
DataProcessor

CovarianceData
Processor

HeatingCross
SectionGenerator

Implemented
moduleNot implemented
module12Advantage for using the FRENDY’s
original nuclear data object• FRENDY has original format of the nuclear data for

efficient management.
• NuclearDataObject class• Minimizing the impact by the change of nuclear data

format• Developer and users are not necessary to consider the nuclear

data format.
• Consideration of a new data format GNDS
• GNDS format can be addressed if another set of parser, writer andconverter classes are implemented.
ENDF-6
formatEndf6Parser/Writer
Resonance
Reconstructor

DopplerBroader

Endf6
Converter

Gnds
Converter

NuclearData
ObjectGndsParser/Writer

GNDS
format

ThermalScattering
DataProcessor

13

Input format of FRENDY• FRENDY treats two types of the input formats.
• FRENDY original input format• NJOY compatible• RECONR, BROADR, GASPR, PURR, UNRESR, THERMR,GROUPR, MATXSR, MODER• Simple and easy input data
• The simplest input: Nuclear data file name and processing
mode are only required for the processing.
• FRENDY has recommended parameters in the source code.
• User can also change (override) parameters.

14

Input format of
FRENDY and NJOY
• Input parameters of

FRENDY consist of “input
data name” and “input
data”.
• Comment line is similar to

C/C++.
• //~ or /* ~ */• Input format of FRENDY

is easy to understand.
• This input format is suitablefor beginners.

[Sample input of FRENDY]
ace_fast_mode // Processing mode
nucl_file_name U235.dat
ace_file_name U235.ace
temp
296.0

[Sample input of NJOY]
reconr/ command
20 21/ input(tape20), output(tape21)'pendf tape for JENDL-4 U235' / identifier for PENDF
9228
/ mat
1.00e-03 0.00
/ err, temp0/
broadr/ command
20 21 22
/ endf, pendf(in), pendf(out)9228 1/ mat, temp no1.00e-03 -5.0E+2/ err, thnmax
296.0/ temp0/
gaspr/ command
20 22 23
/ endf, pendf(in), pendf(out)purr/ command
20 23 25
/ endf, pendf(in), pendf(out)9228 1 10 20 500/ mat, temp no, sig no, bin no, lad no296.0/ temp
1E10 1.0E6 1E4 1E3 300 100 35 10 1.0 0.1 / sig zero0/acer
/ command
20 25 0 30 31/ nendf, npend, ngend, nace, ndir
1 1 1 0.30
/ iopt(fast), iprint(max), itype, suffix'ACE file for JENDL-4 U235' / descriptive character
9228 296.0/ mat, temp
1 1
/ newfor(yes), iopp(yes)1 1 1/ thin(1), thin(2), thin(3)
stop
/

15

Release of FRENDY• FRENDY was released from our website.
• https://rpg.jaea.go.jp/main/en/program_frendy/• Open-source software• 2-Clause BSD license• Presentations of FRENDY training courseand exercise are also found in this website.
• Manual of FRENDY• JAEA-Data/Code 2022-009• https://jopss.jaea.go.jp/pdfdata/JAEA-Data-Code-2022-009.pdf• There are many sample input files in this manual.

16

Development of FRENDY Ver. 2• FRENDY Ver. 1 only generates ACE file.
• Many capabilities are prepared after FRENDY Ver. 1.
• Neutron induced multi-group cross section generation• Perturbation of ACE file for uncertainty quantification• This function was implemented in FRENDY Ver.1.01.001.

• Modification of evaluated nuclear data file
• Uncertainty quantification for probability tables• Improvement of input checker to reduce input errors• FRENDY version 2 was released including thesefunctions in Jan. 2022.

17

Features of new functionsin FRENDY Version 2

18

Multi-group XS generation• FRENDY/MG generates multi-group

XS files from ACE files.
• FRENDY/MG can also generate amulti-group XS library from the existing
ACE library.

• NJOY input is also available for…

multi-group XS files.

Linearization

Probability-tablecalculation
ACE file generation

multi-group XS file generation.
• Input of GROUPR and MATXSRACE lib.

modules are available.
*) A. Yamamoto, K. Tada, G. Chiba, T. Endo, “Multi-groupneutron cross section generation capability for FRENDY
nuclear data processing code,” J. Nucl. Sci. Technol., 2021.
https://doi.org/10.1080/00223131.2021.1921631

FRENDY/MG
Multi-group
XS lib.

FRENDY Version 1• FRENDY/MG*) is used for a

Evaluatednuclear data lib.

19

Major capabilities of multi-group generation• Output format of multi-group cross sections• GENDF and MATXS• Angular/energy distributions• LAW=3, 4, 7, 9, 11, 44, 61, and 66 in ACE file
• All nuclides in JENDL-4.0, -5, ENDF/B-V, B-VII.1, B-VIII.0,
JEFF-3.3, and TENDL-2019, -2021 are available.
• Calculation function of multi-group photoatomic

cross section, KERMA factors, and group-to-group
photon scattering matrix is not implemented.
• GAMINR of NJOY

20

New functions of multi-group generation• Back-ground cross-section set can be

automatically set with minimum number ofbackground cross-sections*).
• A compound of different isotope can be specified
to explicitly consider the resonance interference
effect.
• For example, U-235, U-238, and O-16 in UO2.

• Any energy grid points can be used for ultra-finegroup slowing down calculation.
*) A. Yamamoto, T. Endo, K. Tada, “Adaptive setting of background cross sections forgeneration of effective multi-group cross sections in FRENDY nuclear data processing code,”
J. Nucl. Sci. Technol., 2021. https://doi.org/10.1080/00223131.2021.1944930

21Resonance up-scattering correction (RUC)• Calculation of elastic scattering cross section andscattering kernel using 0 K scattering cross section
data generated by FRENDY.
• Mixed nuclide is also available to explicitly consider the

resonance interference effect.
• For example, U-235 with RUC, U-238 with RUC, and O-16 without

RUC in UO2.

• Sample input files to use RUC are found in the FRENDY

manual.
• Neutronics calculation codes can consider RUC

without any modification.

22

Comparison of XS with and without RUC• Larger differences are found at the large

resonances below 100 eV.

Cross section [b]<Comparison of cross section with and without RUC (U-238, MT=102)>

23

Multigroup generation of secondary gamma
spectrum• FRENDY results show good agreement with NJOY

results.
• All nuclides in JENDL-4.0, JENDL-5, ENDF/B-VII.1,

ENDF/B-VIII.0, JEFF-3.3, and TENDL-2018 are available.
Secondary gamma spectrum of Fe-56 from J5(MT=102, Ein=13.840-14.191MeV, vitamin-j 42g)
1.E-03
1.E-04

1.E-05
1.E-06
1.E-07
1.E-08FRENDYNJOY2016.65

1.E-09
1.E+04 1.E+05 1.E+06 1.E+07 1.E+08
Outgoing energy [eV]24

2D interpolation scheme of FRENDY• ENDF-6 format recommends using unit baseinterpolation for 2D interpolation.
• FRENDY uses MCE for
2D interpolation.

MCE divides into N
equal volume areas.

• Method of Corresponding

Energy• MCE is identical to unit
base interpolation when
the number of divisions N
is 1.
*)*) A. Trkov et al., “ENDF-6 Format Manual,” BNL-203218-2018-INRE. (2018).

25

Difference of 2D interpolation scheme• Difference was found in the secondary gamma

spectrum between the unit base and MCE.
• Close to threshold energy.
• We think that MCE shows a reasonable distribution.
• Default number ofSecondary gamma spectrum of Fe-56 from J5
Ein=12.5-12.8MeV, vitamin-j 42g)
divisions in FRENDY is 2. (MT=22,
1.E-04
unit base• Gamma spectrum does not
mce=2
1.E-06change even if the number
mce=5

of divisions is increased.

1.E-08
1.E-101.E-12
1.0E+05

1.0E+06
Outgoing energy [eV]

1.0E+07

26

Calculation of thermal neutron scatteringcross section• NJOY uses 118 fixed energy grid points.
• These fixed energy grid points are insufficient forneutronics calculation.
• FRENDY used the incident energy grid points ofthe corresponding nuclide.
• Very large processed data size

• Optimization of the number of incident energy

grid points is required to reduce the data size.
• Linearization function is newly added for thermal

scattering law data processing.
• From FRENDY Ver. 2.02

27

Comparison of thermal neutron scatteringcross section• NJOY only calculates the incoherent inelastic XS on 118

energy grids.
• Other energy grids are interpolated using the 5th order Lagrange

interpolation.

• The fixed energy grid is not appropriate for a material ofwhich the cross section is oscillated.
• This difference may have impact on the TRIGA reactor.

<Incoherent inelastic scattering XS (H in ZrH, 400 K)>
Difference at low
energy region isobserved in many
materials.

0%

10

1
1.0E-5+1%(FRENDY-NJOY99)/NJOY99

Cross section [b]1001.0E-4

1.0E-3 1.0E-2 1.0E-1 1.0E+0
Incident neutron energy [eV]

-1%

28

Impact of number of energy grids on k-eff• Investigation of appropriate tolerance value forlinearization.
• Criticality benchmarks which contain H in ZrH were selected.

• Large differences between FRENDY and NJOY were

found.
• The tolerance value for linearization should be less than or equal

to 0.05%.
• FRENDY uses a default tolerance value of 0.02%.
<Relative difference of k-eff in each criticality benchmark>Fixed 118
energy gridHCM-3-1
ICT-3-132
ICT-3-133-0.037%0.026%
0.021%

Old
FRENDY

Tolerance value
0.10%

0.05%

0.02%

0.003% 0.013%-0.002% -0.009%
0.001% -0.012%

0.002%-0.001%-0.004%0.002%-0.006%-0.005%

29

Comparison of incident energy grid points• The number of incident

energy grid points isvery small even if
tolerance value is0.02%.
• Incident energy grid

points and data size of
the ACE file are reducedby more than 80% formany materials when
the tolerance value is0.02%.

<Comparison of the number of incident
neutron energies in each tolerance value>Old
FRENDY

Tolerance value
0.10% 0.05% 0.02%

Al-27
Fe-56
Be in BeO
D in D2O

2,773
9,775
10,272
559

173
174
186
166

234
238
261
208

330
345
387
280

H in H2O
H in ZrH
O in BeO
O in H2O

500
500
10,243
512

157
297
185
152

194
423
256
176

274
653
381
225

O in UO2

3,958

178

246

352

U in UO2

4,626

170

230

326

Zr in ZrH

3,378

172

235

336

30

Uncertainty quantification for probability
tables• FRENDY Ver. 1 requires number of ladders as an

input parameter to calculate probability table.
• An optimal number of ladders has not been investigated.
• Probability table is important to consider self-shielding effectin unresolved resonance region.

• FRENDY Ver. 2 calculates statistical uncertainty ofprobability table.
• Product of probability table and total XS is considered astarget of statistical uncertainty of probability table.

• User can generate probability

table with optimal number ofladders.

<Example of ladder(pseudo resonance structure)>
P’’55
P’

• This function is also contributed toP’’444P’
P

reduce the total processing time.

P’’333
P’
P
P’’222
P’
P
P’’111P’
PE

31

Calculation time of probability table ineach energy grid (U235 from J-40)• Optimal number of ladders is Incident Total

No. ofTime perdifferent in each energy grid.

energytime

ladders

ladder• Criteria of statistical error : 5%

[eV]

[s]

[-]

[ms]

(default value)• Ladder number optimization

effectively reduces
generation time of probability
table.
• Number of ladders becomes

smaller when incident energy isincreased.
• Calculation time to generate
one ladder becomes longer
when incident energy isincreased.

500

4.28

80

53.5

563

3.33

60

55.5

850

3.81

60

63.5

1110

3.52

50

70.4

1300

2.25

30

75.0

4250

3.74

30

124.7

6500

4.53

30

151.0

13000

8.15

40

203.8

26900

16.40

60

273.3

30000

11.50

40

287.5

32

ACE file perturbation tool
• Implementation of a random sampling tool to perturb

cross section and fission spectrum of ACE file*).
• User sets reaction type, energy region, and amount of

perturbation.
• Cross section and fission spectrum are randomly perturbed
using random sampling mode.
[Example ofcross section perturbation]

×1.1

Incident neutron energy

Cross section [b]Cross section• This tool is available from FRENDY Ver. 1.01.001.
[Example of random sampling (1H, (n,γ))]1.0E-3

1.0E-4

1.0E-5
1.0E+5

1.0E+6
1.0E+7
Incident neutron energy [eV]*) K. Tada, et al., “Development of ACE file perturbation tool using FRENDY,” J. Nucl. Sci.
Technol., 60, pp.624-631 (2023). https://doi.org/10.1080/00223131.2022.2130463

33

ENDF modification function• This function removes, adds, exchanges specified

MF/MT data.

MF=1
MF=2, MT=151
MF=3, MT=1
MF=3, MT=2
MF=6

remove

add
MF=1

MF=1

MF=2, MT=151

MF=2, MT=151

MF=3, MT=1

MF=3, MT=1
MF=6

MF=3, MT=2
MF=3, MT=102
MF=6

The modified evaluatednuclear data file must be
checked carefully since
FRENDY does not checkthe new file.
exchange

MF=1
MF=2, MT=151
MF=3, MT=1
MF=3, MT=2
MF=6

34

Linearization function of TAB1 data
• NJOY uses the first interpolation scheme for multiple

interpolation schemes in some MF data.
• This process is not applicable when multiple interpolation

schemes are used in the TAB1 data.
• FRENDY linearizes TAB1 data to handle multiple interpolation
schemes.
• Differences were found between FRENDY and NJOY.

• Linearization function ofSecondary energy distribution(C000 from J40, MF=5/MT=91,
4.E+06
Ein=7.887 MeV)

TAB1 data was prepared asa tool of FRENDY.
solved if the TAB1 data arelinearized before the nuclear
data processing.

g(E→E’)• Such interpolation problem is

3.E+06NJOY

3.E+06

FRENDY

2.E+06
2.E+06
1.E+06
5.E+050.E+00
0.00

0.20

0.40

0.60

0.80

Secondary energy (E’) [eV]

1.00

35

Input checker functions to reduce input errors• FRENDY automatically checks input parametersand processing results.
• FRENDY reads nuclear data files and sets recommended

values.
• Input parameters are compared to recommended values.

• Processing results are also used to verify input parameters.

• These functions help many users• These functions are also available

for NJOY input format.
• FRENDY is also used as the input
checker of NJOY.

100

nmix=2
nmix=1

Cross section [b]to generate correct input file.

< Incoh-Inela XS of H in CH2 >

×1/2

10
1.E-05

1.E-03

1.E-01

Incident Neutron Energy [eV]

1.E+01

36

Input format of FRENDY

37

Basics of FRENDY input format• First line must be processing mode.
• ace_fast_mode： Neutron induced ACE file• ace_tsl_mode： ACE file of thermal scattering law data
• ace_dosi_mode： Dosimetry ACE file
• mg_neutron_mode : Neutron induced multi-group XS file

• Other lines are free format.
• Setting “input data name” and “input data”• Parentheses are used to describe array data, e.g., (1.0 2.0 3.0).
• Text data is surrounded by single or double quotation mark, e.g.,“~” or ‘~’.
• Multiple lines are available for array data and text data.

• Comment line is similar to C/C++.
• //~ or /* ~ */38

Main input parameters• Required parameters• nucl_file_name：Evaluated nuclear data file name
• nucl_file_name_tsl：TSL data file name (TSL only)• Other main input parameters• temp：Temperature (K) (Default: 293.6 K)• ace_file_name：ACE file name (Default: “nucl_file_name”.ace)• ace_dir_file_name ：XSDIR file name (Default: “nucl_file_name”.ace.dir)• suffix_id：Suffix ID of ACE file (Default: 00)• ace_label_data：Comment line of ACE file (one line) (Default: none)• thermal_za_id_name： S(α, β) identifier of MCNP (TSL only) (Default: ZA
value)• Additional parameters• write_pendf_probability_table：PENDF file name after probability table
generation• write_pendf_tsl：PENDF file name after TSL data processing (TSL only)• PENDF (Point-wise ENDF) file is a processing result of NJOY.
• These PENDF file can be used as input PENDF of NJOY.

39

Example of FRENDY input (Neutron induced)
ace_fast_mode // Processing modeFirst line must be set
nucl_file_name../lib/U235.dat
processing mode.
temp /* [K] */300.0
ace_file_name./ace/U235.ace
Relative path or
absolute path
ace_dir_file_name ./xsd/U235.xsdir
ace_label_data“U-235 from JENDL-4.0”
suffix_id
50• Processing conditions of above example• Evaluated nuclear data file name： ../lib/U235.dat• Temperature：
300.0 [K]• ACE file name：./ace/U235.ace• XSDIR file name：./xsd/U235.xsdir• Comment line of ACE file：U-235 from JENDL-4.0• Suffix ID：
0.50

40

Example of FRENDY input (TSL)First line must be set
ace_tsl_mode // Processing mode
processing mode.
nucl_file_name../lib/H001.dat
nucl_file_name_tsl../lib_sab/01_h_in_h2o.txt
TSL data file nameace_label_data“HinH2O from JENDL-4.0”
temp
296.0
Temperature prepared in TSL
ace_file_name./ace_sab/lwtr.ace
data file (nucl_file_name_tsl)
ace_dir_file_name./xsd_sab/lwtr.xsdir
is only available.
suffix_id
50
thermal_za_id_name “lwtr”
S(α, β) identifier of MCNP
(Maximum: 6 characters)• Processing conditions of above example• TSL data file name: ../lib_sab/01_h_in_h2o.txt• S(α, β) identifier of MCNP： lwtr
• lwtr：light water• This name is used in S(α, β) card of MCNP input.

41

Input format of multi-group generation• Input format of multi-group generation is similar tothat of ACE file generation.
• Difference is only processing mode name• Multi-group file can be generated from ENDF and ACE files• Processing of U-235
mg_neutron_mode//Process mode
nucl_file_name ( ../lib/U235.dat ) //ENDF file name• Processing of H in H2O
mg_neutron_mode//Process mode
nucl_file_name ( ../lib/H001.dat ) //ENDF file namenucl_file_name_tsl ( ../lib/01_h_in_h2o.txt )mg_tsl_data_type (hh2o) //This data type is used for MATXS

42

Difference of IWT options• IWT option affects the angular neutron flux

distribution• Angular neutron flux distribution may become jagged when

IWT=0 (variable) *).

• Recommend option: IWT=2• If users want to accurately
calculate the angular neutron
flux distribution.
• IWT must be 2 for the multigroup cross section generation.
• Do not use IWT=1 (constant).
• This option will affect not only
the angular neutron flux
distribution but also k-effective
value.
*) J. L. Conlin, et al., “Continuous-S(α, β) Capability in
MCNP, LA-UR-12-00155, LANL (2012).

Example of angular neutron
flux distribution with IWT=0/2

IWT=0

IWT=2
μ=cosθ

43

IFENG option and weight_option• weight_option (0~2) used in FRENDY is identicalto iwt option in NJOY/ACER module.
• iwt=0 (variable), 1 (constant), 2 (tabulated)• The values of weight_option (iwt) and those of

IFENG are different.
• IFENG=0 (discrete), 1 (skewed), 2 (continuous)

IFENG

44

Future work

45

Development of XS output function• It outputs X-Y data for gnuplot, Matplotlib, and Excel.
• Cross section data and double differential cross section
data from ENDF(PENDF), ACE, and GENDF.
• Pointwise and multi-group data.
• Multi-group cross section can be generated from pointwise data.
• Multi-group cross section generation using flat or 1/E flux.

• This function will be released in next update of

FRENDY Ver. 2 (Ver. 2.03).
Cross section [b]Cross section [b]<Comparison of pointwise XS of U-238> <Comparison of multi-group XS of U-238>

46

Release of FRENDY Version 3• FRENDY Ver. 3 will be released including thesefunctions.
• Calculation function of neutron KERMA factor and DPA

cross section (HEATR of NJOY).
• KERMA: Kinetic Energy Release in Materials• DPA: Displacement Per Atom• Calculation function of multi-group covariance matrix

calculation function (ERRORR of NJOY).
• Calculation function of multi-group photoatomic crosssection, KERMA factors, and group-to-group photon
scattering matrix (GAMINR of NJOY).
• Treatment of the GNDS formatted nuclear data file.
• Using GIDI+ of LLNL
• https://github.com/LLNL/gidiplus

47

Reference Materials

48

Random sampling
Covariance matrix of nuclear data

inp/Nddd_000x
102 2.00E+01 6.43E+00 8.78E-01
102 6.43E+00 4.30E+00 8.67E-01
102 4.30E+00 3.00E+00 8.62E-01．．．

• User has to prepare covariance matrix.
• We are now developing converter from GENDF file of
NJOY/ERROR to input of random sampling tool.

• Generation of perturbation factors using random

sampling method• See “/frendy_20yymmdd/tools/make_perturbation_factor/sample”

49

Uncertainty quantification using random
sampling method
Godiva (HMF-001)
Geom
etry

SphereRadius: 8.7 cm

CompositionU-235: 93.71 wt.%
U-238: 5.27 wt.%
U-234: 1.02 wt.%𝑘eff

1.000 ± 0.001

◆
◆
◆

◆

MCNP6.2
Number of perturbed ACE file: 100
Covariance data: 56groupcov7.1(from SCALE6.2.3)
MT=2,4,16,18,102,452, and 1018
(MT=452: ν, MT=1018: χ)

Godiva [1][1] ICSBEP NEA/NSC/DOC(95)03, Organization for
Economic Co-operation and Development-Nuclear
Energy Agency (OECD-NEA) (September 2016).

50

Calculation results (k-effective uncertainty)𝑘eff -uncertainty due to all nuclides and reactions ∆𝑘/𝑘 [%]Sensitivity analysis (SA) Random sampling method usingof MCNP6.2
perturbation tool
1.11
1.12 [0.98 – 1.24]
Comparison of 𝑘eff -uncertainty due to individual nuclide andreaction ∆𝑘/𝑘 [%]51

Appendix
Processing method in FRENDY

52

Linearization• Evaluated nuclear data library describes crosssections with different interpolation scheme.
• Log-log interpolation, linear-linear interpolation, …

𝜎𝑖• Different interpolation schemes are inconvenient.
• Linearization is required for Doppler broadening.
• Many nuclear calculation codes use only
Add middle point
if 𝜎 cannot belinear-linear interpolation.
represented by𝜎𝑖𝜎𝑖linear-linear
interpolation

𝜎
𝜎
𝜎𝑖+1
𝑥𝑖

𝑥𝑖+1 𝑥𝑖

𝑥

1
𝑖+2

1
𝑖+2

𝜎

𝜎𝑖+1
𝑥𝑖+1 𝑥𝑖

1
𝑖+4

𝑥

1
𝑖+4

𝑥

1
2

𝜎𝑖+1

1
2

𝑥𝑖+1

𝑖+

𝑖+

53

Resonance reconstruction• XS of resolved and unresolved resonance regions are

given as resonance parameters of resonance formula.
• Calculation of XS in resolved resonance region• Calculation of averaged XS in unresolved resonance region• XS in continuous region is given in MF=3.

Cross section [b]1.0×104[Radiative capture XS of U-235 from JENDL-4.0 (0 K)]

1.0×102

Continuous
region1.0×100
1.0×10-2
1.0×10-4

Resolved
resonance region(RR)

Unresolved
resonance region(URR)

1.0×10-4 1.0×10-2 1.0×100 1.0×102 1.0×104 1.0×106

Incident neutron energy [eV]

54

Resonance formulae• ENDF-6 format arrows to use many resonance formulae• Single-Level Breit-Wigner (SLBW)• Multi-Level Breit-Wigner (MLBW)• Reich-Moore• R-Matrix Limited• These resonance formulae are obtained from calculation of

the Schrödinger equation for the multi-particle system.
• The detail description is found in following papers• M. E. Dunn and N. M. Greene, “POLIDENT: A Module for Generating
Continuous-Energy Cross Sections from ENDF Resonance Data,”
NUREG/CR-ORNL/TM-2000/035 (2000).
• A. M. Lane, R. G. Thomas, “R-Matrix Theory of Nuclear Reactions,” Rev. Mod.
Phys., 30, [2], pp.257-353 (1958).
• N. M. Larson, “Updated Users’ guide for SAMMY: Multilevel R-Matrix Fits to
Neutron Data Using Bayes’ Equations,” ORNL/TM-9179/R8, ENDF-364/R2,
p.9 (2008).

55

Single-Level Breit-Wigner• Radiative capture cross section of SLBW𝑁𝑅𝑗𝜋• σ𝛾 = 2 σ𝑗 𝑔𝑗 σ𝑟=1
𝑘1

𝚪𝑛𝑟 𝐸1 𝚪𝛾𝑟
𝐸1
𝐸𝑟 𝐸𝑟 −𝐸1 2 +Γ𝑟 2 Τ4

• Elastic scattering cross section of SLBW
• σ𝑠𝑐 = σ𝑝 +𝑁𝑅𝑗 𝚪𝑛𝑟 𝐸1 2 −2𝚪𝑛𝑟 𝐸1 Γ𝑟 sin2 𝜙𝑙 +2𝚪𝑛𝑟 𝐸1 𝐸𝑟 −𝐸1 sin 2𝜙𝑙𝜋
σ 𝑔 σ
𝑘1 2 𝑗 𝑗 𝑟=1
𝐸𝑟 −𝐸1 2 +Γ𝑟 2 Τ4

• Evaluated nuclear data file contains many parameters, e.g.,

neutron width 𝚪𝑛𝑟 and Γ𝑟 , spin 𝑔𝑗 , resonance energy 𝐸1 .
• Resonance reconstruction calculates cross section atincident energy 𝐸𝑟 and linearizes cross section.

56

Doppler broadening• Evaluated nuclear data file contains the data at 0 K.
• Doppler effect must be considered to calculate

higher temperature cross section, i.e., Doppler
broadened cross section.
• Nuclear data processing system should generate thetemperature-dependent cross section data from the data
at 0K.

57

Doppler effect• Doppler effect is the change in frequency of a wave.
• E.g., change of pitch heard when a vehicle sounding a horn
approaches and recedes from an observer.
• This effect must be considered in reaction of nucleus andincident particle.
• Nucleus is at rest when temperature is 0 K.

• Kinetic energy of nucleus is increased when temperature increases.
• Relative energy between nucleus and incident particle is changed whentemperature changes.

[Example of Doppler effect (pitch heard)]© ICOOON MONO

58

Doppler broadening• Most of evaluated nuclear data files contain crosssections at 0 K.
• Nucleus vibration (Doppler broadening) should beconsidered to calculate cross section at T K.
[Reaction of incident particle
and nucleus]

1,500

1,000

[Equation of Doppler broadening][Example of Doppler broadening]
0K

Peak value
becomes
lower

1200K

Results ofintegration
do notchange

500
2

1 𝛽 ∞
ሾ𝑒−𝛽 𝑣−𝑣𝑟
2
න 𝑑𝑣𝑟 𝑣𝑟 𝜎 𝑣𝑟
𝜎 𝑣, 𝑇 = 22𝑣 𝜋 0−𝑒 −𝛽 𝑣+𝑣𝑟 ൧

0

𝑇: Temperature,
8.4
𝑣：velocity of incident particle, 𝑣𝑟 ：relative velocity

Resonance width
becomes wider

8.6

8.8

9.0

9.2

59

Calculation of Doppler broadened XS• Linearization is required

for easy calculation of
Doppler broadened XS.
• If 𝝈 𝒗𝒓 is not a function of𝑣𝑟 , Doppler broadened XS
is obtained using error
function.
• Error function is provided
in many programming
language as a service
routine.

[Calculation formula of Doppler
broadened XS]
2−𝛽 𝑣−𝑣𝑟
1 𝛽 ∞ሾ𝑒𝜎 𝑣, 𝑇 = 2
න 𝑑𝑣𝑟 𝑣𝑟 2 𝝈 𝒗𝒓2𝑣 𝜋 0−𝑒 −𝛽 𝑣+𝑣𝑟 ൧

𝐸 − 𝐸𝑘
𝐸𝑘+1 − 𝐸𝝈 𝒗𝒓 =
𝜎𝑘+1 +𝜎𝑘
𝐸𝑘+1 − 𝐸𝑘
𝐸𝑘+1 − 𝐸𝑘
1
𝐸𝑟 = 𝑚𝑣𝑟 2
2∞
F𝑛 𝑎 ==

12
න 𝑧 𝑛 𝑒 −𝑧 𝑑𝑧
𝜋 𝑎𝑛−1
F𝑛−2 𝑎 + 𝑎𝑛−1 F1 𝑎
2

[Calculation of error function erf(a)]1 𝑎 −𝑧 2
𝑒𝑟𝑓 𝑎 =
න 𝑒
𝑑𝑧
𝜋 0

60

Gas production XS calculation• Nuclear reactions sometimes yield gases.
• p(proton: 1H), D(deuteron: 2H), T(triton: 3H), 3He, α(4He)• Gas generation affects the embrittlement of material

and the internal gas pressure in a fuel pin.
• ENDF-6 format prepares gas production reaction in

MT=203-207.
• MT=203:(z, Xp), MT=204:(z, Xd), MT=205:(z, Xt), MT=206(z, X3He),

MT=207(z, Xα)• Nuclear data processing system calculates gas production XS even
if gas production XS is not found in evaluated nuclear data file.

• Generated nuclide is also considered.
• For example, 12C(n, n2α)4He reaction generates 3 alpha(4He) particles• 2α+4He=3α61

Self-shielding effect in URR• Resonance parameters in URR are averaged ones.
• Nuclear data processing system cannot reconstruct resonancestructure in URR.
• Consideration of the self-shielding effect in URR is important for
accurate neutronics calculations.
• Self-shielding effect in RR is automatically considered in transport

calculation codes using pointwise XS data.

• How to consider the self-shielding effect in URR?
• NJOY prepares two methods.
• Deterministic method : UNRESR module• Monte Carlo method (probability table method) : PURR module• We strongly recommend to use probability table method.
• “For most purposes, UNRESR has been superseded by PURR.”*)• “The PURR results may be more reliable at low σ0 values than UNRESR
results.”*)*) R. E. MacFarlane and A. C. Kahler, “Methods for Processing ENDF/B-VII with NJOY,”
Nuclear Data Sheets, 111, pp.2739-2890 (2010).

62Consideration of self-shielding effect using
probability table (NJOY/PURR & FRENDY)• Bondarenko-type self-shielded XS σ𝑥 𝐸 is used.
• σ𝑥 𝐸 is obtained by probability table 𝑃𝑖 𝐸 .

• σ𝑥 𝐸 =𝑃𝑖 𝐸 𝜎𝑥,𝑖 𝐸
𝑏𝑖𝑛
σ𝑖=1
𝜎0 +𝜎𝑡,𝑖 𝐸
𝑃𝑖 𝐸
𝜎0 +𝜎𝑡,𝑖 𝐸

σ𝑏𝑖𝑛
𝑖=1

• 𝑥: reaction type 𝑡, sc, 𝑓, 𝛾 , 𝜎0 : back ground XS• Monte Carlo calculation codes directly use 𝑃𝑖 𝐸 .
[Example of probability table]

E=1.2～1.5keV

𝜎𝑥,1

𝜎𝑥,𝟐

𝜎𝑥,𝟑

(10～16 barn)(16～42 barn)(42～176 barn)

𝑃1 = 0.56

𝑃2 = 0.36

𝑃3 = 0.08

63

Generation of probability table (1/5)
1. Determination ofresonance energy 𝑬𝒓,𝒊using random number
2. Determination of resonancewidth using random number
3. Calculation of XS andprobability 𝑃n 𝑇 of 𝜎𝑛−1 ≤𝜎𝑡,𝑗 𝐸𝑗 , 𝑇 < 𝜎𝑛σ

4. Continuing generation ofpseudo resonance structure(Pseudo resonancestructure is called as“ladder”)
5. Generation of table bins of
average XS 𝜎𝑥,𝑛 andcorresponding probability
𝑃n 𝑇 (probability table).
ഥ : Average level spacing
𝐷
𝑅𝑊 : Random number
conforming to the
Wigner distributionfunctionഥ × 𝑅𝑊
𝐸𝑟,𝑖 = 𝐸𝑟,𝑖−1 + 𝐷

E64

Generation of probability table (2/5)
1. Determination of resonance 4. Continuing generation ofenergy using random number
pseudo resonance structure
2. Determination of resonance(Pseudo resonancestructure is called as
width 𝜞𝒓,𝒊 using random“ladder”)
number
5. Generation of table bins of
3. Calculation of XS andaverage XS 𝜎𝑥,𝑛 andprobability 𝑃n 𝑇 of 𝜎𝑛−1 ≤corresponding probability
𝜎𝑡,𝑗 𝐸𝑗 , 𝑇 < 𝜎𝑛𝑃n 𝑇 (probability table).
σ𝑹𝝌,𝒌𝜞𝒓,𝒊 = 𝜞𝒓 ×𝒌𝛤ഥ𝑟 : Average resonancewidth𝑅𝜒,𝑘 : Chi-squared
random numberswith k degree of
freedom

E65

Generation of probability table (3/5)
1. Determination of resonance 4. Continuing generation ofpseudo resonance structureenergy using random number(Pseudo resonance
2. Determination of resonancestructure is called as
width using random number“ladder”)
3. Calculation of XS and
5. Generation of table bins ofprobability 𝑷𝒏 𝑻 of 𝝈𝒏−𝟏 ≤average XS 𝜎𝑥,𝑛 and
𝝈𝒕,𝒋 𝑬𝒋 , 𝑻 < 𝝈𝒏corresponding probability
𝑃n 𝑇 (probability table).
σ
P5
𝜎5
P4
𝜎4
P3
𝜎3
P2
𝜎2
P1
𝜎1
E

66

Generation of probability table (4/5)
1. Determination of resonance 4. Continuing generation ofenergy using random number pseudo resonancestructure
2. Determination of resonanceresonance structure
width using random number (Pseudois called as “ladder”)
3. Calculation of XS and
5. Generation of table bins ofprobability 𝑃n 𝑇 of 𝜎𝑛−1 ≤average XS 𝜎𝑥,𝑛 and
𝜎𝑡,𝑗 𝐸𝑗 , 𝑇 < 𝜎𝑛corresponding probability
𝑃n 𝑇 (probability table).
σ
P’5
𝜎5
P44
P’
𝜎4
P33
P’
𝜎3
P22
P’
𝜎2
P11
P’
𝜎1
E

67

Generation of probability table (5/5)
1. Determination of resonanceenergy using random
number
2. Determination of resonancewidth using random number
3. Calculation of XS andprobability 𝑃n 𝑇 of 𝜎𝑛−1 ≤𝜎𝑡,𝑗 𝐸𝑗 , 𝑇 < 𝜎𝑛4. Continuing generation ofpseudo resonance structure(Pseudo resonancestructure is called as“ladder”)
5. Generation of table bins of
average XS 𝝈𝒙,𝒏 andcorresponding probability
𝑷𝒏 𝑻 (probability table).
P’’55
P’

𝜎5
𝜎4

𝜎3
𝜎2
𝜎1

P’’444P’
P
P’’333
P’
P
P’’222
P’
P

P’’111P’
PE

68

Calculation of Doppler broadened XS in URR• Single-Level Breit-Wigner (SLBW) resonance formula is usedin URR.
• Approximated Doppler broadened XS can be directly
obtained using psi-chi method when SLBW is adopted.
• This approximation method cannot correctly calculate XS at peak andbottom of resonance.
• This approximation has no large impact on probability table generation.
∞
Radiative capture XS
ζ2
ζ
1− 4 𝑥−𝑦 2ψ 𝜁, 𝑥 =න𝑒𝑑𝑦
𝜎1 𝚪𝛾𝑟22 𝜋 −∞ 1 + 𝑦
σ𝑐 𝐸, 𝑇 ≅
𝛙 𝜻, 𝒙
Γ𝑟
∞
𝜁2
𝜁
𝑦
−
𝑥−𝑦 2
Elastic scattering XS𝜒 𝜁, 𝑥 =න𝑒 4𝑑𝑦2
2 𝜋 −∞ 1 + 𝑦
𝜎1 𝚪𝑛𝑟 𝐸1
σ𝑠𝑐 𝐸, 𝑇 ≅𝛙 𝜻, 𝒙Γ𝑟+2𝜎1 𝑘1 𝑎𝑐 𝛘 𝜻, 𝒙 + σ𝑝

69

ACE file generation
0.15

Probability [-]Monte Carlo calculation
codes use cumulative
probability distribution(CDF).

From linear-linear to PDF/CDF
0.10

0.5
0.05

0.00

0.0
0

2

4

6

8 10 12 14 16

Probability bins [-]• Angular and energy

From histogram to PDF/CDF
Probability [-]

0.15

1.0

PDF
CDF

0.10

0.5
0.05
0.00

0.0
0

2

4

6

8 10 12 14 16

Probability bins [-]Probability [-]distributions areconverted to cumulative
probability distribution.
• PDF: Probability Density
Function• CDF : Cumulative
Density Function

1.0

PDF
CDF

Probability [-]• Continuous energy[Example of PDF and CDF]

70

Multi-group XS calculation• Multi-group XS 𝜎𝑖𝑙,𝑔

is obtained using continuous XS

𝜎 𝑖 𝐸 and weighting flux 𝜙𝑙 𝐸 .
𝐸𝑔−1 𝑖𝜎 𝐸 𝜙𝑙 𝐸 𝑑𝐸
𝑔

‫𝐸׬‬

• 𝜎

𝑖

=
𝑙,𝑔

𝐸𝑔−1 𝑖𝜎 𝐸 𝜙𝑙 𝐸 𝑑𝐸
‫𝐸׬‬
𝑔
𝐸𝑔−1
𝜙𝑙 𝐸 𝑑𝐸‫𝐸׬‬𝑔, 𝜎 𝑖 𝑙,𝑔→𝑔′ =𝐸 ′𝑔 −1 𝜋×‫𝐸׬‬‫׬‬0 𝑓 𝐸→𝐸 ′ ,𝜇 𝑃𝑙 𝜇 𝑑𝜇𝑑𝐸 ′𝑔′
𝐸𝑔−1
𝜙𝑙 𝐸 𝑑𝐸‫𝐸׬‬𝑔• NJOY uses the following weighting flux 𝜙𝑙 𝐸 to consider self-shielding effect.
• 𝜙𝑙𝑖 𝐸 =𝐶 𝐸

𝜎t𝑖 𝐸 +𝜎0𝑖

𝑙+1

• 𝐶 𝐸 is set by user input, 𝜎t𝑖 𝐸 is total XS, 𝜎0𝑖 is background XS.

71Multi-group XS file generation using NJOY
Evaluated nuclear
data file

Processing of thermal
scattering law data
(Free gas model)
THERMR

Resonance reconstruction(Linearization)
RECONR

Multi-group XS file generation
GROUPR

Doppler broadening
BROADR

Multi-group nuclear
data file GENDF(Groupwise-ENDF)

Gas production XS calculation
GASPR
Probability table generation
PURR

Format converter for each code
MATXSR/WIMSR/CCCCR

Multi-group XS file72

Consideration of TSL data
• Chemical state of material has large impact onthermal neutron scattering property.
• Crystalline structure and chemical bounding of nuclide• H2O, Polyethylene (CH2), Graphite, Be, ZrH, …

• Evaluated nuclear data file gives Thermal Scattering

Law data to consider this thermal scattering.
• Thermal scattering law data S(α, β) is given as a function of

α and β.
• TSL data is also called as “S(α, β)”.
• 𝛼 = 𝐸 ′ + 𝐸 − 2𝜇 𝐸𝐸′ ൗ𝐴0 𝑘𝐵 𝑇: dimensionless momentum transfer• 𝛽 = 𝐸 ′ − 𝐸 Τ𝑘𝐵 𝑇: dimensionless energy transfer

73

Impact of TSL data on k-effective• Impact of TSL data on k-effective is evaluated.
• Calculation geometry : LCT-006 (JAEA/TCA) from ICSBEP
benchmark• Impart of TSL data on k-effective is 0.5-1.3%Δk• TSL data is very important for thermal reactor.
• Users have to use TSL data if TSL data is prepared in XS library.
[Impact of TSL data on k-effective]
1.015[LCT-006 (TCA)]
VerticalHorizontal

熱中性子散乱則考慮
With TSL data

HWL

熱中性子散乱則非考慮
Without TSL data

C/E value

1.010

1.005

LWL

1.000

Measuring criticalwater level withdifferent pin-pitch
and number of fuels

0.995
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Case No. of LCT-006

74

Thermal scattering• ENDF-6 format handles three types of thermal

scattering in MF=7.
• MF=7/MT=2• Coherent elastic scattering for crystalline materials• Incoherent elastic scattering for partially ordered materials• MF=7/MT＝4• Incoherent inelastic scattering for non-crystalline materials• Represented by S(α, β)75

Coherent elastic scattering• Observed in crystalline materials such as graphite• Bragg diffraction• Evaluated nuclear data file contains number of

Bragg edges and its energy.
• Nuclear data processing systems calculate coherent

elastic scattering XS.
• σ𝑐𝑜ℎ 𝐸, 𝐸 ′ , 𝜇 =1 𝐸𝑖 <𝐸
σ𝑖
𝜎𝑐 𝑓𝑖 e−2𝑊𝐸𝑖 𝛿 𝜇 − 𝜇𝑖 𝛿 𝐸 − 𝐸 ′𝐸[Example of Bragg
diffraction]θθ

Cross section [b]

10.0

1.0

0.1

[Coherent elastic scattering XSof graphite from JENDL-4.0]
1.0×10-4

1.0×10-2

1.0×100

Incident neutron energy [eV]76

Incoherent inelastic scattering• Observed in non-crystalline materials such as

H2O, ZrH, and BeO• Chemical binding effect of a molecule• Non-crystalline materials cannot move like a free atom.

• Thermal scattering law data is used to consider

incoherent inelastic scattering.
[Comparison of free atoms and H in H2O][Free atoms]

H
HH[H in H2O]

H
H

H

H

O

H

O

H

H

O
H

O

H

H

O
H

H

77

Calculation of incoherent inelastic scattering XS• Incoherent inelastic scattering XS is calculatedusing 𝑆 𝛼, 𝛽, 𝑇
𝑑2 𝜎
•
𝑑Ω𝑑𝐸′

𝐸 → 𝐸 ′ , 𝜇, 𝑇

𝑀𝑛 𝜎𝑏𝑛σ= 𝑛4𝜋𝑘𝐵 𝑇

𝐸 ′ −𝛽𝑒 2 𝑆𝑛 𝛼, 𝛽, 𝑇𝐸

• 𝜇 is scattering cosine, 𝑀𝑛 is number of atoms• 𝑆 𝛼, 𝛽, 𝑇 is only available for given temperaturesin TSL data
• ENDF-6 format gives interpolation law LI.
• Interpolated 𝑆 𝛼, 𝛽, 𝑇 is not correct and many nuclear

processing systems, e.g., NJOY and FRENDY, only
process given temperature.
• Nuclear data processing system users have to check the given

temperature.
• Given temperature is written in comment line (MF=1/MT=451).

78

S(α, β) of free gas (free atom)• Free gas scattering law is required to consider up

scattering.
• S(α, β) of free gas: 𝑆 𝛼, 𝛽 =

1
exp
4𝜋𝛼

𝛼2 +𝛽 2−4𝛼

• Evaluated nuclear data contains 0 K data.
• Up scattering does not occur at 0 K.

• Continuous energy Monte Carlo codes treat free gas

scattering law in their codes.
• They do not need up scattering XS data in XS libraries.
• Nuclear data processing systems do not have to process freegas scattering law.

• Multi-group calculation codes need incoherent

inelastic scattering XS using free gas scattering law.
• Nuclear data processing systems have to process free gas

scattering law.