Physics-based Calculation of the Prompt Neutron Spectrum

・ Brief Review of Fission Physics ・ Classification of Problems with Neutron Emission from FFs ・ Sequential Evaporation ・ Angular Anisotropy in the CM-system of FF ・ Neutron emission During Acceleration ・ Scission Neutrons ・ Yrast levels ・ Application of the Multimodal MN-Model to Bimodal Fission ・ My interest in Thorium Utilization ・ Summary

Takaaki Ohsawa ( 大澤孝明 )School of Science and Engineering Kinki University, Osaka, Japan

Basic idea The calculation model of the PFNS should be grounded on the present knowledge of fission physics and be consistent with it.

Stage I: Deformation

Stage II: Saddle-to-Scission

Stage III: Neutron emission

Prompt neutron are emitted from fragments of differentproperty, each of which is formed by the deformationpath it followed .

Stage IV: DN & γ-emission

β

γ n

The fission process is a multi-stage, multi-facet phenomenon.

• The nuclear deformation proceeds by way of a few distinct paths, defined by macroscopic and microscopic effects.

Asymmetric fission

(standard mode)

Symmetric fis

sion

(superlong mode)

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

Stage I: Deformation process

Hypothesis of “Mis-alined Valleys” (Swiatecki & Bjørnholm (1972))

Hartree-Fock-Bogoliubov calc.(Bernard, Girod, Gogny (1991))

• The nucleus transafers from fission (one-body) to fusion (two-body) valley (scission point).

Most scission occurs around here.But scission point isnot uniquely defined. →fluctuation in TKE

• Fission valley and fusion valley are separated by a ridge.

one-body

two-body

ridge

Stage II: Saddle-to-Scission Transition

• Brosa visualized the scission process as stochastic breakup in the neck region of the deformed nucleus.• Several well-defined pre-scission shapes, together with uncertainty in the breakup point, bring about different energy partition.

Standard-1 (S1)

Illustration by K. Nishio

Different Q-values,Different mass distributions,Different average TKEs,Different st. dev. of TKEs, for different fission modes.

Standard-2 (S2)

Superlong (SL)

TKE and TXE Distributions of Primary Fragments (based on the data of H.-H. Knitter et al. for U-235(nth,f) )

Three groups of primary fragment pairswith different excitation energies.

TKE TXE

Since neutron emission starts from these three groups of primary FF pairs, prompt neutron emission calculation should take the fact into account.

Stage III: Neutron Emission

A. From fully-accelerated FFs? Isotropic in the CM-system of the FFs?

Possible effect ofthe Yrast levels?

● Evaporation (equilibrium emission)?

B. During acceleration? (Less kinematic boost by the FFs)

● Non-equilibrium (non-adiabatic) emission? (Neutrons with E>20MeV: H.Märten & D.Seeliger, INDC(GDR)-17 - completely WRONG!)

C. At the moment of scission?

Fragment spin: <J> = 7 - 8

We can classify the problems relevant to prompt neutron emission as follows:

• Madland and Nix proposed to approximate the temperature distribution by a triangular distribution.

• Mawxellian and Watt formulas are characterized by a single temperature → This assumption is justified only when the cooling due to particle emission is small.

T1

T2

T

Max. temp.→Tm

P(T)

1( , )T

2( , )T

3( , )T

Evaporation: Sequential neutron emission from fragments with different temperatures and with different properties.

a) Temperature distribution:

Original M-N model:

a = A/CC = 8 – 11 (adjustable parameter)

LF HF

A/8

A/10

A/11

D.G.Madland & J.R.Nix, Nucl. Sci. Eng. 81, 213 (1982)

b) Nuclear properties of the FFs:

Nowadays, Ignatyuk’s superfluid model with consideration of the shell effects is known to give better description of the LDP, thus eliminating the adjustable parameter C.

Level density systematics

( ) ( )[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.

Level Density : Ignatyuk’s Superfluid Model

-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)

The LDP for FFs for different fragment excitation energies

At higher excitation energies, the structure due to shell effects become less pronounced but still persists.

A/1030MeV

10MeV

The LDP for neutron-rich nuclei

The shell effect persists for neutron-rich nuclei.

The LDP for FFs for each fission mode were calculated considering their exc. energy and neutron-richness .

50Z

82N

0 5 10 15 200

5

10

15

20

25

S2

U-235(n,f) Ein=thermal

Ratio a=Ignatyuk a=A/8

Rat

io

En (MeV)

S1

Effect of the Level Density Parameters on the Spectrum

● The high-energy part of the spectrum is sensitive to the LDP, but the low-energy part is not.● Ignatyuk’s LDP, without any artificial adjustment, gives good fit to the experimental data. 　 This proves that Ignatyuk’s LDP is physically adequate.

0 5 10 15 201E- 7

1E- 6

1E- 5

1E- 4

1E- 3

0.01

0.1

1

S1; a=A/ 8

S1; a=Ignatyuk

S2; a=A/ 8

No.

of

Neu

tron

s (M

eV)

En (MeV)

S2; a=Ignatyuk

U- 235(nth

,f )

Exp.(J ohansson)

70 80 90 100 110 120 130 140 150 160 1700

1

2

3

4

5

6

SL

S1

S2

U-235(nth,f)

Shape parameters:

T-S. Fan et al. Nucl. Phys. 591,161 (1995)

(A

)

A (u)

Nishio et al. (HWHF=6u)

80 100 120 140 1600

1

2

3

4

5

6

7

8

Total

SL

S2

S1

Knitter's parameters usedU-235(n,f) En=thermal

Fiss

ion

Yie

ld (%

)

Mass Number (u)

Average number of neutrons emitted from a fragment for each mode

)]()([1

)(

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

niHiHniLiLiHiL

ni

niHniLni

EEE

EEE

c) Asymmetry in ν(A)

• The number of neutrons emitted from the two FFs are not equal.• The CM-spectra of neutrons from Lf and HF are very different.

Weighted average should be takeninstead of the simple average.

HF LF

mHvH ＝ mLvL

1. Kinematic boost is greater for LF, leading to harder LF-spectra. 2. Higher inverse cross

section for HF enhances emission of slow neutrons.

CM LS

HF

LF

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

Neutron Spectra from LF and HF are very different

MULTIF1

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

Neutron spectra from three fission modes for U-235(nth,f)

Experimental datafor U-235, Ein = 0.4 - 0.5 MeV

0.1 1 100.4

0.6

0.8

1.0

1.2

1.4

Rat

io t

o M

axw

ell (T

=1.3

24 M

eV)

En

Adams(E

0 =0.52MeV)

J ohansson(E

0 =0.53MeV)

J ENDL- 3.3

Nefedov(E

0 =0.4MeV)

U- 235(n,f)

ENDF/ B- 7

0 2 4 6 8 10 120.4

0.6

0.8

1.0

1.2

1.4

Rat

io t

o M

axw

ell

(T=1.3

24 M

eV

)

En (MeV)

Adams(E

0 =0.52MeV)

J ohansson(E

0 =0.53MeV)

J ENDL- 3.3

Nefedov(E

0 =0.4MeV)

U- 235(n,f)

ENDF/ B- 7

• En<0.5 MeV ： Two kinds of exp. data　　　 Higher: Nefedov （ Starostov) [Obninsk]　　　 Lower: Johansson [Studsvik]

• En>2 MeV ： Two kinds of exp. data　　　 Higher: Adams [Harwell] Johansson [Studsvik]　　　 Lower: Nefedov （ Starostov) [Obninsk]

Possibilities:

1. The exp. data are not accurate enough, due to scattered neutrons and/or low detection efficiency.

2. Possible existence of scission neutrons.

3. Angular anisotropy in neutron emission in the CM-system of FF.

4. Neutron emission during acceleration (NEDA), instead of after full acceleration.

5. Possible effect of “yrast levels”.

● What are the reasons for the discrepancy in the region En < 0.5 MeV?

2. Possible Existence of Scission Neutrons

● C. Wagemans, “The Nuclear Fission Process” (1991): <νSCN>/<νp> = (1.1 ±0.3) % <ESCN> = (0.39 ±0.06) MeV for Cf-252(sf)

● O.I. Batenkov, M.V. Blinov et al., IAEA/CM on Phys. of Neutron Emission in Fission, Mito (1989), INDC(NDS)-220: <νSCN>/<νp> ~ 3 %

● N.V. Kornilov et al., ND2007, Nice <νSCN>/<νp> ~ 25 % <ESCN> = 2.08 MeV for U-235 at E0=0.5 MeV

● A.S.Vorobyev, O.A.Shcherbakov et al. NIM 598, 795 (2009): <νSCN>/<νp> < 5 % for U-235 at E0=thermal

■ Uncertainties are still large in the data of fraction and average energy of scission neutrons.

・ With an assumption of 3 % of SCN with nuclear temperature T=0.3 – 0.5 MeV, the spectrum still remains within uncertainty of the experimental data.

Multimodal Madland-Nix model + 3%-SCN

・ SCN fraction as big as 25% is not required to fit to the experimental data.・ Scission neutron is an interesting phenomenon from physics point of view, and further investigation is needed to identify its characteristics.・ But it should not be treated as a convenient tool for fitting.

5% of SCN is too much.

E = Ef + ε + 2(Ef ε)1/2cosθ

2( , , ) ( , )(1 cos ) /(1 / 3)c c b b

b=W(θ)/W(90º) – 1

2

2

2

2

2

2

2( )

( )

( )

2 ( ) 0

2( )

3 / 2 2 ( ) 0

( , )[1 ( ) / 4 ]1( , , )

4 (1 / 3)

1( ) ( ) exp( / )

2 (1 / 3)

( )( )( ) exp( / )

8 (1 / 3)

f

f

f m

f

f m

f

E E c f ff c E E

f

E E T

cE Ef m

E E Tc f

E Ef m

b E E EN E E d

E b

d k T T T dTE T b

E Ebd k T T T dT

E T b

3. Angular Anisotropy of Neutron Emission in the Fragment CM-system

The L-system energy

The angle-dependent neutron spectrum in the CM- system

the anisotropy parameter:

The L-system spectrum:

→ Numerical calculation

• The angular anisotropy of neutron emission enhances the low-energy (E<0.6 MeV) and high-energy (E>4 MeV) parts of the spectrum, and diminishes the intermediate part.• The anisotropy parameter b should be studied more in relation to the spin and energy of the FFs.

x

x

x

x

v

st

k 1

1ln

2

1

10

tePtP

1

0

21

*

31

102 T

B

n

n

eBE

A

4. Neutron Emission During Acceleration (NEDA)

・ Certain fraction of prompt neutrons may be emitted before full acceleration of FF [V.P. Eismont, Sov. At. Energy 19, 1000 (1965)]

●Time after scission t as a function of x = E/Ek, the ratio of the FF-KE relative to its final value Ek

●Decay of the FFs by neutron emission

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

[T. Ericson, Advances in Nuclear Physics 6, 425 (1960)]

0

0.5

1

1.0E- 22 1.0E- 21 1.0E- 20 1.0E- 19 1.0E- 18 1.0E- 17 1.0E- 16 1.0E- 15 1.0E- 14

E/Ek

or P(t

)

Time after scission [s]

分裂片の加速と中性子放出の競合

St.1 HF

St.1 LF

St.2 HF

St.2 LFFragment accelerationFragment acceleration

Decay by neutron emissionDecay by neutron emission

Fair chance of competition between FF acceleration and neutron emission for S2-mode.

Pu-239(nth,f)(doesn’t depend on fission mode)

(strongly depend on fission mode)

Pu-239(nth,f) Maxwell即発中性子スペクトル 分布との比

0.6

0.8

1

1.2

1.4

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

Neutron Energy (MeV)

Rat

io t

o M

axw

ellia

n

Full AccelerationDuring AccelerationD.Abramson(1977)A.Lajtai(1985)

Pu-239(nth,f)

Ratio

to M

axw

ellia

n (T

=1.4

2 M

eV)

Neutron Energy (MeV)

• NEDA effect enhances low-energy part of the spectrum.• The degree of enhancement depends on the neutron emission time.

Fragment angular momentum <J> = 7 - 8ħ

( 1)rotE I I

Yrast line : the lowest available states for a given angular momentum I. (No internal excitation below the line. It works something like the “ground state” for the angular momentum.)

I

E

x

Bn

γ(E2)

Initial excited states

n

n

n

n

n

High-energy neutron emission may be suppressed by the yrast level.

5. Effect of the Yrast level on the decay of a high-spin nucleus

This possibility should be examined byMonte Carlo simulation with considerationof angular momentum conservation.

Application of the Multimodal Madland-Nix Model to Bimodal Fission

Interests in Heaviest Actinides

1. Sudden switchover from asymmetric to symmetric fission

ー What will be the impact of the modal change on the PFNS?2. Recent interest in heavy actinides in astronuclearphysics ー Formation and fission of heavy actinides　　　　　 should be considered in the nucleosynthesis in supernovae.

1. Sudden switchover from asymmetric to symmetric fission

Fm-isotopes

N=158

Cf Es Fm Md No Lr Rf Ha

N=158

P.Moller et al., Nucl. Phys. A492, 349 (1989)

Deformation Energy Surface for Fm-258

Old path

New path leading to spherical fragments (SS)

New path leadingto elongated fragments (S1)

50%50%

Fission Recycling

r-process

2. Nucleosynthesis in Supernovae

Fission Recycling Effects on the Abundance of Elements in the Solar System

I. V. Panov et al., Astronomy Letters 34, 189 (2008)

With neutron multiplication andwith fission fragments

With neutron multiplication butwithout fission fragments

Note: Synthesis of elements in the universe is similar to nuclear transmutationin the reactor core, resulting in similar equations describing the processes.

Hulet et al., Phys. Rev. C40, 770 (1989)

S1 SS

SSS1

For Fm-258(sf), S1 : SS = 50% : 50%

Energy Balance in Fission• Total energy release is high (~250 MeV at max.) for Fm-isotopes.

• Coulomb repulsion energy (≈TKE) is also high due to higher fragment charge.

• SuperShort mode is restricted in a narrow region due to energetic reason.• For U-235, shell regions are separated; this favors asymmetric fission.

80 100 120 140 160 180

200

400

600

800

1000

1200

1400

Fiss

ion

Yie

ld (

arbi

trar

y un

it)

Fragment Mass (u)

Exp. data: van Aarle et al. (1998)

• For Fm-257, fission is governed by S1-mode, hence the PFNS is also characterized by the hard spectrum peculiar to this mode.

Results of Multimodal Madland-Nix Model Calculation for Fm (1)

• For Fm-258, the contributions of SS and S1 are 50-50. (van Aarle et al. 1998)

• However, since more neutrons are emitted from S1-mode, the PFNS is still dominated by S1-mode.

Results of Multimodal Madland-Nix Model Calculation for Fm (2)

• Sudden softening of the spectra for Fm-259 is due to complete switchover to the SuperShort mode (cold fission).• Thus, fission modal change causes a big change in PFNS.

Integral Test of Nuclear DataCore systems: 900 cases were selected out of ICSBEP (Int’l Criticality Safety Benchmark Evaluation Project) Handbook, including various reactor-types, fuels, spectra (thermal, fast, intermediate,…)Codes: • MVP (Continuous Energy Monte Carlo Calculation Code) was used to perform accurate calculation for systems with complicated core geometry • Sensitivity analysis codes CBG, MOSRA-SAGEP, SVS were used to extract information from the benchmark tests

Ref.) G. Chiba, K. Okumura, Benchmark Tests of Nuclear Data Files for Development of JENDL-4, JAEA-Research-089 (2008) O. Iwamoto et al., PHYSOR’08 (2008); T. Mori, Proc. of Nuclear Data Symposium (2005)

In JENDL-4,▪ No big change has been made for major actinides.・ For MA, Maslov’s evaluations in JENDL-3.3 were superseded by new evaluations by O.Iwamoto.

My interest in Thorium Fuel Utilization

1. Molten-Salt Reactor - As a MA Burner - As a Plutonium Burner

Reactor core

HEX

Proposed Concept ofA Small-sized [150 MWe] MSR “FUJI”

Pa-reservoir

U233

Pa233

2. Reactor Physics Experiments Using Thorium - KUCA (Kyoto University Critical Assembly)

• Verification of nuclear data (JENDL, ENDF/B) was done through k-eff, reaction rate distribution, neutron flux distribution.

• Central Test Zone (Th-232) + Surrounding Driver Zone (EU)

• Spectral Shift Experiments by changing moderator-to-fuel ratio

• Accelerator Driven System with Thorium-target (planned)

• Some problems have been found in the thermal and resonance capture cross sections. These problems have already been solved in JENDL-3.3.

Conversion ratio & Pa-233 pile-upin the U-233-fueled & Pu(6%Th)-fueled MSR cores

Without the reprocessing unit,the conversion ratio decreaseswith the burn-up time, levelingoff at an equilibrium value 0.73.

This is due to the pileup ofPa-233 in the core, which leadsto loss of Pa-233 by neutronabsorption.

In-core Accumulation of U-233, U-234 etc.

U-233-fueled core

Pu-fueled core233Pa(n,γ)234Pa→

1- From neutron economy point of view, a considerable fraction* of neutrons are lost by Pa-233(n,γ), rather than by Th-232(n,γ). This also reduces the neutron flux in the core region. (* ~1.9% at 131 days of burn-up)

2- From fuel economy point of view, capture by Pa-233 implies loss of Pa-233 that would otherwise convert into U-233.

Accumulation of Pa-233 in the thermal core is notdesirable, due to two reasons:

For Th-loaded hard-spectrum ADS, Pa-233 can fission to produce energy.

Summary

1. The Multimodal Madland-Nix Model provides a method of calculating the PFNS, based on the multimodal analysis of fission. Therefore it can be applied to evaluation of PFNS for actinides and transactinides as well as for exotic nuclides far from beta- stability line.

3. In order to finalize the remaining question of scission neutrons, and angular anisotropy of neutron emission, multi-parameter coincident measurement of angular and energy distributions of neutrons and FFs are required.

2. The methodology incorporates requirements from basic physics, such as energy conservation and shell effects, as well as other aspects of fission physics, such as mass and TKE distributions.

4. For Th-cycle nuclides, the quantity as well as quality of nuclear data is not so good as for U-Pu cycle. The author hopes some experimentalists will take on the work of measuring the PFNS, as well as mass & TKE distributions for Th-232, Pa-233 and U-233 with improved accuracy.

TXE = ER + Ein + Bn - TKE

• Mass and TKE distributions are consistently described as a superposition of components of a few fission modes of different properties.

• Therefore, naturally, the TXE of FFs are different for different fission modes.

• The shell correction energies demonstrate that the fission process is strongly governed by the fragment shell effects.

• Multimodal random-neck rupture (MM-RNR) model is, at present, the only model that can give consistent description of the 2D- distributions of mass and TKE of FFs.

Np-237(n,f)• Experimental measurements and analyses have been done for (n,f), (sf) and (p,f) for many nuclides over wide range of energy up to Ep=200 MeV.

Gorodisskiy et al., Ann. Nucl. Energy 35 (2008)

P. Siegler, F.-J. Hambsch et al. Nucl. Phys. A679 (2000)

(1) Multimodal Nature : Experimental Evidence TKE Distribution for Various Fragment Mass Intervals

R.Müller et al. Phys. Rev. C29,885 (1984)

S1

S2

SL

1 2 (MeV)(fm)

1.44Z Z

TKED

・ TKE distributions for a fixed mass interval are distinctively different, and represented well with Gaussian functions

・ Charge-center distance deduced from the most probable TKE:

U-235

A D(fm) mode 132 16.92 S1 140 17.37 S2 120 18.93 SL

Some Relevant Problems (1) 　

●Single-modal approach: Madland & Nix, NSE 81(1982): Typical LF & HF

How to consider neutrons emitted from numerous fragments of diverse excitation energy and of different nuclear properties?

●Multimodal approach: Ohsawa et al., Nucl. Phys. A653 (1999); A665 (2000) ▪ 3 - 5 fission modes (Standard-1, -2, -3, Superlong, Super-asym.)

▪ Consider “average fragments” for each mode

The physical properties were determined by seven-point approximation .

( )( ) ( ) ( ) ( , ( ), ( , ), ( , ))f C m

A Ztot

AN E Y A P Z N E E A Z A T Z A

• Madland, “Refined LA-model”, INDC(NDS)-220 (1989), p.201

LS-spectrum:

considering 28 fission fragments (Z,A)

Results: Better agreement with experiments.

●Point-by-point approach

• Märten & Seeliger, NSE, 93 (1986): “Generalized M-N model” TM, σinv(E): different for different A

• Tudora, Ann. Nucl. Energy, 36 (2009) “Point-by-point approach is the most accurate because it takes into account the full range of possible fragmentations.”

However, considering that percent contributionof different modes varies according to fragment mass region,simple point-by-point approach mixesup different modes of fragments of different properties.

If necessary, “Modal point-by-point approach”,which considers many fragment pairs of the same mode, would be a reasonable choice.

Modal point-by-point approachFor mass pairs (~110,~126), S2-mode prevails.

Refinements in the Methodology

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 in ν(A): νL ≠ νH

/( )i iL iL iH iH iL iH

(4) Asymmetry in T : T L ≠ TH

because of difference in deformation

Strong energy dependence inthe neutron transmission coefficients.

However, • The Maxwellian and Watt formula completely ignore the energy dependence. • Here is another reason for essential defect in these formulas.

Inevitably, this brings aboutstrong influence on the energy spectrum of the emitted neutrons.

c) Neutron Transmission Coefficients:

3. νL = νH ?

Experiments give the answer:

Experimental data: U-235(n,f) Np-237(n,f)

E0 νL νH νL νH

0.5 MeV 1.44 1.02 1.59 1.14

5.5 MeV 1.43 1.71 1.59 1.87 Müller et al. Naqvi et al.

Phys. Rev. C29,885 (1984) Phys. Rev. C34, 218 (1986)

● Kornilov described: “The yields of neutrons from the LF and HF differ insignificantly from the mean value (within 10%).” (N.V.Kornilov et al., Phys. At. Nucl. 62(2) 173 (1999) )

U-235(n,f) Müller et al.(1984) Np-237(n,f) Naqvi et al.(1986)

• The increase in νtot is totally accounted for by νH(A) alone.• This is explained as due to diminishing of the shell effects at higher excitation energy.

U-235(n,f) Np-237(n,f)

E0 νL νH νL νH

0.5 MeV 1.44 1.02 1.59 1.14

5.5 MeV 1.43 1.71 1.59 1.87

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

HF

LF

• σ(HF) ＞ σ(LF), E<0.1MeV• This is consistent with the s-wave strength function (shown below).• Low-energy neutrons are more easily emitted from HFs than from LFs. • This softens the spectrum from HFs.

Inverse reaction cross sections for LF and HF

2d5/23p3/2

c) Inverse Reaction Cross Sections:

S-wave strength function

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

Effects of Asymmetry in ν for LF and HF

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.]

N Z

Sub-shell region (86N) : responsible for S2-mode (leading to more deformed pre-scission shapes)

Shell-corrections as functions of deformation and neutron- and proton-numbers

Scission-Point model calculation by B. D. Wilkins et al., Phys. Rev. C14, 76 (1976)

50N 82N 50P

Major shell(spherical)regions

β

E*CN=Bn+En E*

L=aLT2L E*

H=aHT2H

E*L= Eint L + DL

E*H= Eint H + DH

TXE = <ER> + Bn + En ー TKE = aCNT2m = aLT2

L + aHT2H

= (aLRT2 + aH)T2

H where RT=TL/TH : temperature ratio

2T CN CN

L H2 2H T L H T L

,m m

R a aT T T T

a R a a R a

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

(4) Asymmetry in the Nuclear Temperature

・ T. Ohsawa, INDC(NDS)-251 (1991), IAEA/CM ,Vienna, 1990. p.71.・ T. Ohsawa and T. Shibata, Nucl. Data for Sci. Tech., Jülich, p.965 (1992)・ P. Talou, Proc. ND2007, Nice (2008)