New capability of FRENDY for multigroup neutron cross section processing
Akio Yamamoto
Nagoya University
Kenichi Tada
JAEA
2023/5/2, FRENDY Ver.2.01.000
Manual: https://doi.org/10.11484/jaea-data-code-2022-009
Download: https://rpg.jaea.go.jp/main/ja/program_frendy/

1Outline
Background◼ Outline of FRENDY V1 (ACE file generation)◼ Multi-group generation capability of FRENDY
◼

◼
◼
◼

OutlineTheoryVerifications

2

Background
NJOY has been widely used to generate multi-groupcross sections for deterministic core analysis codes◼ Independent efforts to develop nuclear dataprocessing codes have been carried out◼ A nuclear data processing code, FRENDY, is being
developed in JAEA (Japan Atomic Energy Agency)◼ Generation capability of ACE pointwise cross section
has been implemented and released (FRENDY V1)◼ Generation capability of multi-group cross section is
implemented and released as FRENDY V2◼

3

Outline of FRENDY V1 nuclear
data processing code4

5

Present status of nuclear data processing
code developments• Development of nuclear data processing code isstarted in many institute• To process their own nuclear data library• To handle new nuclear data format GNDS
【Nuclear data processing codes development in the world】
CALENDF(CEA)GRUCON(Kurchatov)

Ruler(CIAE)FUDGE(LLNL)AMPX(ORNL)PREPRO (IAEA)GALILEE (CEA),GAIA (IRSN)NJOY(LANL)

NECP-Atlas(Xi’an Jiaotong Uni.)

FRENDY(JAEA)

Existing code
New code

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

6Development of nuclear data processing
code FRENDY• JAEA started developing a new nuclear dataprocessing code FRENDY in 2013
• FRom Evaluated Nuclear Data librarY to any application• To process the nuclear data library with simple input file

• The first goal is processing the nuclear data forcontinuous energy Monte Carlo codes• For MVP, PHITS of JAEA and MCNP of LANL
k-eff , flux, …

Nucleardata library

Nuclear dataprocessing
code
FRENDY

Nuclearapplication code
MVP、MARBLE2、
PHITS

7

Features of FRENDY• Utilization of modern programming techniques• C++, BoostTest library, Git• Improvement of quality and reliability• Consideration of maintainability, modularity, portability

and flexibility• Encapsulate all classes• Minimize the function• Maintain the independence of each module• Processing methods of FRENDY is similar to NJOY99• Incorporating requests from users of nuclear dataprocessing codes• Development of FRENDY is supported by many organizations

and companies in Japan
Ref. K. Tada, et. al., “Development and verification of a new nuclear data processing system
FRENDY,” J. Nucl. Sci. Technol., 54 [7], pp.806-817 (2017).
(http://www.tandfonline.com/doi/abs/10.1080/00223131.2017.1309306)

Oct. 24, 2019, CIAE

Release of FRENDY ver. 1• FRENDY Ver.1 was released from JAEA

website• https://rpg.jaea.go.jp/main/en/program_frendy/• FRENDY Ver.1 generates ACE files• FRENDY Ver.1 is open source software• 2-Clause BSD license• Manual of FRENDY Ver. 1 is published fromJAEA
• JAEA-Data/Code 2018-014• https://jopss.jaea.go.jp/pdfdata/JAEA-Data-Code-2018-014.pdf• The input instructions and the details of processing

method are described8Outline and features of multigroup generation capability of
FRENDY V2

9

Concepts for developments of MG XS
generation capability◼

Use ACE files as input cross section rather than original ENDF
files
◼
◼

◼
◼
◼
◼

◼

◼

FRENDY V1 can generate ACE files
Processing corresponding to RECONR-BROADR-UNRESR(PURR)THERMR in NJOY has been done in ACE cross sections
Highly accurate
Versatile cross section data including delayed neutron information
Formal document for ACE format
Adopted in many continuous energy Monte-Carlo codes; familiar inreactor physics community

Focus on neutron cross sections. Processing of photon crosssection will be a future extension
Developed as a class library of c++ using namespaceconsidering integration to FRENDY.
10

Major capabilities and features◼

Weighting spectrum
◼
◼
◼

◼

1/E
Fission + 1/E + Maxwell (arbitrary cutoff, temperature)
User input

Ultra-fine group spectrum
◼

◼

◼

◼

◼

Direct slowing down calculation or narrow resonanceapproximation
Automated correction of ultra-fine energy grid using ACE
pointwise energy grid
Can treat a material including arbitrary number of isotopes;
allows explicit consideration of resonance interference effectArbitrary mass weight of background moderator nuclide for
evaluation of IR parameters
Arbitrary number of background cross sections

11

Major capabilities and features
Self-shielding for unresolved resonance region using
probability table◼ Automated background cross section settings◼ Resonance up-scattering treatment◼ Treatment of angle/energy distributions
◼

◼
◼

◼

Can treat LAW=3, 4, 7, 9, 44, 61, and 66 in an ACE fileCan treat all nuclides in B-VII.1, B-VIII.0, JEFF-3.3, TENDL2019, 2021, JENDL4.0, and JENDL5

Scattering matrices◼

◼

Can treat all reactions with secondary neutron(s) including
self-shielding and anisotropic scattering
𝑆(𝛼, 𝛽)：free gas model, short collision time approximation,
incoherent inelastic, incoherent elastic, and coherent elastic.
12

Major capabilities and features◼

Fission spectrum, 𝜈 values
◼
◼
◼

◼

PromptDelayed (each group)
Prompt + Delayed
Any number of delayed neutron groups

Arbitrary thermal cutoff◼ Arbitrary multi-group energy structure up to a few
thousands◼ Output format of multi-group cross sections
◼

◼
◼
◼

FRENDY V2 original format (readable output list)
MATXS
GENDF
13

Example of input data(UO2)
mg_neutron_mode

mg_edit_option//Process mode

MATXS

nucl_file_name(n_008-O-016.dat
n_092-U-235.dat
n_092-U-238.dat)
mg_file_name
mix_UO2//Output file namemg_structure ( xmas-nea-lanl-172 )temp
legendre_ordersigma_zero_data

600.0
3(auto)

mg_number_density//8016.50c(4.0e-2

92235.50c
1.0e-392238.50c
2.0e-2)14

Example of input data (H in H2O)
mg_neutron_mode//Processing mode

mg_edit_option(MATXS

GENDF)

nucl_file_namenucl_file_name_tsl

H001.dat
01_h_in_h2o.txt

mg_file_namelwtr_from_endf

legendre_order
mg_label_data
temp
sigma_zero_data

3"HinH2O from JENDL"
300.0(1.0E+10 1000.0 1.0)

mg_number_density
mg_tsl_data_type

1.0
hh2o

mg_flux_calc_mode

NR
15

Theories for multi-group crosssection generation16

Calculation of multi-groupconstants

17

1D cross sections◼

1D cross sections are collapsed by:
𝜎𝑔,𝑥,𝑙 =‫𝐸𝑑 𝐸 𝑙𝜙 𝐸 𝑥𝜎 𝐸∈𝐸׬‬𝑔‫𝐸𝑑 𝐸 𝑙𝜙 𝐸∈𝐸׬‬𝑔where g, x, l, 𝜎𝑥 𝐸 , 𝜙𝑙 𝐸 , 𝐸𝑔 , and 𝐸 are energy group index,
reaction type, Legendre order of angular flux moment, pointwisecross section of reaction x, l-th order angular flux moment, energy
range for energy group g, and incident energy, respectively.

18

2D cross sections◼

2D cross sections are collapsed by:
𝜎𝑔,𝑔′ ,𝑥,𝑙 =‫ 𝐸׬ 𝐸∈𝐸׬‬′ ∈𝐸 ′ 𝜎𝑥,𝑙 𝐸, 𝐸 ′ 𝜙𝑙 𝐸 𝑑𝐸 ′ 𝑑𝐸𝑔𝑔

‫𝐸𝑑 𝐸 𝑙𝜙 𝐸∈𝐸׬‬𝑔where g, g’, x, l, 𝜎𝑥 𝐸 , 𝜙𝑙 𝐸 , 𝐸𝑔 , 𝐸𝑔′ , and E are incident energy group index,
outgoing energy group index, reaction type, Legendre order of angular flux
moment, l-th order angular flux moment, energy range for energy group g,
energy range for energy group g’, and incident energy, respectively.
◼

Legendre moments
are calculated by:
1
𝜎𝑥,𝑙 𝐸, 𝐸 ′ =‫׬‬−1 𝜎𝑥 𝐸, 𝐸 ′ , 𝜇 𝑃𝑙 𝜇 𝑑𝜇1

‫׬‬−1 𝑃𝑙 𝜇 𝑑𝜇where 𝜎𝑥 𝐸, 𝐸 ′ , 𝜇 , 𝜇, and 𝑃𝑙 𝜇 are the double differential cross section ofincident energy E, outgoing energy E’, scattering angle 𝜇, scattering angle(direction cosine), l-th Legendre polynomial.

19

Treatment of MTs◼

◼

All types of cross sections for neutrons in an ACE file areautomatically considered in FRENDY V2 and their multi-groupcross sections are generated
Some of (missing) MTs are automatically generated:
Automatically generated mt
number if not exist in an ACE file
4 (inelastic)
18 (fission)
103 (n,p)104 (n,d)105 (n,t)
106 (n, 3He)
107 (n, a)

Summed mt numbers

51-91
19, 20, 21, 38
600-649
650-699700-749
750-799
800-84920

Ultra-fine group spectrum calculation◼

Two options can be used for ultra-fine group
spectrum to collapse pointwise cross sections◼
◼

◼

Narrow resonance (NR) approximation
Ultra-fine group slowing down calculation

Spectrum by the NR approximation
𝑊 𝐸
𝜙𝑙 𝐸 =σ𝑡 𝐸 + σ𝑏 𝑙+1
𝑊 𝐸
𝜙𝑙 𝐸 =
𝑙+1
σ𝑘 𝜎𝑡,𝑘 𝐸 𝑁𝑘 + σ𝑏 σ𝑘 𝑁𝑘where l, 𝜙𝑙 𝐸 , 𝑊 𝐸 , σ𝑡 𝐸 , 𝜎𝑡,𝑘 𝐸 , σ𝑏 , and 𝑁𝑘 , are order of Legendre moment, neutron
spectrum (Pl flux), weight, total microscopic cross section (of k-th nuclide), microscopic
background cross section, and, number density of k-th nuclide, respectively.
21

Ultra-fine group spectrum calculation◼

Weight can be chosen from
◼
◼
◼

◼

1/E
Fission + 1/E + Maxwell
User input

1/E
𝑊 𝐸 =

◼

1
𝐸

Fission + 1/E + Maxwell𝐸−𝑇𝑓

𝑊 𝐸 = 𝐸e
𝐸 ≥ 𝐸𝑓
1
𝑊 𝐸 =𝐸𝑓 > 𝐸 > 𝐸𝑡
𝐸𝐸
−𝑇 𝐵
𝑊 𝐸 = 𝐸𝑒 𝑚 𝐸𝑡 ≥ 𝐸
where 𝐸, 𝑇𝑓 , 𝑇𝑚 , 𝐵, 𝐸𝑓 , and 𝐸𝑡 are incident energy [eV], fission temperature [eV], Maxwell
temperature [K], Boltzmann constant [eV/K], energy boundary between fission and 1/E spectra,
and energy boundary between 1/E and thermal Maxwell spectra, respectively.
22

Ultra-fine group spectrum calculation◼

◼
◼

The NR spectrum may not be accurate for wide resonances ofheavy nuclides (e.g., 6.8eV large resonance of U238)
Slowing down calculation provides more accurate results
Slowing down equation
𝛴𝑡 𝐸 𝜙 𝐸 = 𝑆

𝑓𝑖𝑠𝑠𝑖𝑜𝑛𝐸 +න
𝐸 ′ >𝐸
𝑓𝑖𝑠𝑠𝑖𝑜𝑛෍𝑘

𝑁𝑘 𝜎𝑠,𝑘 𝐸′

𝜙 𝐸′

𝑑𝐸 ′
1 − 𝛼𝑘 𝐸 ′where E, 𝛴𝑡 𝐸 , 𝜙 𝐸 , 𝑆
𝐸 , 𝑁𝑘 , 𝜎𝑠,𝑘 𝐸 , and 𝛼𝑘 are energy at the LAB
system, total cross section, scalar flux, fission source, number density of k-th
nuclide, elastic scattering cross section of k-th nuclide, and 𝛼𝑘 =𝐴𝑘 −1 2
(𝐴𝑘 is𝐴𝑘 +1 2the relative mass of k-th nuclide to neutron).
◼

Assumptions◼
◼

◼

Inelastic scattering is neglected
A fictitious background (moderator) nuclide having a constant scatteringcross section (σb)
Zero absorption cross section for moderator fictitious moderator nuclide23

Ultra-fine group spectrum calculation◼

In numerical calculation:𝑔−1
𝑓𝑖𝑠𝑠𝑖𝑜𝑛𝛴𝑡,𝑔 𝜙𝑔 = 𝑆𝑔+෍
෍ 𝑁𝑘 𝜎𝑆,𝑘,𝑔′
′
𝑔 =1
𝑘

𝜙𝑔′

𝑑𝐸𝑔′
1 − 𝛼𝑘 𝐸𝑔′

where g and 𝑑𝐸𝑔′ are the index of ultra-fine energy group andenergy width of g’-th ultra-fine energy group.
◼

The above equation is numerically solved by therecurrent formula proposed by Kier.
◼

Details are described in “Handbook of Nuclear Engineering”
Chap.9, Lattice Physics Computation.

24Ultra-fine group spectrum calculation◼

The NR spectrum and slowing down spectrum are combined
even if slowing down calculations are carried out since slowing
down calculation cannot accurately consider:◼
◼

Fast spectrum shape due to fission spectrum
Thermal Maxwell spectrum shape◼

The NR spectrum is used for 𝐸 > 𝐸ℎ𝑖𝑔ℎ , 𝐸 < 𝐸𝑙𝑜𝑤◼

Slowing down spectrum is used for 𝐸ℎ𝑖𝑔ℎ ≥ 𝐸 ≥ 𝐸𝑙𝑜𝑤◼

◼

𝐸ℎ𝑖𝑔ℎ and 𝐸𝑙𝑜𝑤 are user specified parameters, e.g., 𝐸ℎ𝑖𝑔ℎ =104 eV, 𝐸𝑙𝑜𝑤 = 1 eV.
For multi-group P0/P1 flux, the NR spectrum is used for wholeenergy range to avoid discontinuity between NR/slowing down
spectra25

Ultra-fine group spectrum calculation◼

◼

Slowing down calculation can be carried out considering
multiple nuclides
Total cross section is calculated by:
Σ𝑡 𝐸 = ෍ 𝜎𝑡,𝑘 𝐸 𝑁𝑘 + σ𝑏 ෍ 𝑁𝑘 .
𝑘◼

𝑘Note that the background cross section is multiplied by σ𝑘 𝑁𝑘 inthis calculation. Namely, a mixture is considered as a “fictitious
resonance nuclide”.

26Ultra-fine group spectrum calculation◼

Choice of background cross section and its mass◼

◼

◼

Infinite dilute background cross section is always necessary (1010
barn)
Smaller background cross sections representing shielded conditions
should be appropriately provided by input. Number and point ofbackground cross sections depend on the behavior andinterpolation scheme of self-shielding factor.

Mass of the background nuclide can be specified
◼

◼
◼
◼

Mass is defined by the relative mass to the neutron
Default: hydrogen, 0.999167
Can consider heavier nuclide such as oxygen.
Can be used for intermediate resonance (IR) parameter estimation27

Ultra-fine group spectrum calculation◼

Example of IR estimation procedure (O16 & U238)
1) Generate effective cross section of U238 using slowing down calculation.
The background cross section of 𝜎𝑏0 and default mass (𝑎0 =0.999167) forthe background nuclide are used. The generated microscopic cross section
is 𝜎0 .
2) Generate effective cross section of U238 using slowing down calculation.
The background cross section of 𝜎𝑏0 and mass of oxygen (𝑎1 =15.8575)
for the background nuclide are used. The generated microscopic crosssection is 𝜎1 .
3) Generate effective cross section of U238 using slowing down calculation.
The background cross section of 𝜎𝑏1 (e.g., 𝜎𝑏1 = 1.05 × 𝜎𝑏0) and default
mass (𝑎0 =0.999167) for the background nuclide are used. The generated
microscopic cross section is 𝜎2 .
4) The IR parameter is calculated by:
𝜎1 − 𝜎0𝑓𝐼𝑅 = 1 + 𝜎 − 𝜎20
𝜎𝑏1 − 𝜎𝑏0
𝜎𝑏028

Automated background XS settings
Choice of background cross section in slowing down
calculation requires experience and knowledge◼ An automated sequence to choose appropriate
background XS is implemented in FRENDY V2.
◼

◼

◼
◼

Minimize interpolation error and number of backgroundcross sections
Adaptively add/eliminate background cross sections
Interpolation error is measured by self-shielding factor orreaction rate29

Automated background XS settings
Steps (1) - (4)Steps (5), (6), (7)Steps (5), (6), (7)Steps (8) - (12)Steps (13) - (16)

Steps (13) - (16)

Steps (13) - (16)•https://doi.org/10.1080/00223131.2021.1944930

30

Automated background XS settings◼

Example of self-shielding factor (U-238, capture crosssection, XMAS 172 energy groups, >1.0eV, 300 K).
1.0
0.9

Self-shielding factor

0.8
0.7
0.6
0.5

0.4
0.3
0.2
0.1
0.0
1.0E-03

1.0E-01

1.0E+01
1.0E+03
1.0E+05
1.0E+07
Background cross section [barn]•https://doi.org/10.1080/00223131.2021.1944930

1.0E+0931

Automated background XS settings◼

Adaptively set background cross sections for all
nuclides in JENDL-4.0 or -4.0u.
300

250

Atomic mass

200

150

100

50

0
1.0E-03

1.0E-01

1.0E+01 1.0E+03 1.0E+05 1.0E+07
Background cross section [barn]•https://doi.org/10.1080/00223131.2021.1944930

1.0E+0932

Resonance up-scattering treatment◼

In the epi-thermal energy range, neutron scattering
is usually treated as the asymptotic kernel:
𝐾𝑎𝑠,𝑖 𝐸 → 𝐸 ′1
𝛼𝑖 𝐸 ≤ 𝐸 ′ ≤ 𝐸= ൞ 1 − 𝛼𝑖 𝐸
0 𝐸 ′ < 𝛼𝑖 𝐸 𝐸 ′ > 𝐸

where 𝛼𝑖 = 𝐴𝑖 − 1 2 / 𝐴𝑖 + 1 2 and 𝐴𝑖 is the relative mass of target
nuclide i to a neutron. In the asymptotic kernel, transfer probability
takes a constant value for 𝛼𝑖 𝐸 ≤ 𝐸 ′ ≤ 𝐸.

33

Resonance up-scattering treatment◼

However, especially for heavy nuclides, up-scattering
due to thermal motion of nucleus may no be
neglected in epi-thermal energy range:
𝐾𝑟𝑢,𝑖,𝑙 𝐸 → 𝐸 ′
=

1

5
𝛽𝑖2

∞

න 𝑡𝜎𝑠,𝑖
𝜎𝑠,𝑖,𝑙 𝐸 4𝐸 0

𝑘𝑇 2
𝐸
𝑡2
𝛽𝑖
𝑡 exp
−
𝛹𝑙 𝑡 𝑑𝑡
𝐴𝑖
𝑘𝑇 𝐴𝑖

where 𝜎𝑠,𝑖,𝑙 𝐸 , 𝜎𝑠,𝑖 𝐸 , 𝛽𝑖 , 𝑘, and 𝑇 are the l-th order total elastic scattering crosssection at T K, the total elastic scattering at 0 K of nuclide i at incident energy E,
𝛽𝑖 = (𝐴𝑖 + 1)/𝐴𝑖 , the Boltzmann constant, and temperature, respectively.

𝛹0 𝑡 = 𝐻 𝑡+ − 𝑡 𝐻 𝑡 − 𝑡− erf 𝑡 + 𝜀𝑚𝑖𝑛 − erf 𝜀𝑚𝑎𝑥 − 𝑡+𝐻(𝑡 − 𝑡+ ) erf 𝑡 + 𝜀𝑚𝑖𝑛 − erf 𝑡 − 𝜀𝑚𝑖𝑛where 𝐻 and erf represent the Heaviside and error functions, respectively, and 𝜀𝑚𝑖𝑛 =min(𝐸, 𝐸′)(𝐴𝑖 + 1)/𝑘𝑇, 𝜀𝑚𝑎𝑥 = max(𝐸, 𝐸′)(𝐴𝑖 + 1)/𝑘𝑇, 𝑡± = (𝜀𝑚𝑎𝑥 ± 𝜀𝑚𝑖𝑛 )/2.

34

Resonance up-scattering treatment
Incident energy

3.0

300 K
600 K
900 K
Asymptotic

Scattering kernel

2.5

2.0
1.5
1.0
0.5
0.0
35.0

35.5

36.0

36.5
Energy (eV)

37.0

37.5

38.0

35

Resonance up-scattering treatment𝑛+1

෍ 𝑁𝑖 𝜎𝑡,𝑖,𝑔 𝜙𝑔

Δ𝐸𝑔

𝑖

=

෍

0

෍ 𝐾𝑟𝑢,𝑖,0,𝑔′→𝑔 𝑁𝑖 𝜎𝑠,𝑖,𝑔′ 𝜙𝑔′ Δ𝐸𝑔′

𝑔′<𝑔250𝑒𝑉 𝑖

+

෍

𝑛+1

෍ 𝐾𝑟𝑢,𝑖,0,𝑔′→𝑔 𝑁𝑖 𝜎𝑠,𝑖,𝑔′ 𝜙𝑔′

Δ𝐸𝑔′

𝑔250𝑒𝑉 ≤𝑔′<𝑔 𝑖𝑛+ ෍ ෍ 𝐾𝑟𝑢,𝑖,0,𝑔′→𝑔 𝑁𝑖 𝜎𝑠,𝑖,𝑔′ 𝜙𝑔′ Δ𝐸𝑔′
𝑔′ >𝑔 𝑖10-5 eV

1 eV

250 eV

Energy range consideringresonance up-scattering

20 MeV36

Resonance up-scattering treatment◼

Mosteller benchmark results
計算手法
MethodAsymptotic kernel

RUSK
FTC Diff. (%)

HZP

HFP

FTC

HZP

HFP

FTC

Lee, et al. (2008)

CASMO-5

1.29990 1.28907-2.15

1.29878 1.28696

-2.36

9.4

Xu, et al., (2019)

EPRI-CPM 69G

1.29831 1.28719-2.22

1.29790 1.28569

-2.44

10.0

Ouisloumen, et al., (2015)

PARAGON1.29964 1.28952-2.01

1.29902 1.28803

-2.19

8.8

本研究 V2FRENDY

CBZ/XMAS 172G

1.28392 1.27232-2.37

1.28258 1.26995

-2.58

9.2

CBZ/SHEM 361G 1.29643 1.28614-2.06

1.29564 1.28444

-2.24

9.1

CBZ/XMAS 172G

1.28392 1.27232-2.37

1.28291 1.27037

-2.56

8.4

CBZ/SHEM 361G 1.29643 1.28614-2.06

1.29658 1.28636

-2.04

-0.7

本研究(RUSK-1D)

37

Unresolved Resonance treatment◼

◼

◼

Self-shielding factor for unresolved resonance region is
calculated using the probability table“Self-shielded” point-wise cross sections in the unresolved
resonance region are used for ultra-fine group calculation andsuccessive processing
Two options for the probability table in an ACE file
◼
◼

◼

◼

IFF=0, probability table for cross section
IFF=1, probability table for self-shielding factor

For IFF=0, effective cross section is once generated and thenconverted to self-shielding factor
Cross section/self-shielding factor is calculated by:𝑤𝑖 𝜎𝑥,𝑖𝜎𝑏 + 𝜎𝑡,𝑖
𝜎𝑥 =
𝑤𝑖
σ𝑖
𝜎𝑏 + 𝜎𝑡,𝑖where 𝜎𝑥 , 𝜎𝑥,𝑖 , 𝜎𝑏 , 𝜎𝑡,𝑖 , and 𝑤𝑖 are the effective cross section of reaction x,
the i-th band cross section of reaction x, background cross section, i-thband total cross section, and the band probability (weight).
σ𝑖38

Unresolved Resonance treatment◼

P1 weighted total cross section is calculated by:σ𝑖
𝜎𝑡𝑃1 =◼

σ𝑖

𝑤𝑖 𝜎𝑡,𝑖𝜎𝑏 + 𝜎𝑡,𝑖𝑤𝑖
𝜎𝑏 + 𝜎𝑡,𝑖22Note on infinite dilute cross sections for total cross section
◼

◼
◼

◼

◼

Two different definition of infinite dilute cross sections: point-wise total
cross section, average total cross section obtained by the resonance laddercalculation in a probability table calculation
These two values are different
In FRENDY V2, point-wise total cross section is used for the infinite-dilutetotal cross section.
In the case of IFF=1, 𝜎𝑡,𝑖 is calculated by the point-wise total cross sectionin FRENDY (MG generation) while the average total cross section obtainedin a ladder calculation is used in FRENDY (ACE generation)
The above inconsistency is the cause of difference of shielded cross section
39
between FRENDY(ACE) and FRENDY(MG)Cross sections with emitting neutrons◼

Following LAWs in an ACE file can be treated

LAW
None (Elastic
scattering, mt=2)

Angular distribution
AND block

3 (Inelastic scattering,
mt=51-90)

AND block4, 7, 9, 11, 66 (Fission,(n,2n) reactions, etc.)
44, 61 (Continuum
inelastic scattering,
mt=91)AND blockEnergy distribution
Analytically calculated
from angular
distribution
Analytically calculated
from angular
distribution andparameter from DLW
block
DLW block

DLW block

DLW block40

Cross sections with emitting neutrons(Elastic scattering)◼

◼

In the elastic scattering, only angular distribution for an incident
energy is given since outgoing energy is analytically calculated
The multi-group elastic scattering cross section is calculated by:
𝜎𝑠,𝑙,𝑔→𝑔′ =‫𝑠𝜎 𝐸 𝑙𝜙 𝑔∈𝐸׬‬,𝑙,𝐸→𝑔′ 𝑑𝐸‫𝐸𝑑 𝐸 𝑙𝜙 𝑔∈𝐸׬‬𝜎𝑠,𝑙,𝐸→𝑔′ = න

𝐸 ′ ∈𝑔′=න𝐸 ′ ∈𝑔′′𝜎𝑠 𝐸, 𝐸 ′ , 𝜇 𝐸, 𝐸 ′ 𝑃𝑙 𝜇 𝐸, 𝐸 ′ 𝑑𝐸 ′𝜎𝑠 𝐸, 𝐸 , 𝜇 𝐸, 𝐸′

𝑃𝑙 𝜇 𝐸, 𝐸′

1𝑑𝜇𝑑𝜇 𝐸, 𝐸 ′𝑑𝐸 ′where E, E’, 𝜎𝑠,𝑙,𝑔→𝑔′ , 𝜙𝑙 𝐸 , 𝜎𝑠 𝐸, 𝐸 ′ , 𝜇 , and 𝑃𝑙 𝜇 are the incident energy (LAB),
outgoing energy (LAB), multi-group scattering cross section (l-th order anisotropic), lth order angular flux moment (Pl flux), scattering cross section from E to E’, scattering
angle 𝜇, and l-th order Legendre polynomial 𝑃𝑙 𝜇 .
41Cross sections with emitting neutrons(Elastic scattering)◼

The following relations are used for numerical integration:𝜇′ 𝐶𝑀 =

𝐸′
2× 𝐸 − 1+𝛼
1−𝛼

,

1−𝐴 2
𝛼=
1+𝐴 2

−1 𝜇′ 𝐶𝑀 < −1
𝜇𝐶𝑀 = 𝜇′ 𝐶𝑀 −1 ≤ 𝜇′ 𝐶𝑀 ≤ 1 .
1 𝜇′ 𝐶𝑀 > 1

𝜇𝐿𝐴𝐵 =

1 + 𝐴𝜇𝐶𝑀

1 + 𝐴2 + 2𝐴𝜇𝐶𝑀

𝑝𝑑𝑓𝐿𝐴𝐵,𝑖 = 𝑝𝑑𝑓𝐶𝑀,𝑖 ×

𝐴2 + 2𝐴𝜇𝐿𝐴𝐵,𝑖,𝑎𝑣𝑒𝑟𝑎𝑔𝑒

3+1 2

𝐴2 𝐴 + 𝜇𝐿𝐴𝐵,𝑖,𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑛−1

𝜎𝑠,𝑙,𝐸→𝑔′ = ෍ 𝑝𝑑𝑓𝐿𝐴𝐵,𝑖 × 𝑃𝑙 𝜇𝐿𝐴𝐵,𝑖,𝑎𝑣𝑒𝑟𝑎𝑔𝑒 × 𝜇𝐿𝐴𝐵,𝑖,𝑢𝑝𝑝𝑒𝑟 − 𝜇𝐿𝐴𝐵,𝑖,𝑙𝑜𝑤𝑒𝑟𝑖=042

Cross sections with emitting neutrons(LAW=3, level scattering)◼

In the inelastic scattering(LAW=3), only angular distribution for
an incident energy is given since outgoing energy is analyticallycalculated:
2

𝐴
𝐸𝐶𝑜𝑢𝑡 =
𝐴+1

× 𝐸𝐿𝑖𝑛 −

𝐴+1
𝐴

𝑄

where
𝐸𝐶𝑜𝑢𝑡 : outgoing energy of a neutron in the CM system,
𝐸𝐿𝑖𝑛 : incident energy of a neutron in the LAB system,
𝐴: relative atomic mass of target nuclide to a neutron,
𝑄 : Q-value of inelastic scattering where 𝑄 < 0. The values of
𝐴 2
and
𝐴+1

𝐴+1
𝐴

𝐸𝐿𝑜𝑢𝑡 = 𝐸𝐶𝑜𝑢𝑡 +𝑄 are given in the DLW block of an ACE file.

𝐸𝐿𝑖𝑛 + 2𝜇𝐶 𝐴 + 1

𝐸𝐿𝑖𝑛 𝐸𝐶𝑜𝑢𝑡

𝐴+1 2

𝜇𝐶 : direction cosine in the CM system.
43

Cross sections with emitting neutrons(LAW=3)◼

Legendre coefficients are calculated by the following equation
when angular distribution is given in the laboratory system:+𝜇𝐿,𝑔′ ,𝑢𝑝𝑝𝑒𝑟

𝑝𝐿,𝑙,𝑔′ 𝐸𝐿𝑖𝑛 = න+𝜇𝐿,𝑔′ ,𝑙𝑜𝑤𝑒𝑟

𝑝𝐿 𝐸𝐿𝑖𝑛 , 𝜇𝐿 𝑃𝑙 𝜇𝐿 𝑑𝜇𝐿

where
𝑝𝐿,𝑙,𝑔′ 𝐸𝐿𝑖𝑛 : Legendre coefficient for incident energy 𝐸𝐿𝑖𝑛 andoutgoing energy for g’,
𝑝𝐿 𝐸𝐿𝑖𝑛 , 𝜇𝐿 : probability density in the LAB system.
𝜇𝐿,𝑔′ ,𝑢𝑝𝑝𝑒𝑟 : upper value of direction cosine in the LAB systemcorresponding outgoing energy group g’,
𝜇𝐿,𝑔′ ,𝑙𝑜𝑤𝑒𝑟 : lower value of direction cosine in the LAB systemcorresponding outgoing energy group g’.

44Cross sections with emitting neutrons(LAW=3)◼

Legendre coefficients are calculated by the following equation
when angular distribution is given in the center of mass system+𝜇𝐿,𝑔′ ,𝑢𝑝𝑝𝑒𝑟

𝑝𝐿,𝑙,𝑔′ 𝐸𝐿𝑖𝑛 = න+𝜇𝐿,𝑔′ ,𝑙𝑜𝑤𝑒𝑟

𝑝𝐶 𝐸𝐿𝑖𝑛 , 𝜇𝐶 𝐽 𝐸𝐿𝑖𝑛 , 𝜇𝐶 𝑃𝑙 𝜇𝐿 𝑑𝜇𝐿

where 𝑝𝐶 𝐸𝐿𝑖𝑛 , 𝜇𝐶 and 𝐽 𝐸𝐿𝑖𝑛 , 𝜇𝐶 is the probability density in
the CM system and the Jacobian, respectively.
𝜇𝐶 =

𝜇𝐿 =

𝐽 𝐸𝐿𝑖𝑛 , 𝜇𝐶 =𝐴 + 1 2 𝐸𝐿𝑜𝑢𝑡 − 𝐸𝐿𝑖𝑛 1 + 𝑅 2
2𝑅𝐸𝐿𝑖𝑛1 + 𝑅𝜇𝐶
1 + 𝑅2 + 2𝑅𝜇𝐶
𝑑𝜇𝐶
=
𝑑𝜇𝐿

,

𝑅=𝐴

1−

𝐴 + 1 −𝑄
𝐴𝐸𝐿𝑖𝑛

3
2
1 + 𝑅 + 2𝑅𝜇𝐶 2
𝑅2 𝑅 + 𝜇𝐶

45

Cross sections with emitting neutrons(LAW=44/61, Kalbach-87 etc.)◼
◼
◼

LAW=44/61 is used for mt=91 (continuum inelastic scattering)
Coupled angular and energy distribution is given
Legendre coefficients are calculated by the following equation
when angular distribution is given in the LAB system:+1

𝑝𝐿,𝑙 𝐸𝐿𝑖𝑛 , 𝐸𝐿𝑜𝑢𝑡= න 𝑝𝐿 𝐸𝐿𝑖𝑛 , 𝐸𝐿𝑜𝑢𝑡 , 𝜇𝐿 𝑃𝑙 𝜇𝐿 𝑑𝜇𝐿 ,
−1

where
𝑝𝐿,𝑙 𝐸𝐿𝑖𝑛 , 𝐸𝐿𝑜𝑢𝑡 : Legendre coefficient,𝑝𝐿 𝐸𝐿𝑖𝑛 , 𝐸𝐿𝑜𝑢𝑡 , 𝜇𝐿 : probability density in the LAB system,
𝐸𝐿𝑖𝑛 : incident energy in the LAB system,
𝐸𝐿𝑜𝑢𝑡 : outgoing energy in the LAB system,
𝜇𝐿 : direction cosine in the LAB system,
𝑃𝑙 𝜇𝐿 : Legendre polynomial of the l-th order.

46

Cross sections with emitting neutrons(LAW=41/61)◼

Legendre coefficients are calculated by the following equation
when angular distribution is given in the CM system:+1

𝑝𝐿,𝑙 𝐸𝐿𝑖𝑛 , 𝐸𝐿𝑜𝑢𝑡

=න

𝜇𝐿𝑚𝑖𝑛

𝑝𝐶 𝐸𝐿𝑖𝑛 , 𝐸𝐶𝑜𝑢𝑡 , 𝜇𝐶 𝐽 𝑃𝑙 𝜇𝐿 𝑑𝜇𝐿

where
𝑝𝐶 𝐸𝐿𝑖𝑛 , 𝐸𝐶𝑜𝑢𝑡 , 𝜇𝐶 : probability density distribution in the CM system,
𝐸𝐶𝑜𝑢𝑡 : outgoing energy in the CM system,
𝜇𝐶 : direction cosine in the CM system,𝑑𝐸𝐶𝑜𝑢𝑡 𝑑𝜇𝐶
𝐽 = 𝑑𝐸𝑜𝑢𝑡 𝑑𝜇 : Jacobian.
𝐿
𝐿47

Cross sections with emitting neutrons(LAW=41/61)◼

The following relations are used for numerical integration
𝐸𝐿𝑜𝑢𝑡,𝑚𝑎𝑥 = 𝐸𝐿𝑖𝑛

𝜇𝐿𝑚𝑖𝑛 =𝐽=

𝐸𝐶𝑜𝑢𝑡,𝑚𝑎𝑥
1
+
𝐴+1
𝐸𝐿𝑖𝑛

2

𝑜𝑢𝑡,𝑚𝑎𝑥
1𝐸𝐶
1 + 𝑐2 −
2𝑐
𝐸𝐿𝑜𝑢𝑡

𝐸𝐿𝑜𝑢𝑡
=
𝐸𝐶𝑜𝑢𝑡

1
1 + 𝑐 2 − 2𝑐𝜇𝐿

,

1
𝐸𝐿𝑖𝑛
1 𝜐𝐿𝑖𝑛𝜐𝐿𝑖𝑛 1
1
𝜐𝑐𝑚
𝑐=
=
=
=
𝜐
=
,
𝑐𝑚 𝑜𝑢𝑡𝐴 + 1 𝐸𝐿𝑜𝑢𝑡 𝐴 + 1 𝜐𝐿𝑜𝑢𝑡 𝐴 + 1 𝜐𝐿𝑜𝑢𝑡
𝜐𝐿
𝜐𝐿𝑜𝑢𝑡

𝜇𝐶 = 𝐽 𝜇𝐿 − 𝑐 ,48

Cross sections with emitting neutrons(other reactions)◼

LAW=4 (continuous tabular)
◼

◼

◼

◼

The outgoing energy distribution is given in a tabulated form for each
incident energy grid.
The table is interpolated based on the option specified by INTT
(1/2=histogram/linear-linear).
The incident and outgoing energies are specified in the laboratory system.

LAW=7 (simple Maxwellian fission spectrum)◼

The incident and outgoing energies are specified in the laboratory system.

𝑓 𝐸 → 𝐸′ =

𝐸′
𝐸′
exp −
, 0 ≤ 𝐸′ ≤ 𝐸 − 𝑈
𝐼
𝜃(𝐸 )

where
𝐸−𝑈
𝐼: normalization factor to make ‫׬‬0𝑓 𝐸 → 𝐸 ′ 𝑑𝐸′ = 1,
𝜃(𝐸): effective temperature as a function of incident energy,
𝑈: a constant to define the upper limit of outgoing energy.

49

Cross sections with emitting neutrons(other reactions)◼

LAW=9 (evaporation spectrum)◼

The incident and outgoing energies are specified in the laboratory system.

𝐸′𝐸′
𝑓 𝐸 → 𝐸 ′ = exp −, 0 ≤ 𝐸′ ≤ 𝐸 − 𝑈
𝐼
𝜃(𝐸)
where
𝐸−𝑈
𝐼: normalization factor so that ‫׬‬0𝑓 𝐸 → 𝐸 ′ 𝑑𝐸′ = 1,
𝜃(𝐸): effective temperature as a function of incident energy,
𝑈: a constant to define the upper limit of outgoing energy.
◼

LAW=11 (energy dependent Watt spectrum)◼

The incident and outgoing energies are specified in the laboratory system.

𝑓 𝐸 → 𝐸′ =𝐸′
exp − 𝑎𝐼sinh𝑏𝐸′ , 0 ≤ 𝐸 ′ ≤ 𝐸 − 𝑈

where
𝐸−𝑈
𝐼: normalization factor so that ‫׬‬0𝑓 𝐸 → 𝐸 ′ 𝑑𝐸′ = 1,
𝑎, 𝑏: parameters as a function of incident energy,
𝑈: a constant to define the upper limit of outgoing energy.

50Cross sections with emitting neutrons(other reactions)◼

Definition of ENDF for LAW=66 (n-body phase space
distribution), e.g., D2O evaluation of B7.1
𝑓 𝐸𝐶 → 𝐸𝐶′ , 𝜇𝐶 = 𝐶𝑛 𝐸 ′ 𝐶 𝐸𝐶𝑚𝑎𝑥 − 𝐸𝐶′ ,4𝐶3 =,𝜋 𝐸𝐶𝑚𝑎𝑥 2
105
𝐶4 =
7,
𝑚𝑎𝑥 2
32 𝐸𝐶
256
𝐶5 =
,
14𝜋 𝐸𝐶𝑚𝑎𝑥 5
where
𝑛: number of bodies in phase space, defined in an ACE file,
𝐸𝐶𝑚𝑎𝑥 : maximum possible outgoing energy in the CM system,
𝐸𝐶 : incident energy in the CM system,
𝐸𝐶′ : outgoing energy in the CM system,
𝜇𝐶 : scattering angle (cosine direction) in the CM system.
51

Cross sections with emitting neutrons(other reactions)◼

In FRENDY V2, following relation is used for LAW=66:

𝑓 𝐸𝐿 → 𝐸𝐿′ , 𝜇𝐿 = 𝐶𝑛 𝐸𝐿′ 𝐸𝐶𝑚𝑎𝑥 − 𝐸𝐿∗ + 𝐸𝐿′ − 𝜇𝐿 𝐸𝐿∗ 𝐸𝐿′
𝐸𝐿∗ =

3𝑛
−4
2

,

𝐸𝐿
,
𝐴+1 2

where
𝐸𝐿 : incident energy in the LAB system,𝐸𝐿′ : outgoing energy in the LAB system,
𝜇𝐿 : scattering angle (cosine direction) in the LAB system.
𝐸𝑔′ 𝑢

𝑓𝑙 𝐸 → 𝑔′ = න

𝐸𝑔′ 𝑙

1

න 𝑓 𝐸𝐿 → 𝐸𝐿′ , 𝜇𝐿 𝑃𝑙 (𝜇𝐿 )𝑑𝜇𝐿 𝑑𝐸𝐿′
−1

where
𝐸𝑔′ 𝑢 : upper energy boundary of g’-th energy group,𝐸𝑔′ 𝑙 : lower energy boundary of g’-th energy group.
52

Thermal cross sections

53

Thermal cross sections◼

Three different reactions are considered◼

Incoherent inelastic
Free gas model◼ Short collision time (SCT) approximation◼ 𝑆 𝛼, 𝛽data in an ACE file◼

◼

Incoherent elastic◼

◼

𝑆 𝛼, 𝛽 data in an ACE file

Coherent elastic◼

𝑆 𝛼, 𝛽 data in an ACE file54

Free gas model◼

The thermal incoherent inelastic scattering for free gas(unbounded atom) is given by

2
′
𝜎
𝐸
1
𝛼
+
𝛽
𝑏
𝜎 𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎 𝐸, 𝐸 ′ , 𝜇 =exp −,2𝑘𝑇 𝐸 4𝜋𝛼4𝛼𝐸 ′ + 𝐸 − 2𝜇 𝐸𝐸 ′
𝛼=
,
𝐴𝑘𝑇
𝛽=

𝐸′ − 𝐸
𝑘𝑇

,

where
𝐴+1 2
𝜎𝑏 = 𝜎𝑓𝑟𝑒𝑒 𝐴2: the incoherent inelastic
scattering cross section for the bounded state,
𝜎𝑓𝑟𝑒𝑒 : elastic scattering cross section for
unbounded (free) state obtained by pointwise
elastic scattering cross section,
𝐴: relative mass of a nucleus to a neutron,𝑘: Boltzmann constant [eV/K],
𝑇: Temperature [K],𝐸: incident energy in the LAB system,
𝐸 ′ : outgoing energy in the LAB system,
𝜇: scattering cosine angle in the LAB system.

55

Free gas model◼

The Legendre moments can be calculated by𝜎𝑙𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎 𝐸, 𝐸 ′1= න 𝜎 𝑖𝑛𝑐 𝐸, 𝐸 ′ , 𝜇 𝑃𝑙 𝜇 𝑑𝜇 ,
−1

where
𝑃𝑙 𝜇 : Legendre polynomial of l-th order.
◼

◼

The above numerical integration is difficult since 𝜎 𝑖𝑛𝑐 𝐸, 𝐸 ′ , 𝜇shows steep variation on 𝜇 since 𝛼 can be zero when 𝐸 ≈ 𝐸′.
NJOY uses adaptive numerical integration (discretization ischanged on-the-fly based on the behavior of 𝜎 𝑖𝑛𝑐 𝐸, 𝐸 ′ , 𝜇 )56Free gas model◼

FRENDY/MG utilize the following relation to avoid numericaldifficulty:
𝜎𝑙𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎 𝐸, 𝐸 ′

𝑎+𝑏

2𝑐
1
𝛽
=න
exp − 𝑢 +
4
𝑢
𝑎−𝑏 𝑏

2

𝑎 − 𝑢2𝑃𝑙
𝑑𝑢 ,𝑏𝐸′ + 𝐸
𝑎=
,
𝐴𝑘𝑇
𝐸𝐸 ′
𝑏=
,
𝐴𝑘𝑇
𝜎𝑏 1
𝐸′
𝑐=
,
4𝑘𝑇 4𝜋 𝐸
𝑢2 = 𝛼◼

Adaptive numerical integration with the Simpson formula is used57

Free gas model◼

Comparison of integrand𝜎𝑏 𝐸 ′ 1
𝛼+𝛽 2
𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎′𝜎𝐸, 𝐸 , 𝜇 =exp −,2𝑘𝑇 𝐸 4𝜋𝛼
4𝛼

NJOY

2𝑐
1
𝛽
exp − 𝑢 +𝑏4
𝑢2

FRENDY V2

E=0.01 [eV], E’=0.00999 [eV], A=1, T=300 [K], 𝜎𝑏 =1.0

58

Free gas model◼

◼

Numerical integration on 𝜇 still gives some discretization error
even if adaptive numerical integration is used
To reduce the numerical integration error, the analytic solution isused. The analytic solution for P0 component is:𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎
𝜎0,𝑎𝑛𝑎𝐸, 𝐸 ′

𝜎𝑓 𝜂 2
𝐸
𝐸′
=
቎exp
−
𝐸 2
𝑘𝑇 𝑘𝑇
+ erf 𝜂

𝐸
𝐸′
𝐸
𝐸′
erf 𝜂
−𝜌
± erf 𝜂
+𝜌
𝑘𝑇
𝑘𝑇
𝑘𝑇
𝑘𝑇

𝐸′
𝐸
𝐸′
𝐸
−𝜌
∓ erf 𝜂
+𝜌
𝑘𝑇
𝑘𝑇
𝑘𝑇
𝑘𝑇

where erf is the error function, 𝜂 =

𝐴+1
,
2 𝐴

𝜌=

቏

𝐴−1
,
2 𝐴

and upper/lower signs

correspond to 𝐸 < 𝐸′ or 𝐸′ < 𝐸, respectively.

59

Free gas model◼

Multi-group scattering matrix is obtained by:
𝑢𝑝𝑝𝑒𝑟

𝐸𝑔

𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎
𝜎𝑙,𝑔→𝑔
=
′

𝑢𝑝𝑝𝑒𝑟
𝑔
𝑙𝑜𝑤𝑒𝑟
𝐸𝑔′

‫׬ 𝐸 𝑙𝜙 𝑟𝑒𝑤𝑜𝑙𝐸׬‬𝑔𝐸 ′𝜎𝑙𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎 𝐸, 𝐸 ′ 𝑑𝐸 ′ 𝑑𝐸

𝑢𝑝𝑝𝑒𝑟

𝐸𝑔

‫𝐸𝑑 𝐸 𝑙𝜙 𝑟𝑒𝑤𝑜𝑙𝐸׬‬𝑔=◼

σ𝐸𝑖 ∈𝑔 Δ𝐸𝑖 𝜙𝑙 𝐸𝑖 σ𝐸𝑗 ∈𝑔′ 𝜎𝑙𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎 𝐸𝑖 , 𝐸𝑗′ Δ𝐸′𝑗
σ𝐸𝑖 ∈𝑔 Δ𝐸𝑖 𝜙𝑙 𝐸𝑖,

Then normalized by analytic solution as:𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎𝜎𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎 0,𝑔→𝑔′ ,𝑎𝑛𝑎

𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎𝜎ො𝑙,𝑔→𝑔= 𝜎𝑙,𝑔→𝑔′′

𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎
𝜎0,𝑔→𝑔′60

Short collision time (SCT)
approximation
◼

◼

◼
◼

When the energy transfer is large in the thermal energy range,
the short time collision (SCT) approximation gives better results
In FRENDY V2, the SCT approximation can used with 𝑆 𝛼, 𝛽treatment. Namely, scattering outside the energy range of
𝑆 𝛼, 𝛽 is treated by the SCT approximation when UseSCT
option is used.
The SCT approximation is not used with the free gas model.
In the SCT approximation, cross section is given by:

𝜎 𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎 𝐸, 𝐸 ′ , 𝜇𝜎𝑏 𝐸 ′ 1
𝑇
𝛼− 𝛽
=
exp −
2𝑘𝑇 𝐸 4𝜋𝛼 𝑇𝑒𝑓𝑓
4𝛼

2

𝑇
𝑇𝑒𝑓𝑓

−

𝛽+ 𝛽
2

where 𝑇 and 𝑇𝑒𝑓𝑓 are temperature and the effective temperature in [K].

61

Short collision time (SCT)
approximation◼

The SCT approximation can be formulated as:

𝜎𝑙𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎 𝐸, 𝐸 ′
𝑎+𝑏

2𝑐
1
𝛽
=න
exp − 𝑢 +
4
𝑢
𝑎−𝑏 𝑏
◼

2

𝑇

1
𝑇
×
exp
1−
𝑇𝑒𝑓𝑓
4
𝑇𝑒𝑓𝑓

1
𝑇
exp
1−
𝑇𝑒𝑓𝑓
4
𝑇𝑒𝑓𝑓

◼

2

𝑎 − 𝑢2𝑃𝑙𝑑𝑢𝑏

The difference between the free gas model and SCT is thefollowing factor in integration
𝑇

◼

𝛽
𝑢−
𝑢

𝛽
𝑢−
𝑢

2

Identical numerical integration scheme can be used with thefree gas treatment
Normalization by the analytic solution is not applied62

𝑆 𝛼, 𝛽 data in ACE file(incoherent inelastic)◼

◼

The 𝑆 𝛼, 𝛽 data except for the free gas model and the SCT
approximation are stored in the thermal scattering block of athermal ACE file.
In the case of incoherent inelastic scattering, the following data
exist in an ACE file◼
◼

◼

Cross sections for incident energy grid points. (ITIE block)Energy/angle distribution table for each incident energy grid points (ITXE
block)

Cross section is given by:𝑁𝜇𝜎𝑙𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎 𝐸, 𝐸 ′= ෍ 𝜎 𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎 𝐸, 𝐸 ′
𝑖=1

1
𝑃 𝜇
𝑁𝜇 𝑙 𝑖where
𝜎 𝑖𝑛𝑐 𝐸, 𝐸 ′ : incoherent inelastic scattering cross section given in ITIE block,and outgoing energies are given by the LAB system,
𝑁𝜇 : number of equi-probable cosine bins,
𝜇𝑖 : direction cosine of i-th bin in the LAB system.
63

𝑆 𝛼, 𝛽 data in ACE file(incoherent inelastic)◼

To describe the energy transfer probability, one of the following
three options are used in an ACE file:
◼

◼

◼

◼

◼
◼

◼

1) Equally distributed energy grid points (ifeng=0 in ACE file, iwt=1 in
ACER)
2) Skewed distribution of energy grid points (ifeng=1 in ACE file, iwt=0 in
ACER)
3) Continuous distribution energy grid points (ifeng=2 in ACE file, iwt=2 in
ACER)

Most precise option is 3). However, this option can be used in
MCNP6.2 or later.
Option 2) is the default of FRENDY (ACE generation).
FRENDY V2 can treat all options. However, use of option 3) isrecommended from the viewpoint of accuracy.
Use of option 2) may affect the shape of thermal spectrum (aswell as MCNP).
64

𝑆 𝛼, 𝛽 data in ACE file(incoherent inelastic)◼

To reduce the energy discretization error of option 1) and 2),
the following energy dependencies are assumed in FRENDY V2
𝜎𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎𝐸′
𝐸 𝑓𝑖 exp −, 𝐸 < 𝐸′𝑘𝑇

𝜎 𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎 𝐸, 𝐸 ′ =𝜎 𝑖𝑛𝑐,𝑖𝑛𝑒𝑙𝑎 𝐸 𝑓𝑖 erf

𝑤𝑖
𝐸

𝐸′

𝐸′
, 𝐸 > 𝐸′𝑘𝑇
, 𝐸 < 𝐸′‫𝑖 𝐸׬‬,𝑢 exp − 𝑘𝑇 𝑑𝐸 ′
𝑖,𝑙
𝑤𝑖
𝑓𝑖 =
, 𝐸 > 𝐸′
𝐸𝑖,𝑢𝐸′
‫ 𝐸׬‬erf 𝑘𝑇 𝑑𝐸 ′𝑖,𝑙𝑤𝑖 :weight for i-th energy interval, which is specified by iwt,
𝐸𝑖,𝑙 : lower energy boundary of i-th outgoing energy interval,
𝐸𝑖,𝑢 : upper energy boundary of i-th outgoing energy interval.

65

𝑆 𝛼, 𝛽 data in ACE file(coherent elastic)◼

◼

Coherent elastic scattering occurs due to interference of
scattering by crystalline solids, e.g., graphite. Sharp Braggedges are observed in scattering cross sections. No energychange occurred (only self-scattering). Only flight direction ofneutron is changed.
Angular dependence of coherent elastic scattering cross section
is given by:
𝜎 𝑐𝑜ℎ,𝑒𝑙𝑎 𝐸, 𝜇=

෍𝑖∈ 𝜇𝑖 ≥−1

𝜇𝑖 = 1 − 2𝐸𝜎 𝑐𝑜ℎ,𝑒𝑙𝑎 𝐸 =𝑏𝑖𝛿 𝜇 − 𝜇𝑖
𝐸

𝐸𝑖
𝐸

෍

𝑏𝑖

𝑖∈ 𝜇𝑖 ≥−1where
𝑏𝑖 : size of step, defined later,
𝐸𝑖 : the energy of i-th Bragg edge.

66

𝑆 𝛼, 𝛽 data in ACE file(coherent elastic)◼

Coherent elastic scattering cross section is given by:
𝜎 𝑐𝑜ℎ,𝑒𝑙𝑎 𝐸

𝑃 𝑖
=
𝐸

, 𝐸𝑒𝑙𝑎 𝑖 ≤ 𝐸 < 𝐸𝑒𝑙𝑎 𝑖 + 1 ,𝜎 𝑐𝑜ℎ,𝑒𝑙𝑎 𝐸 = 0, 𝐸 < 𝐸𝑒𝑙𝑎 1 ,𝑃 𝑁𝑒𝑙𝑎𝑐𝑜ℎ,𝑒𝑙𝑎𝜎𝐸 =, 𝐸𝑒𝑙𝑎 𝑁𝑒𝑙𝑎 < 𝐸
𝐸
where
𝑃 𝑖 : constant given in an ACE file for the interval between i-th and(i+1)-th Bragg edges,
𝐸𝑒𝑙𝑎 𝑖 : energy of i-th Bragg edge,
𝑁𝑒𝑙𝑎 : total number of Bragg edges.
◼

𝜎𝑙𝑐𝑜ℎ,𝑒𝑙𝑎 𝐸

Legendre component is calculated by:
1

=න

1

𝜎 𝑐𝑜ℎ,𝑒𝑙𝑎 𝐸, 𝜇 𝑑𝜇 =−1

෍𝑖∈ 𝜇𝑖≥−1𝑏𝑖න𝑃𝑙 𝜇𝑖 𝛿 𝜇 − 𝜇𝑖 𝑑𝜇 =𝐸−1෍𝑖∈ 𝜇𝑖 ≥−1

𝑏𝑖
𝑃 𝜇
𝐸 𝑙 𝑖
67

𝑆 𝛼, 𝛽 data in ACE file(incoherent elastic)◼

◼

Scattering with no energy change is occurred in hydrogenous
solids due to zero phonon term, e.g, H in ZrH.
Cross section is given by:𝑁

1
𝜎𝑙𝑖𝑛𝑐𝑜ℎ,𝑒𝑙𝑎 𝐸 = 𝜎 𝑖𝑛𝑐𝑜ℎ,𝑒𝑙𝑎 𝐸
෍𝑃𝑙 (𝜇𝑖 )
𝑁
𝑖=1
where
𝑁: total number of discrete cosine directions,
𝜇𝑖 : the i-th discrete cosine direction given in an
ACE file.
◼

𝜎 𝑖𝑛𝑐𝑜ℎ,𝑒𝑙𝑎 𝐸 and 𝜇𝑖 is tabulated as function of incident energy
in ACE file.

68

𝑆 𝛼, 𝛽 データ (B-VIII.0)
FRENDY Description-------------------------------------------------------------------------------------al
Al
be
Be
bebeo
Be in BeO
benz
Benzene(caution: not implemented in FRENDY V2)
csic
C in SiC
c5o2h8 C5O2H8(caution: not implemented in FRENDY V2)
dd2o
D in D2O
dortho D in Ortho-D
dpara
D in Para-D
fe
Fe
graph
Graphite
poly
Polyethylene
hh2o
H in H2O
hice
H in Ice (H2O)
hortho H in Ortho-H
hpara
H in Para-H
hyh2
H in YH2
hzrh
H in ZrH
lch4
Liquid CH4 (Methane)nun
N in UNobeo
O in BeO
od2o
O in D2O
oice
O in Ice (H2O)
ouo2
O in UO2
sch4
Solid CH4 (Methane)sisic
Si in SiC
uun
U in UNuuo2
U in UO2
yyh2
Y in YH2
zrzrh
Zr in ZrH

69

Fission spectrum, Nu-value,fission cross sections

70

Fission spectrum◼

◼

In an ACE file, energy distributions for prompt and delayed
neutrons are given in the DLW block.
The fission spectrum for delayed neutron is calculated by
𝜒𝐷,𝑖 𝑔 → 𝑔′=

‫𝐷𝜈 𝑔∈𝐸׬‬,𝑖 𝐸 𝜎𝑓 𝐸 𝜙 𝐸 ‫𝐸׬‬′ ∈𝑔′ 𝑓𝐷𝑖 𝐸 → 𝐸 ′ 𝑑𝐸 ′ 𝑑𝐸‫𝐷𝜈 𝑔∈𝐸׬‬,𝑖 𝐸 𝜎𝑓 𝐸 𝜙 𝐸 𝑑𝐸
𝜈𝐷,𝑖 𝐸 = 𝜈𝐷 𝐸 𝑔𝑖 𝐸

where
𝜒𝐷,𝑖 𝑔 → 𝑔′ : delayed neutron spectrum for delayed group i, from group g to g’,𝜈𝐷 𝐸 : delayed neutron nu-value,𝜈𝐷,𝑖 𝐸 : delayed neutron nu-value for delayed group i,𝑔𝑖 (𝐸): fraction of neutron for delayed group i,
𝜎𝑓 𝐸 : fission cross section,
𝜙 𝐸 : neutron spectrum (scalar flux),
𝑓𝐷𝑖 𝐸 → 𝐸 ′ : energy distribution of delayed neutron group i, from incident
energy E to outgoing energy E’, where ‫ 𝐸 → 𝐸 𝑖𝐷𝑓 ׬‬′ 𝑑𝐸′ = 1.

71

Fission spectrum◼

Total delayed neutron spectrum is calculated by𝜒𝐷 𝑔 → 𝑔′ =◼

σ𝑖 ‫𝐷𝜈 𝑔∈𝐸׬‬,𝑖 𝐸 𝜎𝑓 𝐸 𝜙 𝐸 ‫𝐸׬‬′ ∈𝑔′ 𝑓𝑖 𝐸 → 𝐸 ′ 𝑑𝐸 ′ 𝑑𝐸
σ𝑖 ‫𝐷𝜈 𝑔∈𝐸׬‬,𝑖 𝐸 𝜎𝑓 𝐸 𝜙 𝐸 𝑑𝐸

The fission spectrum for prompt neutrons is calculated by𝜒𝑃 𝑔 → 𝑔′ =‫𝐸׬ 𝐸 𝜙 𝐸 𝑓𝜎 𝐸 𝑃𝜈 𝑔∈𝐸׬‬′ ∈𝑔′ 𝑓𝑃 𝐸 → 𝐸 ′ 𝑑𝐸 ′ 𝑑𝐸
‫𝐸𝑑 𝐸 𝜙 𝐸 𝑓𝜎 𝐸 𝑃𝜈 𝑔∈𝐸׬‬

where
𝜈𝑃 𝐸 : prompt neutron nu-value,𝑓𝑃 𝐸 → 𝐸 ′ : energy distribution of prompt neutron, from incident
energy E to outgoing energy E’, where ‫ 𝐸 → 𝐸 𝑃𝑓 ׬‬′ 𝑑𝐸′ = 1.

72

Fission spectrum◼

The fission spectrum for total (prompt+delayed) neutrons is
calculated by𝜒 𝑔 → 𝑔′=‫ 𝐸׬ 𝐸 𝜙 𝐸 𝑓𝜎 𝐸 𝑃𝜈 𝑔∈𝐸׬‬′ ∈𝑔′ 𝑓𝑃 𝐸 → 𝐸 ′ 𝑑𝐸 ′ 𝑑𝐸 + σ𝑖 ‫𝐷𝜈 𝑔∈𝐸׬‬,𝑖 𝐸 𝜎𝑓 𝐸 𝜙 𝐸 ‫ 𝐸׬‬′ ∈𝑔′ 𝑓𝑖 𝐸 → 𝐸 ′ 𝑑𝐸 ′ 𝑑𝐸
‫ 𝐸𝑑 𝐸 𝜙 𝐸 𝑓𝜎 𝐸 𝑃𝜈 𝑔∈𝐸׬‬+ σ𝑖 ‫𝐷𝜈 𝑔∈𝐸׬‬,𝑖 𝐸 𝜎𝑓 𝐸 𝜙 𝐸 𝑑𝐸73

Nu-values
◼
◼

◼

◼

The total and/or delayed nu-values are given in an ACE file.
In FRENDY MG generation, the prompt nu-values are not usedeven if they are given in the ACE file.
The prompt nu-value is evaluated by subtracting the delayed
nu-value from the total nu-value.
Any number of delayed neutron groups can be used in FRENDY
V2.

𝜈𝑇 𝑔 =

𝜈𝐷 𝑔 =‫𝐸𝑑 𝐸 𝜙 𝐸 𝑓𝜎 𝐸 𝑇𝜈 𝑔∈𝐸׬‬,‫𝐸𝑑 𝐸 𝜙 𝐸 𝑓𝜎 𝑔∈𝐸׬‬‫𝐸𝑑 𝐸 𝜙 𝐸 𝑓𝜎 𝐸 𝐷𝜈 𝑔∈𝐸׬‬‫𝐸𝑑 𝐸 𝜙 𝐸 𝑓𝜎 𝑔∈𝐸׬‬𝜈𝑃 𝑔 = 𝜈𝑇 𝑔 − 𝜈𝐷 𝑔 .

,

where
𝜈𝑇 𝑔 : total nu-value for group g,
𝜈𝐷 𝑔 : delayed nu-value for group g,
𝜈𝑃 𝑔 : prompt nu-value for group g.

74

Delayed neutron constants◼

The delayed neutron fraction of each group is calculated by:
𝑟𝐷,𝑖 =‫𝐷𝜈 𝑔∈𝐸׬‬,𝑖 𝐸 𝜎𝑓 𝐸 𝜙 𝐸 𝑑𝐸

σ𝑖 ‫𝐷𝜈 𝑔∈𝐸׬‬,𝑖 𝐸 𝜎𝑓 𝐸 𝜙 𝐸 𝑑𝐸

where

𝑟𝐷,𝑖 : delayed neutron fraction of group i,
normalized as σ𝑖 𝑟𝐷,𝑖 = 1.
◼

The delayed neutron decay constant of each group is edited
from ACE file75

Outputs/Edits76

Outputs/Edits◼

FRENDY/MG has the following outputs◼

Edit files: self-explanatory and readable
◼
◼
◼
◼

◼
◼
◼
◼
◼

◼
◼

◼
◼

◼

◼

1D multi-group cross sections
2D multi-group cross sections
Nu-values for total, prompt, and delayed neutrons
Fission spectra for total, prompt, and delayed neutrons
Multi-group scalar flux (P0 flux) used for weighting
Multi-group current (P1 flux) used for weighting
Ultra-fine group scalar flux
Ultra-fine group 1D cross sections
KRAM format multi-group cross sections
MATXS format multi-group cross sections
Debug output

Cross sections in the KRAM cross section format
MATXS format
GENDF format

Can be controlled by input
77

KRAM format◼

◼

The KRAM format was used in the KRAM code, which is a MOC
transport solver.
The cross sections generated in the KRAM format is suitable forordinary transport calculations since various cross sections areprocessed to be consistent with common transport solvers.
𝜎𝑡,𝑔 = 𝜎1,𝑔
𝜎𝑓,𝑔 = 𝜎18,𝑔
𝜈𝑔 =σ ′ 𝜎18,𝑔→𝑔′
𝑔

𝜎18,𝑔

+𝜈𝐷,𝑔

117

𝜎𝑎,𝑔 = 𝜎18,𝑔 + ෍ 𝜎𝑖,𝑔 + 1 − 𝑛5,𝑔 𝜎5,𝑔 − 𝜎16,𝑔 + 𝜎24,𝑔 + 𝜎30,𝑔 + 𝜎41,𝑔 − 2 𝜎17,𝑔 + 𝜎25,𝑔 + 𝜎42,𝑔 − 3𝜎37,𝑔
𝑖=102

91

𝜎𝑠,𝑔→𝑔′ =෍𝑖=2,5,16,17,24,
25,30,37,41,42

𝜒𝑔 =𝜎𝑖,𝑔→𝑔′ + ෍ 𝜎𝑗,𝑔→𝑔′
𝑗=51

σ𝑔′ 𝜎18,𝑔′ →𝑔 𝜙𝑔′ + 𝜒𝐷,𝑔 σ𝑔′ 𝜈𝐷,𝑔′ 𝜎18,𝑔′ 𝜙𝑔′
σ𝑔 σ𝑔′ 𝜎18,𝑔′ →𝑔 𝜙𝑔′ + σ𝑔′ 𝜈𝐷,𝑔′ 𝜎18,𝑔′ 𝜙𝑔′where 𝜎𝑖,𝑔 , 𝜎𝑖,𝑔→𝑔′ , 𝑛5,𝑔 , subscript D show 1D microscopic cross section for mt=i, 2D microscopic
cross section for mt=i from energy group g to g’, the number of released neutron for g-th energy
group for mt=5, and delayed neutron parameters, respectively.

78

KRAM format◼

Following relations are used to calculate mt=18, 103-107 when
mt=18, 103-107 do not exist but the following reactions exist in
an ACE file.
𝜎18,𝑔 =

෍𝜎𝑖,𝑔𝑖=19,20,21,38
649

𝜎103,𝑔 = ෍ 𝜎𝑖,𝑔
𝑖=600
699

𝜎104,𝑔 = ෍ 𝜎𝑖,𝑔
𝑖=650
749

𝜎105,𝑔 = ෍ 𝜎𝑖,𝑔
𝑖=700
799

𝜎106,𝑔 = ෍ 𝜎𝑖,𝑔
𝑖=750
849

𝜎107,𝑔 = ෍ 𝜎𝑖,𝑔
𝑖=800

79

MATXS format
◼

◼

◼

◼

◼

◼

The MATXS is a versatile cross section format for multi-groupcross sections.
The FRENDY/MG code can output the generated multi-groupcross sections in the MATXS format.
The output cross sections are chosen to be consistent withGROUPR/MATXSR in the NJOY code.
Part of the edited cross sections in FRENDY/MG are not used in
the cross section in the MATXS format.
For example, though self-shielded cross sections are generated
for all reactions with energy transfer in FRENDY/MG, only theelastic scattering and fission matrix cross sections are output as
the self-shielded cross sections in the MATXS format
In FRENDY/MG, full shielded cross sections (1D/2D) without
truncation can be output in the MATXS format by input data
(FullMATXS) though file size is larger.
80

MATXS format◼

1D cross sections◼

◼

All reaction types having 1D cross section is output in the MATXS
file.
In addition to the cross section, the following quantities are also
output:◼
◼
◼
◼
◼

P0 multi-group flux
P1 multi-group flux (current)Group averaged inverse velocity (mt259)
Lethargy width (mt258)
Average direction cosine (mu-bar) (mt251)

1=𝑣𝑔‫𝑔∈𝐸׬‬1
𝜙 𝐸 𝑑𝐸𝑣 𝐸,‫𝐸𝑑 𝐸 𝜙 𝑔∈𝐸׬‬
𝐸𝑔,𝑢𝑝𝑝𝑒𝑟𝛥𝑢𝑔 = 𝑙𝑛,𝐸𝑔,𝑙𝑜𝑤𝑒𝑟𝛴𝑠1,𝑔 σ𝑔′ 𝛴𝑠1,𝑔→𝑔′
𝜇ҧ ==.
𝛴𝑠0,𝑔 σ𝑔′ 𝛴𝑠0,𝑔→𝑔′

81

MATXS format◼
◼

All 1D cross sections are output for the infinite-dilute condition
The following cross sections and parameters are output forshielded conditions:
◼
◼

◼
◼
◼
◼
◼
◼

◼

◼

P0 multi-group flux
P1 multi-group flux (current)
P0 weighted total cross section
P1 weighted total cross section
Elastic scattering
Fission cross sections (mt=18, or 19, 20, 21, 38), if exist
Inelastic scattering (mt=51)
Capture cross section (mt=102)

All 2D cross sections with energy transfer are output in theinfinite-dilute condition.
For the shielded conditions, only the elastic scattering andfission matrix (nftot, mt=18) cross sections are output in the
MATXS cross section file.

82

MATXS format
Three options available for MATXS file
◼ Ordinary MATXS (option MATXS)◼

◼

◼

Simple MATXS format◼

◼

Compatible with MATXS file generated by NJOY2016
Compatible with MATXS file generated by NJOY99

Full MATXS format◼

All shielded cross sections (including anisotropic scattering)83

MATXS format
0v
matxs *frendy/mg
*
1
1d
1
2
9
1 5000902d92238.51c : generated by frendy/mg at 2023/03/02 08:58:21
3d
n
nscat
ntherm u238
172
1
1
1
1
3
0
4d
1.96403e+07 1.73325e+07 1.49182e+07 1.38403e+07 1.16183e+07
1.00000e+07 8.18731e+06 6.70320e+06 6.06531e+06 5.48812e+06 4.49329e+06
3.67879e+06 3.01194e+06 2.46597e+06 2.23130e+06 2.01896e+06 1.65299e+06
1.35335e+06 1.22456e+06 1.10803e+06 1.00259e+06 9.07180e+05 8.20850e+05
6.08101e+05 5.50232e+05 4.97871e+05 4.50492e+05 4.07622e+05 3.01974e+05
Skip1.50000e-02 1.00000e-02 6.90000e-03 5.00000e-03 3.00000e-03 1.00001e-05
5d u2382.36006e+02
6.00000e+02 1.00000e+10
1
46
33
0
6.00000e+02 5.20000e+01
182
75
6.00000e+02 1.00000e+10
2
1
1
86
6d
nwt0
nwt1
ntot0
ntot1
nelas
ninel
n2n
n3n
nftot
n4n
n01
n02
n03
n04
n05
n06
n07
n08
n09
n10
n11
n12
n13
n14
n15
n16
n17
n18
n19
n20
n21
n22
n23
n24
n25
n26
n27
ncn
ng
mt258
invel
mubar
nutot
nudel
chid
nupmt
Data follows

84

GENDF format
◼

◼

◼

◼
◼

The GENDF format is used as the output file of the GROUPR
module of NJOY.
It is also used as an interface file format among several
modules, such as DTFR, CCCCR, MATXSR, POWR, and WIMSR.
In FRENDY V2, the GENDF output in text (ASCII) format is
implemented.
The GENDF format utilizes a similar format with that of ENDF.
The major structure of the GENDF format is as follows:
◼

◼
◼

◼

General information, background cross section, energy group structure(mf=1, mt=451)
One-dimensional cross sections, nu-value, flux (mf=3)
Decay constant of delayed neutron, delayed neutron spectrum, if exist(mf=5)
Two-dimensional cross sections (mf=6)85

GENDF format
Three options available for GENDF file
◼ Ordinary GENDF (option GENDF)◼

◼

◼

Simple GENDF format◼

◼

Compatible with MATXS file generated by NJOY2016
Compatible with MATXS file generated by NJOY99

Full GENDF format◼

All shielded cross sections (including anisotropic scattering)86

GENDF format
92235.60c : frendy/mg at 2021/04/01 09:17:24
0 0 0
9.223500+4 2.330250+203-1
19228 1451
2.936059+2 0.000000+0
172
0
178
09228 1451
0.000000+0 1.00000+10 1.000000+3 1.000000+0 1.000010-5 3.000000-39228 1451
5.000000-3 6.900000-3 1.000000-2 1.500000-2 2.000000-2 2.500000-29228 1451
Skip
6.065307+6 6.703200+6 8.187308+6 1.000000+7 1.161834+7 1.384031+79228 1451
1.491825+7 1.733253+7 1.964033+7 0.000000+09228 1451
9228 0 09.223500+4 2.330250+22
301729228 3 1
2.936059+2 0.000000+02
1
12
19228 3 1
2.015037-1 2.015035-1 3.397897-2 6.815823-3 7.842878-4 2.197617-69228 3 11.235639+4 1.235639+4 7.042329+3 4.546633+3 6.523738+3 4.067116+39228 3 1
2.936059+2 0.000000+02
1
1229228 3 1
1.804654-2 1.804655-2 8.497675-3 3.290681-3 2.441188-4 1.490679-69228 3 1
1.887062+3 1.887062+3 1.880066+3 1.873107+3 1.876361+3 1.865753+39228 3 1
Data follows

0
1
2
3
4
31
32
0
1
2
3
4
5
6
7

87

Verifications

88

Comparison of microscopic
reaction rate
◼

◼

◼
◼
◼

172 group microscopic cross sections obtained by NJOY2016
and FRENDY/MG are multiplied by LWR, small fast reactor, 1/E
spectra to obtain 1-group reaction rates
All nuclides in JENDL-4 -4u, 5, ENDF-B7.1, -8.0, JEFF-3.3,
TENDL-2021
300K
1010, 1 barn
Compared reaction
◼
◼
◼
◼
◼

Total
Absorption
Fission
Production
Scattering (P0, P1)

89

JENDL4, 300K, 1010 barn
LWR

1/E

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05

FR

LWR

1/E

Difference |(MG-NJ)/NJ|

FR

Difference |(MG-NJ)/NJ|

Difference |(MG-NJ)/NJ|

1.0E+00

1.0E+00
LWR

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06

1.0E-06
0

50

100

150

200

250

Difference of production reaction rate

50

100

150

200

250

FR

Difference |(MG-NJ)/NJ|

1.0E-05
1.0E-06

Mass number

100

200

250

300

150

200

250

300

Difference of P1 reaction rate
1.0E+00
LWR

1.0E-04150

50

Mass number

1/E

1.0E-03100

1.0E-05

Difference of P0 reaction rate

1.0E-0250

1.0E-04

0

FR

1/E

LWR

Difference |(MG-NJ)/NJ|

LWR

0

1.0E-03300

1.0E+00

1.0E-01

1/E

1.0E-02

Mass number

1.0E+00FR

1.0E-01

1.0E-06
0300

Mass number

Difference |(MG-NJ)/NJ|

Difference of fission reaction rate

Difference of absorption reaction rate

Difference of total reaction rate
1.0E+00

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06FR

1/E

1.0E-01

1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06

0

50

100

150

Mass number

200

250

300

0

50

100

150

200

250

Mass number

90

300

JENDL4, 300K, 1 barn
1/E

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05

LWR

1.0E+00

FR

LWR

1/E

Difference |(MG-NJ)/NJ|

FR

Difference |(MG-NJ)/NJ|

Difference |(MG-NJ)/NJ|

1.0E+00

1.0E+01
LWR

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05

1.0E-06

1.0E-06
0

50

100

150

200

250

Difference of production reaction rate

50

100

150

200

250

FR

Difference |(MG-NJ)/NJ|

1.0E-05
1.0E-06
150

100

200

250

300

150

200

250

300

Difference of P1 reaction rate
1.0E+02

LWR

1.0E-04

Mass number

50

Mass number

1/E

1.0E-03100

1.0E-05

Difference of P0 reaction rate

1.0E-0250

1.0E-04

0

FR

1/E

Difference |(MG-NJ)/NJ|

LWR

0

1.0E-03300

1.0E+00

1.0E-01

1/E

1.0E-02

Mass number

1.0E+00FR

1.0E-01

1.0E-06
0300

Mass number

Difference |(MG-NJ)/NJ|

Difference of fission reaction rate

Difference of absorption reaction rate

Difference of total reaction rate
1.0E+00

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06

LWR

1.0E+01FR

1/E

1.0E+00
1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06

0

50

100

150

Mass number

200

250

300

0

50

100

150

200

250

Mass number

91

300

B7.1, 300K, 1010 barn
LWR

1/E

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05

FR

LWR

1/E

Difference |(MG-NJ)/NJ|

FR

Difference |(MG-NJ)/NJ|

LWR

Difference |(MG-NJ)/NJ|

1.0E+00

1.0E+00

1.0E+00

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06

1.0E-06

0

50

100

150

200

250

Difference of production reaction rate

50

100

150

200

250

FR

Difference |(MG-NJ)/NJ|

1.0E-05
1.0E-06

Mass number

100

200

250

300

150

200

250

300

Difference of P1 reaction rate
1.0E+00
LWR

1.0E-04150

50

Mass number

1/E

1.0E-03100

1.0E-05

Difference of P0 reaction rate

1.0E-0250

1.0E-04

0

FR

1/E

LWR

Difference |(MG-NJ)/NJ|

LWR

0

1.0E-03300

1.0E+00

1.0E-01

1/E

1.0E-02

Mass number

1.0E+00FR

1.0E-01

1.0E-06
0300

Mass number

Difference |(MG-NJ)/NJ|

Difference of fission reaction rate

Difference of absorption reaction rate

Difference of total reaction rate

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06FR

1/E

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06

0

50

100

150

Mass number

200

250

300

0

50

100

150

200

250

Mass number

92

300

B8.0, 300K, 1010 barn
LWR

1/E

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05

FR

LWR

1/E

Difference |(MG-NJ)/NJ|

FR

Difference |(MG-NJ)/NJ|

Difference |(MG-NJ)/NJ|

1.0E+00

1.0E+00
LWR

1.0E-01

1.0E-02
1.0E-03

1.0E-04
1.0E-05

1.0E-06

1.0E-06
0

50

100

150

200

250

Difference of production reaction rate

50

100

150

200

250

FR

Difference |(MG-NJ)/NJ|

1.0E-05
1.0E-06
150

100

200

250

300

150

200

250

300

Difference of P1 reaction rate
1.0E+00

LWR

1.0E-04

Mass number

50

Mass number

1/E

1.0E-03100

1.0E-05

Difference of P0 reaction rate

1.0E-0250

1.0E-04

0

FR

1/E

LWR

Difference |(MG-NJ)/NJ|

LWR

0

1.0E-03300

1.0E+00

1.0E-01

1/E

1.0E-02

Mass number

1.0E+00FR

1.0E-01

1.0E-06
0300

Mass number

Difference |(MG-NJ)/NJ|

Difference of fission reaction rate

Difference of absorption reaction rate

Difference of total reaction rate
1.0E+00

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06FR

1/E

1.0E-01
1.0E-02
1.0E-03

1.0E-04
1.0E-05
1.0E-06

0

50

100

150

Mass number

200

250

300

0

50

100

150

200

250

Mass number

93

300

JEFF3.3, 300K, 1010 barn
LWR

1/E

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05

FR

LWR

1/E

Difference |(MG-NJ)/NJ|

FR

Difference |(MG-NJ)/NJ|

Difference |(MG-NJ)/NJ|

1.0E+00

1.0E+00
LWR

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06

1.0E-06

0

50

100

150

200

250

Difference of production reaction rate

50

100

150

200

250

FR

Difference |(MG-NJ)/NJ|

1.0E-05
1.0E-06

Mass number

100

200

250

300

150

200

250

300

Difference of P1 reaction rate
1.0E+00
LWR

1.0E-04150

50

Mass number

1/E

1.0E-03100

1.0E-05

Difference of P0 reaction rate

1.0E-0250

1.0E-04

0

FR

1/E

LWR

Difference |(MG-NJ)/NJ|

LWR

0

1.0E-03300

1.0E+00

1.0E-01

1/E

1.0E-02

Mass number

1.0E+00FR

1.0E-01

1.0E-06
0300

Mass number

Difference |(MG-NJ)/NJ|

Difference of fission reaction rate

Difference of absorption reaction rate

Difference of total reaction rate
1.0E+00

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06FR

1/E

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06

0

50

100

150

Mass number

200

250

300

0

50

100

150

200

250

Mass number

94

300

TENDL2021, 300K, 1010 barn
LWR

1/E

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05

FR

1/E

Difference |(MG-NJ)/NJ|

FR

Difference |(MG-NJ)/NJ|

Difference |(MG-NJ)/NJ|

1.0E+00

1.0E+00
LWR

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06

1.0E-06
0

50

100

150

200

250

300

Difference of production reaction rate

50

100

150

200

250

300

Difference |(MG-NJ)/NJ|

1.0E-04
1.0E-05
1.0E-06
50

100

150

200

Mass number

100

250

300

350

150

200

FR

250

300

350

Mass number

Difference of P1 reaction rate
LWR

1.0E-03050

1.0E+02

1/E

1.0E-02

LWR

1.0E-05

Difference of P0 reaction rateFR300

1.0E-04

0

FR

1/E

Difference |(MG-NJ)/NJ|

LWR

250

1/E

1.0E-03350

1.0E+00

1.0E-01FR

1.0E-02

Mass number

1.0E+00

LWR
1.0E-01

1.0E-06
0350

Mass number

Difference |(MG-NJ)/NJ|

Difference of fission reaction rate

Difference of absorption reaction rate

Difference of total reaction rate
1.0E+00

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06

1.0E+01

1/E

1.0E+00
1.0E-01
1.0E-02

1.0E-03
1.0E-04
1.0E-05
1.0E-06

0

50

100

150

200

Mass number

250

300

350

0

50

100

150

200

Mass number

95

350

JENDL5, 300K, 1010 barn
LWR

1/E

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05

FR

LWR

1/E

Difference |(MG-NJ)/NJ|

FR

Difference |(MG-NJ)/NJ|

Difference |(MG-NJ)/NJ|

1.0E+00

1.0E+00
LWR

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06

1.0E-06
0

50

100

150

200

250

Difference of production reaction rate

50

100

150

200

250

FR

Difference |(MG-NJ)/NJ|

1.0E-05
1.0E-06
150

100

200

250

300

150

200

250

300

Difference of P1 reaction rate
1.0E+02
LWR

1.0E-04

Mass number

50

Mass number

1/E

1.0E-03100

1.0E-05

Difference of P0 reaction rate

1.0E-0250

1.0E-04

0

FR

1/E

Difference |(MG-NJ)/NJ|

LWR

0

1.0E-03300

1.0E+00

1.0E-01

1/E

1.0E-02

Mass number

1.0E+00FR

1.0E-01

1.0E-06
0300

Mass number

Difference |(MG-NJ)/NJ|

Difference of fission reaction rate

Difference of absorption reaction rate

Difference of total reaction rate
1.0E+00

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06

LWR

1.0E+01FR

1/E

1.0E+00
1.0E-01
1.0E-02

1.0E-03
1.0E-04
1.0E-05
1.0E-06

0

50

100

150

Mass number

200

250

300

0

50

100

150

200

250

Mass number

96

300

Note on processing◼

FRENDY V2◼

◼

All nuclides are processed without errors, except for Pu238
of TENDL-2021 (inconsistent nuclear data for number ofdelayed neutron groups).

NJOY2016 (latest at 2020/12) :◼

◼

All nuclides are processed without errors by increasing
dimension size (next page)
Compilation for 64-bit execution module may be necessary(using Visual C++)

97

Note on processing==================================================
acefc.f90
27c27< integer,parameter::nxcmax=600--> integer,parameter::nxcmax=500
86c86< integer,parameter::nxss=100000000--> integer,parameter::nxss=20000000
8004,8005c8004,8005< integer,parameter::nwscr=20000< integer,parameter::ndise=20000--> integer,parameter::nwscr=5000> integer,parameter::ndise=5000
8019c8019< nwords=10*ntrp--> nwords=5*ntrp==================================================
purr.f90
44c44< integer,parameter::jx=100000--> integer,parameter::jx=10000
133c133< nermax=5000--> nermax=1000
150c150< maxscr=1000000--> maxscr=20000
1094c1094< nthr=300--> nthr=140
1854c1854< dmin= 150000--> dmin= 100000==================================================

98

Modification of NJOY2016◼

◼
◼

When multiple temperatures or background cross sections are
specified, 10d block of matxs file is not correct.
To avoid this issue, treatment by 10d block is bypassed.
In groupr.f90, subroutine getff add statement with yellow
marked. (start – end)
! find group where e-dependence begins
ngn1=ngn+1
do ig=1,ngn1
if (econst.gt.egn(ig)*(1+small)) jconst=ig-1
enddo
econst=egn(jconst+1)
if (jconst.le.2) econst=0!do not use constant spectrum starteconst = 0
jconst = 0!do not use constant spectrum end
return99

Thermal spectrum in homogeneous
medium (H-free gas)◼

◼

MCNP6.2, FRENGY/ACE/MG-GENESIS(MG), NJOY(99.364)GENESIS(NJ)JENDL4.0, 600K
H free 600K
4.5E-01

4.0E-01

Specrum

3.5E-01
3.0E-01
2.5E-01
2.0E-01
1.5E-01
1.0E-01
5.0E-02

0.0E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
-2.0%
-4.0%
-6.0%
-8.0%
-10.0%
1.E+02

Energy (eV)
MCNP

MG

NJ

(MG-MC)/MC(NJ-MC)/MC100

Thermal spectrum in homogeneous
medium (H-H2O)◼

◼

MCNP6.2, FRENGY/ACE/MG-GENESIS(MG), NJOY(99.364)GENESIS(NJ)JENDL4.0, 600K
H H2O 600K
4.5E-01
4.0E-01

3.5E-01
Specrum

3.0E-01
2.5E-01
2.0E-01
1.5E-01
1.0E-01
5.0E-02
0.0E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
-2.0%
-4.0%
-6.0%
-8.0%
-10.0%
1.E+02

Energy (eV)
MCNP

MG

NJ

(MG-MC)/MC(NJ-MC)/MC101

Thermal spectrum in homogeneous
medium (H-ZrH)◼

MCNP6.2, FRENGY/ACE/MG-GENESIS(MG), NJOY(99.364)GENESIS(NJ)JENDL4.0, 600K
H ZrH 600K

Specrum◼

5.0E-01
4.5E-01
4.0E-01
3.5E-01
3.0E-01
2.5E-01
2.0E-01
1.5E-01
1.0E-01
5.0E-02
0.0E+00
1.E-04MCNP

1.E-03

1.E-02MG

1.E-01

1.E+00

Energy (eV)NJ
(MG-MC)/MC

1.E+01

10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
-2.0%
-4.0%
-6.0%
-8.0%
-10.0%
1.E+02(NJ-MC)/MC102

Thermal spectrum in homogeneous
medium (H-Polyethylene)◼

◼

MCNP6.2, FRENGY/ACE/MG-GENESIS(MG), NJOY(99.364)GENESIS(NJ)JENDL4.0, 300K
H poly 300K
4.0E-01
3.5E-01

Specrum

3.0E-01
2.5E-01
2.0E-01
1.5E-01
1.0E-01

5.0E-02
0.0E+00
1.E-04MCNP

1.E-03

1.E-02MG

1.E-01

1.E+00

Energy (eV)NJ
(MG-MC)/MC

1.E+01

10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
-2.0%
-4.0%
-6.0%
-8.0%
-10.0%
1.E+02(NJ-MC)/MC103

Thermal spectrum in homogeneous
medium (C-Free)◼

◼

MCNP6.2, FRENGY/ACE/MG-GENESIS(MG), NJOY(99.364)GENESIS(NJ)JENDL4.0, 600K
C free 600K
6.0E-01

20.0%

Difference due toweighting spectrum

5.0E-01

15.0%

10.0%
Specrum

4.0E-01

5.0%

3.0E-01

0.0%

-5.0%

2.0E-01-10.0%
1.0E-01-15.0%

0.0E+00
1.E-04MCNP

1.E-03

1.E-02MG

1.E-01

1.E+00

Energy (eV)NJ
(MG-MC)/MC

1.E+01-20.0%
1.E+02(NJ-MC)/MC104

Thermal spectrum in homogeneous
medium (C-Graphite)◼

◼

MCNP6.2, FRENGY/ACE/MG-GENESIS(MG), NJOY(99.364)GENESIS(NJ)JENDL4.0, 600K
C graphite 600K
6.0E-01

20.0%
15.0%

5.0E-01

Specrum

10.0%
4.0E-01

5.0%

3.0E-01

0.0%
-5.0%

2.0E-01-10.0%
1.0E-01-15.0%

0.0E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01-20.0%
1.E+02

Energy (eV)
MCNP

MG

NJ

(MG-MC)/MC(NJ-MC)/MC105

k-effective of UO2 pincell
5wt% UO2, 600K, PWR pincell◼ GENESIS (MOC), 172gr., P3◼ 𝑆(𝛼, 𝛽)：H-free gas, H-H2O, H-ZrH, H-Polyethylene◼ Summary of k-effective◼

H-temp.

S(a,b)FRENDY/MG

NJOY99Difference(MG-NJ)/NJ

600K

Free gas1.4089470

1.4089180

0.002%

H2O

1.4075497

1.4075113

0.003%

H-Zrh

1.4075781

1.4075327

0.003%

Free gas1.4125241

1.4125083

0.001%

Poly

1.4102028

1.4101741

0.002%

300K

106

Neutron spectrum in pincell◼

Spectrum in H2O
Difference of neutron flux, UO2 pincell

Neutron flux (per unit lethargy)

6.0
5.0
4.0

Mod.(NJOY-GENESIS)
Mod.(MG-GENESIS)

3.0
2.0
1.0
0.0
1.0E-03

1.0E-01

1.0E+01 1.0E+03 1.0E+05 1.0E+07
Energy [eV]

Differeicen of neutron flux (NJOYMG)/NJOY

Neutron flux, UO2 pincell
0.10
0.08
0.06
0.04

Mod.

0.02
0.00
-0.02
-0.04
-0.06
-0.08
-0.10
1.0E-03

1.0E-01

1.0E+01 1.0E+03 1.0E+05 1.0E+07
Energy [eV]107

UO2 assembly+reflector
5wt% UO2, 600K, PWR fuel assembly◼ GENESIS, 172 gr., P3, 𝑆(𝛼, 𝛽) : free gas◼ K-effective
◼

◼
◼

◼

NJOY-GENESIS
MG-GENESIS：1.12372：1.12368

All reflective BC

Fission rate distribution

1.6

1.0E-031.2

5.0E-041.0
0.8

0.0E+00

0.6

NJOY-GENESIS

0.4

MG-GENESIS-5.0E-04

Difference (MG-NJOY)/NJOY

0.2
0.0

Difference (MG-NJOY)/NJOY

Relative fission rate

1.4

Water reflector-1.0E-03
1

2

3

4

5

6

7 8 9 10 11 12 13 14 15 16 17Fuel rod number in row

108Comparison of cross sections
H1 Thermal free gas scattering
H1, mt=2, 600K, Ein=0.1eV
1.0E+01
Cross section [barn]

1.0E+00

NJOY

MG

1.0E-01

1.0E-02
1.0E-03
1.0E-04
1.0E-05

1.0E-06
1.0E-03

1.0E-02JENDL4.0u
1/E spectrum, NR approximation

1.0E-01
Energy [eV]

1.0E+00

1.0E+01109

Comparison of cross section
H1 elastic scattering (P1)
H1, mt=2, 600K, Ein=1.0-0.9MeV, P1
1.0E+01

Cross section [barn]

1.0E+00
1.0E-01

NJOY

MG

1.0E-02
1.0E-03

1.0E-04
1.0E-05
1.0E-06

1.0E-07
1.0E+00
JENDL4.0u
1/E spectrum, NR approximation

1.0E+02
1.0E+04
Outgoing energy [eV]

1.0E+06110

Comparison of cross section
U235 fission
U235, mt=18, 600K, sigz=1200barn
1.0E+04
MG

Cross section [barn]

NJOY
1.0E+03
1.0E+02
1.0E+01

1.0E+00
1.0E-03 1.0E-01 1.0E+01 1.0E+03 1.0E+05 1.0E+07
Energy [eV]JENDL4.0u
1/E spectrum, NR approximation

111Comparison of cross section

U235 fission spectrum (prompt+delayed)
U235, Chi-total, 600K
Fission spectrum (gr. integrated)

1.0E+00
1.0E-01
1.0E-02

NJOY

MG

1.0E-03
1.0E-04
1.0E-05
1.0E-06

1.0E-07
1.0E+00

1.0E+02JENDL4.0u
1/E spectrum, NR approximation

1.0E+04
Energy [eV]

1.0E+06

1.0E+08112

Comparison of cross section
U238 (n,2n)
U238, mt=16, 600K, sigz=1.0e+10barn

Cross section [barn]

1.0E+01

1.0E+00

NJOY

MG

1.0E-01

1.0E-02
1.0E+06
JENDL4.0u
1/E spectrum, NR approximation

1.0E+07
Energy [eV]

1.0E+08113

Comparison of cross section
U238 inelastic scattering (mt=51)
U238, mt=51, 600K, sigz=1.0e+10barn
1.0E+01

Cross section [barn]

NJOY

MG

1.0E+00

1.0E-01

1.0E-02
1.0E+04

1.0E+05JENDL4.0u
1/E spectrum, NR approximation

1.0E+06
Energy [eV]

1.0E+07

1.0E+08114

Comparison of cross section
U238 inelastic scattering (mt=91)
U238, mt=91, 600K, Ein=19.6-17.3MeV
1.0E+00

Cross section [barn]

1.0E-01

1.0E-02

NJOY

MG

1.0E-03
1.0E-04

1.0E-05
1.0E-06

1.0E-07
1.0E+04
1.0E+05
1.0E+06
1.0E+07
JENDL4.0u
Outgoing energy [eV]
1/E spectrum, NR approximation

1.0E+08115

Comparison of cross section
U238 capture
U238, 600K, mt=102

Cross section [barn]

1.0E+03

NJOY(bgxs=1e+10barn)
MG(bgxs=1e+10barn)
NJOY(bgxs=52barn)
MG(bgxs=52barn)

1.0E+02
1.0E+01
1.0E+00
1.0E-01
1.0E-02
1.0E+00

1.0E+01JENDL4.0u
1/E spectrum, NR approximation

1.0E+02
1.0E+03
Energy [eV]

1.0E+04

1.0E+05116

Summary and future works

117Summary◼

◼

Generation capability of multi-group cross sections in FRENDY
V2 is described.
FRENDY V2 can be downloaded from:◼

◼

https://rpg.jaea.go.jp/main/en/program_frendy/

Reference of FRENDY V2
◼

◼
◼

◼
◼

◼

K. Tada, Y. Nagaya, S. Kunieda, K. Suyama, T. Fukahori, "Development andverification of a new nuclear data processing system FRENDY," J. Nucl. Sci.
Technol., 54, pp.806-817 (2017).
https://www.tandfonline.com/doi/abs/10.1080/00223131.2017.1309306
A. Yamamoto, K. Tada, G. Chiba, T. Endo, "Multi-group neutron crosssection generation capability for FRENDY nuclear data processing code," J.
Nucl. Sci. Technol., 58, pp.1165-1183 (2021).
https://www.tandfonline.com/doi/full/10.1080/00223131.2021.1921631
K. Tada, R. Kondo, T. Endo, A. Yamamoto, "Development of ACE file
perturbation tool using FRENDY," J. Nucl. Sci. Technol. (2022).
https://www.tandfonline.com/doi/full/10.1080/00223131.2022.2130463
118

Appendix

119Potential issues of NJOY2016◼

Difference of absorption reaction in Cs137 of JENDL4.0: Due to inconsistency for
mt=30, which provides zero cross sections at mf=3 in nuclear data. In the 7d
section of the MATXS file, zero cross sections are automatically eliminated, but
not in 6d in the NJOY processing. FRENDY/ACE/MG avoids this issue by
bypassing 1d cross section processing whose all elements are zero.
◼

◼

In NJOY, self-shielding factors are calculated only for total, fission, capture, andelastic scattering. However, some nuclides have large resonances in otherreactions. For example, in the case of C000 (J4) and Ni059 (J4), self-shielding in
mt=107 is not negligible in the high energy region. However, infinite dilute crosssections are used in the GROUPR of NJOY. FRENDY/MG explicitly considers selfshielding for all reactions.
◼

◼

Issue of NJOY, MATXSR.

Issue of NJOY. In GROUPR, self-shielding is calculated only for mt=1, 2, 18, 102. For
2d cross section, only for mt=2. Therefore, even if GENDF is used, this issue will not
be resolved.

Eu156 of B7.1, 8.0: Some difference is observed in all reactions. Smooth crosssections for 1 - 7eV show difference due to different interpolation scheme◼

Issue of NJOY. Log-log in NJOY using limited energy points, FRENDY/ACE does not
perform interpolation - direct calculation at all energy points.
120

Potential issues of NJOY2016◼

Pm148g of JEFF3.3: Some difference is observed in all reactions. Smooth crosssections for 1.5 – 10 eV show difference due to different interpolation scheme.
◼

◼

Cm247 of JEFF3.3: Some difference is observed in all reactions. Smooth crosssections for 50 – 500 eV show difference.
◼

◼

Same as above.

Ca040 of B8.0: peak value of mt=600 around 1 MeV is not appropriately
reconstructed in NJOY due to numerical issue of the Coulomb function.
◼

◼

Same as above.

Issue of NJOY

Cs135 in JEFF3.3: inconsistent definition of mt=30 for mf=3 (zero) and mf=6
(nonzero small value). FRENDY/ACE/MG sets zero values for mt=30, but
NJOY21 provides some value, which seems to be incorrect. The same issue
exists for Cs135 and Cs137 of JENDL4.0.
◼

Issue of NJOY, MATXSR

121

Potential issues of NJOY2016◼

◼
◼

Ra226 of JEFF-3.3: nu-value is incorrect in NJOY.
Be7 of ENDF-7.1: mt=1 of the 7th group in XMAS 172 energy group is zeroP1 reaction rates: Differences are generally larger than those of the P0 reaction
rate. These differences mainly come from insufficient numerical integration
accuracy of NJOY in the thermal energy region.
◼

◼

◼

◼

◼

◼

Deficiency of NJOY

F017, Ne019 of TENDL2019: reconstructed pointwise cross sections by
NJOY2016 are not accurate (due to zero cross section at the upper energy ofresonance range in nuclear data file)
Sc45 (mt=1) in B7.1, Sn124 (mt=102), Po210 (mt=102) in B8.0: Boundary
between unresolved/resolved energy range is not appropriately set in NJOY.
O018 of TENDL2019: scattering matrix of mt=91 is half of the appropriate
valuesO018 of TENDL2019: pointwise cross section reconstruction for mt=102 is
inaccurate
Reaction name in MATXS file correspond to mt=749 is “tcd” in NJOY2016, while“tcn” is correct.
122

Potential issues of nuclear data◼

Zero cross sections between resolved/unresolved resonance◼

◼

Sum of prompt and delayed nu-values is not consistent with total nuvalue◼
◼

◼

Th231, ENDF-7.1, 8.0
Pa231, Cm247, JEFF-3.3

Some metastable nuclides in B8.0: Differences of P0 reaction rate are observed.
Elastic cross section (mt=2) is not adequately processed in the MATXS file of
NJOY.
◼
◼

◼

TENDL-2019, C011 mt=2, F017 mt=1, Ne019 mt=2Issue of nuclear data (Q-value of mt=2 is not zero in B8.0)
Al26m, Ag118m, Sn121m, Te121m, Te131m, I132m, Ce137m, Re186m, Ir197m,
Hg197m, Bi210m

Delayed nu-value for mf=1 mt=451 exists but mf=5 mt=455 does notexist. No delayed nu-value is given in the ACE file (both of FRENDY and
NJOY).

123