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Fission Product Yield Calculations by the Hauser-Feshbach Statistical Decay and Beta Decay
Shin OkumuraLaboratory for Advanced Nuclear Energy, Tokyo Institute of Technology
in collaboration withToshihiko Kawano, Patrick Jaffke, and Patrick Talou,
Los Alamos National Laboratory
Tadashi Yoshida, and Satoshi ChibaTokyo Institute of Technology
June 11, 2018 Varenna, Italy 1
Theoretical approaches
2
• Wide variety of fission products (more than 1,000 nuclides) are produced during the fission process of actinides such as 235U
• Many theoretical approaches to understand the dynamics from the compound nucleus formation to the scission• e.g. Macroscopic and microscopic models
• Many theoretical models and codes to study the fission observables through Monte Carlo simulations for the de-excitation of the primary fission fragments• e.g. CGMF, FIFFRELIN, FREYA, GEF...
0 0.02 0.04 0.06 0.08
0.1
Y P(A
)
Primary Fission Fragment Yield
0 0.5
1 1.5
2 2.5
3 3.5
4
<ν>
Prompt Neutron Multiplicity
0 0.02 0.04 0.06 0.08
0.1
Y I(A
)
Independent Fission Product Yield
0 0.1 0.2 0.3 0.4 0.5
70 80 90 100 110 120 130 140 150 160
Y C(A
)
Mass Number
Cumulative Fission Product Yield
Zl, A’l
Fission fragment and fission product yield (FPY)
CN
Zl, AlZh, Ah
Zh, A’h
Z’l, A’’l
Z’h, A’’h
n
γ
γ
β
nγDecay heat Cumulative fission
product yield
Independent fission product yield
Primary fission fragment yield
Beta delayed neutron
3
nn
β
Decay data
Hauser-Feshbachtheory
• Excitation energy• Spin and parity • YP(Z,A,M)…etc
• YI(Z,A,M)• "̅, "̅ A• %̅• &(() ..etc
• Yc(Z,A,M)• "̅*• Decay heat (β, γ) ..etc
Fission theoryCompound nucleus
Prompt neutron
Hauser-Feshbach statistical decay for fission fragments
4
HF3D: Hauser-Feshbach Fission Fragment Decay model• Treat a primary fission fragment as a compound nucleus• Distributions of primary fission fragment characterized by an excitation energy, spin
and parity are integrated deterministically (no Monte Carlo)• Calculate many fission observables in a consistent manner as a function of incident
neutron energy (Ein < 5MeV)
Beta decay calculation
• Starting from independent FPY calculated with HF3D
• Summation calculation of beta decay using the decay data library (ENDF/B-VIII)
Neutron emission multiplicity
u Probability of nucleus having the state of spin J and parity Π, R(J, Π)
u Distribution of excitation energy, G(E)
J. Nucl. Sci. & Technol. Article
example, the average number of prompt neutrons ⌫ is calculated as
⌫ =NX
k=1
Yk
⇣⌫(k)l + ⌫(k)
h
⌘, (5)
whereN is the total number of fragment pairs, and the neutron multiplicities ⌫(k)l,h are given
by integrating the neutron evaporation spectrum �(k)l,h from the light or heavy fragment in
the center-of-mass system,
⌫(k)l,h =
ZdEx
X
J⇧
Zd✏ R(J,⇧)G(Ex)�
(k)l,h (J,⇧, Ex, ✏) , (6)
where R(J,⇧) is the probability of nucleus having the state of spin J and parity ⇧, and
G(Ex) is the distribution of excitation energy. They satisfy the normalization condition
ofP
J⇧ R(J,⇧) = 1 andRG(Ex)dEx = 1.
For the spin and parity distributions we follow our previous work16, in which the
spin-parity population distribution is expressed by
R(J,⇧) =J + 1/2
2f 2�2(U)exp
⇢�(J + 1/2)2
2f 2�2(U)
�, (7)
where the parity distribution is just 1/2, �2(U) is the spin cut-o↵ parameter, U is the
excitation energy corrected by the pairing energy � as U = Ex � �, and f is a scaling
factor determined later by comparing the calculated results with experimental data.
We estimate the average excitation energies in each fragment, El and Eh, with the
anisothermal model16,34,35 that is characterized by an anisothermal parameter RT defined
as the ratio of e↵ective temperatures in the fission fragments
RT =Tl
Th=
sUl
Uh
ah(Uh)
al(Ul), (8)
where a(U) is the shell-e↵ect corrected level density parameter at the excitation energy
of U . In reality, TKE in Eq. (3) could have a distribution characterized by the width
�TKE, which is empirically known to be about 8–10 MeV36,37. This width propagates to
the width of TXE through �TXE = �TKE, then perturbs the excitation energies of two
fragments. Assuming Gaussian for the TXE distribution, each fragment has an excitation
7
σ2(U): spin cut-off parameterU: Excitation energyf: scaling factor
Z, A-1Z, A
R(J)
Tota
l Exc
itatio
n En
ergy
Sn(A)
Sn(A-1)
Z, A-2
G(Ex)
Okumura et al. J. Nucl. Sci. Technol., in press.
Excitation energy sharing between fragments
5
J. Nucl. Sci. & Technol. Article
example, the average number of prompt neutrons ⌫ is calculated as
⌫ =NX
k=1
Yk
⇣⌫(k)l + ⌫(k)
h
⌘, (5)
whereN is the total number of fragment pairs, and the neutron multiplicities ⌫(k)l,h are given
by integrating the neutron evaporation spectrum �(k)l,h from the light or heavy fragment in
the center-of-mass system,
⌫(k)l,h =
ZdEx
X
J⇧
Zd✏ R(J,⇧)G(Ex)�
(k)l,h (J,⇧, Ex, ✏) , (6)
where R(J,⇧) is the probability of nucleus having the state of spin J and parity ⇧, and
G(Ex) is the distribution of excitation energy. They satisfy the normalization condition
ofP
J⇧ R(J,⇧) = 1 andRG(Ex)dEx = 1.
For the spin and parity distributions we follow our previous work16, in which the
spin-parity population distribution is expressed by
R(J,⇧) =J + 1/2
2f 2�2(U)exp
⇢�(J + 1/2)2
2f 2�2(U)
�, (7)
where the parity distribution is just 1/2, �2(U) is the spin cut-o↵ parameter, U is the
excitation energy corrected by the pairing energy � as U = Ex � �, and f is a scaling
factor determined later by comparing the calculated results with experimental data.
We estimate the average excitation energies in each fragment, El and Eh, with the
anisothermal model16,34,35 that is characterized by an anisothermal parameter RT defined
as the ratio of e↵ective temperatures in the fission fragments
RT =Tl
Th=
sUl
Uh
ah(Uh)
al(Ul), (8)
where a(U) is the shell-e↵ect corrected level density parameter at the excitation energy
of U . In reality, TKE in Eq. (3) could have a distribution characterized by the width
�TKE, which is empirically known to be about 8–10 MeV36,37. This width propagates to
the width of TXE through �TXE = �TKE, then perturbs the excitation energies of two
fragments. Assuming Gaussian for the TXE distribution, each fragment has an excitation
7
• Evaporated prompt neutrons take away a lot of excitation energy from primary fission fragments
• Total excitation energy can be calculated by the energy balance of the reaction from TKE
0.6
0.8
1
1.2
1.4
1.6
120 125 130 135 140 145 150 155 160
RT
A
Manailescu et al.Litaize et al.RT= 1.0 (A<132) and RT= 1.2 (A>=132)RT= 1.0 (A<122), slope, RT= 1.2 (A>=132)RT=1.2
0
0.5
1
1.5
2
2.5
3
3.5
4
60 80 100 120 140 160 180
Num
ber o
f Neu
trons
Mass Number
ExperimentsRT 1.20
Litaize et al.Manailescu et al.
RT= 1.0 (A<132) and RT= 1.2 (A>=132)RT= 1.0 (A<122), slope, RT= 1.2 (A>=132)
"̅ENDF = 2.41"̅HF3D = 2.38
CN
Zl, Al
Zh, Ah
Zl, A’l
Zh, A’h
n
γ
γn
n
• The average excitation energies in each fragment with the anisothermal model parameter defined by the ratio of the effective temperatures in two fragments
• Constant RT=1.2 reproduces the sawtooth structure of "̅ A for 235U(n,f) system
Inputs for HF3D calculations for fission fragments
• From experimental data
• Mass dependency of primary fission fragment
yield Y(A) and total kinetic energy TKE(A)
• Energy dependence of Y(A) and TKE(A)
• Incident neutron energy dependence of Y(A) and TKE
• From the other models
• Charge distribution from Wahl systematics (ZPmodel) and its incident neutron energy
dependence
• Ingredients for Hauser-Feshbach calculations
• Neutron optical potential by Koning-Delaroche
• Level density systematics based on KTUY05 mass
model
• Kopecky and Uhl γ-ray strength function for the E1
transition, and standard Lorentzian models for higher
multipolarity
6
0
0.02
0.04
0.06
0.08
0.1
60 80 100 120 140 160 180
Prim
ary
Fiss
ion
Frag
men
t Yie
ld
Mass Number
this workZenyalov (2006)
Simon (1990)Straede (1987)
Baba (1997)Pleasonton (1976)
Hambsch
100
110
120
130
140
150
160
170
180
190
200
120 130 140 150 160 170 180
Tota
l Kin
etic
Ene
rgy
[MeV
]
Mass Number
Zeynalov (2006)Baba (1997)
Simon (1990)Dyachenko (1967)
Hambschpresent
10-4
10-3
10-2
10-1
100
101
80 90 100
Inde
pend
ent F
issi
on P
rodu
ct Y
ield
[%]
Mass Number
Rudstam (1990), GaGeAsSeBrKr
RbSr
HF3D
10-4
10-3
10-2
10-1
100
101
110 120 130 140 150
Inde
pend
ent F
issi
on P
rodu
ct Y
ield
[%]
Mass Number
Rudstam (1990), AgCdIn
SnSbTe
IXeCsBa
HF3D
Calculated independent FPY (after prompt emission by HF3D) for 235U(nth,f)
7
Isotope distribution
of Light FPsHeavy FPs
Rudstam G et. al., Radiochimica Acta. 1990
CN
Zl, Al
Zh, Ah
Zl, A’l
Zh, A’h
n
γ
γn
n
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
60 80 100 120 140 160 180 0
0.5
1
1.5
2
2.5
3
3.5
4
Fiss
ion
Yiel
d
Mass Number
JENDL/FPY-2011ENDF/B-VII.1
HF3D
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
60 80 100 120 140 160 180 0
0.5
1
1.5
2
2.5
3
3.5
4
Fiss
ion
Yiel
d
Mass Number
Nishio 1998Batenkov 2004
HF3D ν(A)
10-4
10-3
10-2
10-1
100
101
110 120 130 140 150
Cum
ulat
ive
Fiss
ion
Prod
uct Y
ield
[%]
Mass Number
Rudstam (1990), AgCdIn
SnSbTe
IXeCsBa
HF3D10-3
10-2
10-1
100
101
80 90 100
Cum
ulat
ive
Fiss
ion
Prod
uct Y
ield
[%]
Mass Number
Rudstam (1990), GaGeAsSeBrKr
RbSr
HF3D
Calculated cumulative FPY (after beta decay) for 235U(nth,f)
8
CN
Zl, A’l
Zh, A’h
n
γ
γn
n
Z’l, A’’l
Z’h, A’’h
β
n
γ
β
Isotope distribution of Light FPs
Heavy FPs
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
60 70 80 90 100 110 120 130 140 150 160 170 180
Cum
ulat
ive
Yiel
d
Mass Number
JENDL/FPY-2011ENDF/B-VII.1
HF3D cumulative
Calculated decay heat and delayed neutron by beta decay for 235U(nth,f)
9
• Summation calculation of beta decay using ENDF/B-VII decay data library
# decay heat total (MeV/fission) Isotope HF3D ENDF/B-VII JENDL/FPY-201139-Y-98-0 1.0029E-01 6.2298E-02 2.9617E-0238-Sr-97-0 7.1420E-02 6.0794E-02 6.0945E-0239-Y-97-1 5.5401E-02 3.6437E-02 5.9791E-0238-Sr-96-0 5.2162E-02 5.4041E-02 5.4186E-0237-Rb-95-0 4.9638E-02 3.0397E-02 3.0479E-0239-Y-99-0 3.7535E-02 3.1491E-02 3.1468E-0240-Zr-99-0 3.1358E-02 2.8238E-02 2.8297E-0255-Cs-142-0 2.5797E-02 3.2072E-02 3.2160E-0237-Rb-92-0 2.5157E-02 2.4540E-02 2.4606E-0236-Kr-93-0 2.3157E-02 1.0814E-02 1.0843E-02
# delayed neutronIsotope HF3D ENDF/B-VII JENDL/FPY-2011
37-Rb-95-0 9.6787E-04 5.9269E-04 5.9428E-0437-Rb-94-0 4.9311E-04 3.9672E-04 3.9778E-0435-Br-90-0 3.8195E-04 4.4369E-04 4.4488E-0435-Br-91-0 3.3934E-04 3.4418E-04 3.4510E-0437-Rb-96-0 3.2717E-04 2.5629E-04 2.5698E-0435-Br-89-0 2.3806E-04 2.1677E-04 2.1735E-0437-Rb-97-0 2.3125E-04 7.7580E-05 7.7789E-0533-As-85-0 2.0059E-04 2.7429E-04 2.1779E-0453-I-139-0 1.8703E-04 2.0876E-04 2.0932E-0439-Y-99-0 1.6211E-04 1.3601E-04 1.3591E-04
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
100 101 102 103 104
t ⋅ D
H(ga
mm
ma)
(MeV
/fiss
ion)
Cooling time (Sec.)
ORNLthLowellth
ENDF-/VIIJENDL/FPY-2011
HF3D
0
0.2
0.4
0.6
0.8
1
100 101 102 103 104
t ⋅ D
H(be
ta) (
MeV
/fiss
ion)
Cooling time (Sec.)
ORNLthLowellth
ENDF/B-VIIJENDL/FPY-2011
HF3D
0
0.001
0.002
0.003
0.004
0.005
100 101 102 103 104
t ⋅ D
N (/
fissi
on)
Cooling time (Sec.)
KeepinthENDF-/VII
JENDL/FPY-2011HF3D
Energy dependence of fission observables by HF3D and beta decay
10
2
2.5
3
3.5
0 1 2 3 4 5
Aver
age
Num
ber o
f Neu
trons
Incident Neutron Energy [MeV]
ExperimentsENDF/B-VIII
JENDL-4HF3D
0
0.005
0.01
0.015
0.02
0.025
0 1 2 3 4 5
Aver
age
Num
ber o
f Del
ayed
Neu
trons
Incident Neutron Energy [MeV]
HF3DExprimentsJENDL4.0
Prompt neutron Delayed neutron
235U(n,f)HF3D model can calculate the all observables by the incident neutron energy dependent.
Energy dependency of cumulative FPY after beta decay
11
CN
Zl, A’l
Zh, A’h
n
γ
γn
n
Z’l, A’’l
Z’h, A’’hβ
n
γ
β
Connect with Langevin model (Primary fission fragment)
12
• Primary fission fragment YI(A) and TKE distribution TKE(A) by 4D Langevin model are used as inputs
• HF3D model calculation for fission fragment decay
• Evaluation of 4D Langevin’s fission fragment distributions
110
120
130
140
150
160
170
180
190
200
210
110 120 130 140 150 160 170 180
TKE
[MeV
]
A
All fission eventAverage
0
0.02
0.04
0.06
0.08
0.1
60 80 100 120 140 160 180
Yiel
d
Mass Number
5 gauss fitLangevin 4D
HambshPleasonton
0
0.02
0.04
0.06
0.08
0.1
60 80 100 120 140 160 180
Yiel
d
Mass Number
Langevin 4D + HF3D (post)Langevin 4D + HF3D (pre)
JENDL-FPY/2011
0
0.1
0.2
0.3
0.4
0.5
60 80 100 120 140 160 180
Cum
ulat
ive
Yiel
d
Mass Number
JENDL/FPY-2011HF3D
Summary
13
HF3D: Hauser-Feshbach Fission Fragment Decay model
• Treat fission fragment as a compound nucleus
• Staring with the initial configuration of the primary fission fragment pairs characterized by YP(Z,A,M), excitation energy, spin and parity
• Simultaneously reproduces fission observables in an energy dependent manner
e.g. P("̅), "̅, "̅(A), YI(A, Z, M), YC(A,Z,M), Isomeric Ratio
• Connection with microscopic and macroscopic models (primary fission fragment (Y(A) and TKE)) is feasible
Beta decay
• Beta decay calculation using the decay data library to obtain cumulative yields, which are energy dependent
• Decay heat and delayed neutron calculations with the summation method to evaluate the independent FPY
0 0.02 0.04 0.06 0.08
0.1(a)
Y P(A
)
Primary Fission Fragment Yield
0 0.5
1 1.5
2 2.5
3 3.5
4
(b)
<n>
ExperimentsHF3D
0 0.02 0.04 0.06 0.08
0.1(c)
Y I(A
)
JENDL/FPY-2011HF3D
0 0.1 0.2 0.3 0.4 0.5
70 80 90 100 110 120 130 140 150 160
(d)
Y C(A
)Mass Number
JENDL/FPY-2011HF3D
Acknowledge
14
K. Nishio, Japan Atomic Energy Agency (JAEA)
The grant “Development of prompt-neutron measurement in fission by surrogate reaction method and evaluation of neutron-energy spectra,” entrusted to JAEA by Ministry of Education, Culture, Sports, Science and Technology, Japan.
The National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396.