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.