IAEA-ND CM on “Prompt fission neutron spectra of major actinides”, 24-27. Nov. 2008

Application of Multimodal Madland-Nix Model

・ Evaluation of PFNS in JENDL-series ・ Multimodal Random Neck-rupture Model : An Outline ・ Refinements in the Madland-Nix Model 1) Multimodal fission, 2) Level density parameter considering the shell effect, 3) Asymmetry in ν for LF and HF, 4) Asymmetry in T for LF and HF ・ Possible early neutrons ： Neutron emission during acceleration （ NEDA) Takaaki Ohsawa ( 大澤孝明 )

Dept. of Electric & Electronic EngineeringSchool of Science and EngineeringKinki University, Osaka, Japan

Prompt fission neuron spectra in JENDL-3.3 and JENDL/AC2008

JENDL-3.3 JENDL/AC2008Th-232 Maxwellian [TM=Howerton-Doyas’ syst.] CCONE (O.Iwamoto)Pa-231 Maxwellian (taken from ENDF/B-V) CCONE U-233 Multimodal M-N (T.Ohsawa) Multimodal M-N [E≤5MeV], CCONE [E>5MeV] U-235 Multimodal M-N [E≤5MeV], Multimodal M-N [E≤5MeV], Preeq. spectrum by FKK model CCONE [E>5MeV]U-238 Multimodal M-N Multimodal M-N [E≤5MeV], CCONE [E>5MeV] Np-237 Maxwellian [TM: Baba(2000),Boikov(1994)] CCONEPu-239 Multimodal M-N Multimodal M-N [E≤5MeV] CCONE [E>5MeV] Pu-241 Maxwellian [TM=Smith’s systematics] CCONE [E>5MeV] Am-241 Maslov’s evaluation (1996) Multimodal M-N [E≤6MeV], CCONE [E>6MeV]Am-242m Maslov’s evaluation (1997) Multimodal M-N [E≤6MeV], CCONE [E>6MeV]Cm-243 Maslov’s evaluation (1995) Multimodal M-N [E≤6MeV], CCONE[E>6MeV]Cm-245 Maslov’s evaluation (1996) Multimodal M-N [E≤6MeV], CCONE[E>6MeV]

JENDL-3.3JENDL-3.3 JENDL/AC2008JENDL/AC2008 JENDL-4 JENDL-4

・ Released 　　　　　 March 2008・ Ac –Fm (Z=89-100)・ 79 nuclides 　　

・ Released 　　 May 2002・ 62 nuclides

Evaluated Nuclear Data for Actinides in the JENDL-series

・ Will be released in 2010・ Slight revision(?)

　 New 17 nuclides (T1/2 >1d) added

Program CCONE (by O. Iwamoto, JAEA)

Main features・” All-in-one” code for evaluation of nuclear data・ Witten in C++ for ease of extension & modification ・ Architecture based on object oriented programming・ Coupled-channel theory・ Hauser-Feshbach theory including Moldauer effect・ DWBA for direct excitation of vibrational states・ Two-component exciton model (Kalbach)・ Multi-particle emission from the CN with spin- and parity-conservation・ Double-humped fission barriers with consideration of collective enhancement of the level density・ Madland-Nix model (original implementation)

cf. Osamu Iwamoto, J. Nucl. Sci. Technol. 44, 687 (2007)

Multimodal Random Neck-rupture Model　　　　 [U.Brosa, S.Grossmann, A.Müller]

Random Neck- Rupture Model Random Neck- Rupture Model

Multichannel Fission Model Multichannel Fission Model

Multimodal Random Neck-Rupture Model 　 (BGM model)Multimodal Random Neck-Rupture Model 　 (BGM model)

[S.L.Whetstone,1959] [e.g. E.K.Hulet et al. 1989]

“hybrid”

Multimodal Random Neck-rupture Model

Several distinct deformation paths ⇒ several pre-scission shapes

Neck-rupture occursrandomly according tothe Gaussian function

S1

S2SL

[U. Brosa et al.1990]

2 2Rupture probability ( ) exp{ 2 [ ( ) ( )] / }rW A z z T

Standard-1

Standard-2

Superlong

Example: 235U(n,f)

3 modes overlapping → largest σ

Standard-1

Standard-2

Superlong

[H.-H. Knitter et al. Z. Naturforsch,42a,760(1987)]

Mas

s Y

ield

TK

Eσ

(TK

E)

2 modes overlapping → larger σ

single mode prevails → smaller σ

　　 　　　　 Justification of the MM-RNR model 　 on the basis of deformation energy surface calc.

Be

ta-

def

orm

atio

n

N Z

Spherical nucleus

N=86 (Meta-stable deformation; S2)

N=82 (S1) Z=50 (S1)

[B. D. Wilkins et al., Phys. Rev. C14,1832 (1976)]

The nascent HF is likely to be formed close to these hollows

Application of the Multimodal RNR Model

Multi-channelFissionModel

Random Neck-Rupture Model

Multimodal RNR Model

Madland-Nix (LA) Model

SummationCalculation

MultimodalMadland-Nix Model

Multimodal Analysis of DNY

T.Ohsawa et al., Nucl. Phys. A653, 17 (1999).

T. Ohsawa & F.-J. Hambsch, Nucl. Sci. Eng. 148, 50 (2004)

Fluctuations Observed in the Fission Yield in the Resonance Region for U-235 [F.-J. Hambsch]

Precursors are localized, because they have a structure of closed shell + loosely bound neutrons outside of the core.

Fluctuation in the Precursor Yields in the Resonance Region of U-235

The precursor yields in the LF-S2-region are considerably decreased.This brings about decrease in the delayed neutron yield at the resonance.

　 Fluctuation in the Delayed Neutron Yields for U-235

-3.5%

cf. T. Ohsawa and F.-J. Hambsch, Nucl. Sci. Eng. 148, 50 (2004)

271

n1

d i ii

Y P

Sudden decrease in the 4 - 7MeV region

Slight decrease in thermal & resonance

regions

CM-spectrum:

)exp(2

)(3/2 T

EE

TE

M

]/)(2sinh[)/exp()(

)/exp()( 2/1

WfWWf

Wf TEETETE

TEE

1. Maxwellian

2. Watt

3. Madland-Nix (LA) model

mT

m

C dTTTTkT 0

2)/exp()(

)(2)(

2

2

*

0

/2)(

m

totfnnr

m

mm

aT

EEBEE

TT

TTTTTP

　　 Models of PFNS

S.S.Kapoor et al., Phys. Rev. 131, 283 (1963)

2

2

( )

20( )

( , , )

1( ) ( ) exp( / )

2

f m

f

f

E E T

C

f m E E

N E E

d k T T T dTE T

LS-spectrum:

Märten & Seeliger 　Hu Jimin

4. Cascade Evaporation Model

5. Hauser-Feshbach ModelBrowne & DietrichGerasimenko

6. Monte Carlo Simulation

Lemaire, Talou, Kawano, Chadwick, MadlandDostrovsky, Fraenkel (1959)

Criteria for choosing a model for evaluation: 1. Accuracy 2. Simplicity 3. Predictive power

Improvements in the Method

Original Madland-Nix Model χtot= ½{ χL+ χH }

Multimodal Madland-Nix Model

(2) LDP : Shell effects on the LDP (Ignatyuk’s model)

(1) Multimodal Fission:　 Energy partition in the fission process is very different for different fission modes

( ) [ ( )] /[ ]tot n i i i n i ii i

E w E w

(3) Asymmetry inν: νL≠νH

/( )i iL iL iH iH iL iH

(4) Asymmetry in T : T L ≠ TH

because of the difference in deformation

(1) Multimodal Fission Model

Each different deformation path leads to different scission configuration, therefore to different energy partition.

S1

S2SL

Asymmetric fission

(standard mode)

Symmetric fis

sion

(superlong mode)

Hartree-Fock-Bogoliubov calc.by H.Goutte et al., Phys. Rev. C71, 024316(2005)

134

236

102

118

14195

118

Standard-1

Standard-2

Superlong

81.6%

18.3

%

0.007%

ER=194.5MeVTKE=187MeV

ER=184.9MeVTKE=167MeV

ER=190.9MeVTKE=157MeV

Multimodal Fission Process 235U(n,f), Ein=thermal

Average fragment mass

ER : calc. with TUYY mass formula (Tachibana et al., Atomic & Nucl. Data Tables, 39, 251 (1988) )TKE : Knitter et al., Naturforsch, 42a, 786 (1987)

Decomposition of Primary FF Mass Distribution

<TXE>= 14.0 MeV24.4 MeV

40.5 MeV24.4 MeV

14.0 MeV

S1-spectrum – softestS2-spectrum – harderSL-spectrum － hardest

Comparison with experimentfor U-235(nth,f)

( )

iL iL iH iHi

iL iH

●Modal spectrum :

●Total spectrum :

wi : mode branching ratio

1, 2,

1, 2,

i i ii S S SL

toti i

i S S SL

w

w

This evaluation is contained in JENDL-3.3 & JENDL/AC2008 and will also be in JENDL-4.

0 5 10 15 201E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

U-235(n,f)

En=0　MeV

En=2　MeV

En=5　MeV

　

　

No.of　Neutrons　(1/MeV)

Neutron　Energy　(MeV)

Spectra　for　Different　Incident　Energies

0 5 10 15 200.5

1.0

1.5

2.0

2.5

3.0

En=0　MeV

En=2　MeV

En=5　MeV

U-235(n,f)

　

　

Ratio　to　Thermal　Fission

Neutron　Energy　(MeV)

At higher incident energiesthe spectrum becomesharder due to1. Higher excitation energies of the FFs.2. Increase in the S2- component.

　　 　 (2) Shell Effects on LDP for FF

( ) ( )[1 ( ) / ]U a A f U Wa U

exp LD( , ) ( , )W M Z A M Z A

)exp(1)( UUf

2a A A

●Shell effects on the LDP vary according to the mass and excitation energy of the FFs.

Ignatyuk’s LDP

-10.054 MeV

Excitation-energy dependence :

0.154 5103.6 Asymptotic value :

Shell correction :

2condU E E ta Effective excitation energy :

20 03 , 12 /cond critE a A

Eq.(1) is a transcendental eq.→Solve it numerically! （ IGNA3 code ）

(1)

Effect of the Level Density Parameter on the Spectrum

LDP has a great effecton the spectrum, esp. inthe higher energy region.

(3) Asymmetry in ν for LF and HF: νL(A) ≠ νH(A)

Saw-tooth structure

)]()([1

)(

)]()()[2/1()(

niHiHniLiLiHiL

ni

niHniLni

EEE

EEE

Madland- Nix:

New modalspectra:

This is important because the neutron spectra from the LF and HF are very different!

HF LF

mHvH ＝ mLvL

1. The LF travels faster than the HF.

Two effects2. Low energy neutrons are more easily emitted from HFs than from LFs.

CM LS

HF

LF

S.S.Kapoor et al.,Phys.Rev. 131,283 (1963)

Consideration of non-equality νL(A) ≠ νH(A) brings about a difference of ~10% at maximum in the spectrum

(4) Asymmetry in the Nuclear Temperatures ・ T. Ohsawa, INDC(NDS)-251 (1991), IAEA/CM on Nuclear Data for Neutron Emission in the Fission Process, Vienna, 1990. p.71.・ T. Ohsawa and T. Shibata, Proc. Int. Conf. on Nucl. Data for Science and Technology, Juelich, 1991, p.965 (1992), Springer-Verlag.・ P. Talou, ND2007, Nice (2008)

●Total excitation energy of the FF:

TXE = Eint (L) + Edef (L) + Eint (H) + Edef (H)

at the scission-point

= E*(L) + E*(H)

at the moment of neutron emission

The nuclear temperatures of the two FFs at the moment of neutron emission are generally not equal, if the deformation is different at scission.

E*CN=Bn+En E*

L=aLT2L E*

H=aHT2H

E*L= Eint L +Edef L

E*H= Eint H +Edef H

TXE =<ER> + Bn + En ー TKE = aCNT2m

= aLT2L + aHT2

H

=(aLRT2 + aH)T2

H

where RT=TL/TH : temperature ratio

CN CNL H2 2

T L H

T

H LT

2

,m m

a aT T T T

a

R

R a a R a

Mode 　　 Standard-1 　　 　 Standard-2 　 　　　Superlong

Nuclides 　 Zr-102 　 Te-134 　　 Sr-95 　 Xe-141 　　 Pd-118 　 Pd-118

ER 　　　　　　 　　 194.49 184.86 190.95

TKE 　 　　　　　　 187 167 157

E* 　　 8.39 10.51 11.74 9.11 22.89 22.89

LDP 　 11.43 8.89 10.31 13.25 11.79 11.79

1.05 1.31 1.47 1.14 2.86 2.86

TL,i, TH,i 0.86 1.09 1.06 0.83 1.39 1.39

L,i H,iν , ν

Basic Fission Data for U-235(nth,f)

Multimodal mass distribution model

TUYYMass Formula Ignatyuk’s Level

Density ModelOptical Model(ELIESEⅢ )

Modified UnchangedCharge Distribution Model

Charge distribution

Total energyrelease LDP for

LF & HF

Inverse reactioncross sections

for LF & HF

TKE for Each Mode

ν L, ν H

Prompt Neutron Spectrum Calculation Code (FISPEK-O)

Total & Modal Neutron Spectra

Systematicsin Modal TKE

Mass distribution &mode branching ratios

Typical LF/ HF & their yields for each mode

TOTAL CODESYSTEM

　 Possible Early Neutrons

　 Neutron Emission During Acceleration (NEDA)

x

x

x

x

v

st

k 1

1ln

2

1

10

t = time after scissionx = E/Ek : ratio of the FF-KE relative to its final value Ek

s0 = charge-center distancevk = final velocity=[2{(M-m)Mm} ・ 1.44(Z-z)z/s0]1/2

s0

t

・ Certain fraction of prompt neutrons may be emitted before full acceleration of FF [V.P. Eismont,1965]

Z, Mz,m

Time Scale of Neutron Emission

)(10)/exp(2 21

3/1

sTBBU

An

n

Neutron emission time from an excited nucleus of excitation energy U and binding energy Bn [T. Ericson, Advances in Nuclear Physics 6, 425 (1960)]

If n-emission time > acceleration time t　　　　　　　　　　　　　　 → NE after full acceleration　　　　　　　　　　　　　　　　　　　　　　　　　　　 < t → NE during acceleration

NEDA is possible, at least in the Standard-2 fission

● Define two parameters:

　・ NEDA factor ： fraction of neutrons emitted　　　　　　　　　　　　　　　　　　　　　　　　 　　 during acceleration　・ Timing factor TF ： the ratio E/Ek at which 　　　　　　　　　　　　　　　　　　　　　　　　　　 　 　 neutrons are emitted

● Then find the best set of parameters that reproduce the experimental data.

Empirical Examination Parametric survey

0 1 2 30.1

0.2

0.3

0.4

Cm-245(n,f), En=thermal

　

　

Drapchinsky's Measurement Full acceleration 0.9N(TF=1)+0.1N(TF=0.7) 0.8N(TF=1)+0.2N(TF=0.7) 0.7N(TF=1)+0.3N(TF=0.7) 0.6N(TF=1)+0.4N(TF=0.7) Maslov's calc.

No.

of

Neu

tron

s (1

/MeV

)

Neutron Energy (MeV)

Results of parameter search ：　 Best fit set of values that reproduce the experimental data for Cm-245(nth,f) is NEDA=0.3, TF=0.7

NEDA factor increases with excitation energy

Concluding Remarks1. Madland-Nix model, refined by considering 1) multimodal nature of the fission process 2) appropriate LDP with inclusion of the shell effect 3) asymmetry in ν for LF & HF 4) asymmetry in T for LF & HF provides a good representation of the spectra for major actinides in the first chance fission region where multimodal analyses have been done.

2. In order to further improve the accuracy and extend the predictive power of the method, it is necessary to have a better knowledge on the systematics of the multimodal parameters for more fissioning systems.

3. Mode detailed study should be undertaken in order to solve the pre-scission/scission neutrons or neutron emission during acceleration.

Justification of the Triangular Temperature Distribution with Sharp Cutoff

2* * 2

( ) ( *) *

( *) 2

( )

when ( *) constant

E aT dE aTdT

P T dT P E dE

P E aTdT

P T T

P E

･

The approximate validity ofthis model is based on aspecific relationship between the FF neutron separation energyand the width of the initial distribution of FF excitation energy. [Terrell; Kapoor et al.]

Mis-alined Valleys [W.J.S. Swiatecki & S. Bjornholm, Phys. Rep.4C, 325 (1972) ]

Hartree-Fock-Bogoliubov calc. [J.F. Bernard, M. Girod, D. Gogny, Comp. Phys. Comm. 63, 365 (1991)]

　 Mis-alined Fission and Fusion Valleys

Scission occurs some-where around here.

・ Fission and fusion valleys　 are separated by a ridge.・ The nucleus gets over the　 ridge somewhere from the fission to fusion valley.

T.-S.Fan et al., Nucl. Phys. A591,161 (1995)

Pre-scission shapes

S1

SL

S2

Average number of neutrons emitted from a fragment foreach mode

* */ /L H L HE E Partition of the TXE

1

0

( ) ( ) exp( / )ck T T d

1E-3 0.01 0.1 1 100

10000

20000

30000

40000

50000

HF

Rea

ctio

n C

ross

Sec

tion

(mb)

Neutron Energy (MeV)

Zr-101 (S2) Cs-141 (S2) Tc-107 (S1) Te-135 (S1)

LF

Am-241(n,f) , En= 2MeV

The inverse reaction cross sections for HFs are higher than those for LFs in the low energy region. (according to the optical model calc.)

Gauss-Legendrequadrature over ε and T

Gauss-Laguerre quadrature

LS-spectrum:2

2

( )

2

( )

0

1( , , ) ( )

2

( ) exp( / )

f

f

m

E E

f c c

f m E E

T

N E E dE T

k T T T dT

　 NEDA increases with excitation energy

General systematic relations :

ER = 0.2197(Z2/A1/3)- 114.37 TKEViola = 0.1189(Z2/A1/3) + 7.3 TXE = ER -TKE + Bn + En

= 0.1008(Z2/A1/3)　-　121.67 +Bn + En

As Z2/A1/3 increases, TXE increases, which, in turn, means more NEDA effects for heavier actinides.

Y (A, Af , Ef*) = CS1[G(A, AS1, µS1s) + G(A, Af -AS1, µS1s)]

+ CS2[G(A, AS2, s) + G(A, Af -AS2, s)] + CSLG(A, Af /2, µSLs) Parameters : CS1 = 59.3 - 0.263 Nf - 0.017(Af -235.7) Ef

*, CS2 = 2.66(169.9 - Nf) + 0.19(Af - 232.6) Ef

*, CSL = 0.01exp(0.46 Ef

*), AS1 = 82.3 + 0.293Nf + 0.1Zf - 0.03 Ef

*, AS2 = 141.0 - 0.053 Ef

* , s = 5.7 - 0.24(149.9 - Nf ) + 0.12 Ef

*, µSL = 1.4, µS1 = 1.884 -0.0094Nf + 0.267exp[-(Nf -142.5)2] + 0.114exp[-|Nf -146.8|], C = 100/( CS1 + CS2 + CSL/2 )

Five-Gaussian Representation of Fragment Mass Distribution by Wang & Hu

80 90 100 110 120 130 140 150 1600.01

0.1

1

10

Superlong (0.3%)

F

issi

on

yie

ld (

%)

Primary fragment mass number

　B　C　D　E

Am-241(n,f) En=2MeV

Total

Standard-1 (21.0%)

Standard-2 (78.6%)

Decomposition of Fission Fragment Mass Distribution

Z=50 N=82N=50

54.6MeV29.1MeV31.5MeVTXE=

80 100 120 140 1600

2

4

6

8

Cm-245(n,f), En=thermal

TOTAL S1 S2 SL

Fis

sio

n Y

ield

(%

)

Mass Number (u)

Location of Delayed Neutron Precursors （ Heavy Fragment Region ）

N=82

Z=50

139Te

138Te

137Te

136Te

135Te

n

n

n

n

If E*(En)＞Bn + <η >then neutron emission occurs.

Exc. energy of the residual nucleus:E(i-1)

*(En) =E(i-1)*(En)ー(Bn + <η >)

Precursor

Precursor

Precursor

Non-precursor

Non-precursor

The non-precursor becomesa precursor.

The precursor becomes a non-precursor.

If E(k)*(En)＞Bn +<η >then neutron emission occurs.

At high incident energy, more and moreprecursors are lost.

1.4s(β )

2.49s（β ）

17.5s（β ）

0.343s（β ）

At higher energies, successive neutron emission from “would-be” precursors (primary FFs) leads to loss of actual precursors

T. Ohsawa et al., Proc. Int. Conf. on Nucl. Data for Sci. & Eng., Nice, France (2007)