Fission Dynamicsbased on Langevin Equations

Y. Aritomo1, Y. Miyamoto1, S. Tanaka1, M. Ohta2, A. Iwamoto3, K. Nishio3

1Faculty of Science and Engineering, Kindai University, Osaka, Japan2Hirao School, Konan University, Kobe, Japan

3Advanced Science Research Center, Japan Atomic Energy Agency, Ibaraki, Japan

Nuclear Fission Dynamics 2019

YITP WORK SHOP

26 October – 08 November2019, YITP, Kyoto, Japan

V.V. Pashkevich (BLTP, Dubna, Russia) 1998

Cassini ovaloids

R(x) = R0(1 + n n Pn(x))

Motivation 1

PES calculation: symmetrical and two asymmetrical fission valleys

n=1~20 minimization 極小化

Nuclei feel

such sensitive potential energy

during Fission process ?

Fission path

always passes through

Bottom of potential energy surface?

Fission

Potential landscape

Static analysis

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

z

d

298Fl

Smoluchowskiequation

( )

friction reduced ;

mass inertia ;

ondistributiy probabilit ; ;,

tqP probability distribution

( )( )

( ) ( )tqPq

TtqP

q

tqV

qtqP

t;,;,

;,1;,

2

2

+

=

What happens

during Dynamical process

Motivation 2

c.m. distance c.m distance

two-center parametrization

(Maruhn and Greiner,

Z. Phys. 251(1972) 431)

),,( dz

Nuclear Shape

(δ1=δ2 )

( , , )q z d

Radial of compound nucleus

A1 + A2

0.0

0.5

1.0

1.5

2.0

0

20

40

-0.5

0.0

0.5

z Corresponds to c.m. distance

Deformation of fragments

Mass asymmetry

𝑧 =𝑧0𝑅

3 − 2𝛿

3 + 𝛿

oblate

prolate

Nuclear shape on z-δ plane

a > b

a < b

Y. Miyamoto (Kindai)

Asy.Sym.

Spontaneous Fission

In Langevin calculation, the excitation energy is necessary

254Fm 258Fm

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

z

d

246Fm = 0

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

z

d

264Fm = 0

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

z

d

246Fm = 0

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

z

d

264Fm = 0

264Fm α=0246Fm α=0

Asy.

Sym.

Direction

Fission process

Cal.

A.V. Karpov,

P.N. Nadtochy,

D.V. Vanin, and G.D. Adeev,

PRC 63 (2001) 054610

elongation

Fission process

Langevin Dynamics in 4-dimensional Model of Nucleus-

Nucleus Collisions J. Blocki, O. Mazonka, J. Wilczynski, Z. Sosin, and A. Wieloch

Acta Physica Polonica B 31 1513 (2000)

elongation

Fission process

2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

d

z0

Distance of two center (fm)

z0 independents on δ

Fusion-Fission process

V. Zagrebaev and W. Greiner

J. Phys. G. 31, 825-844 (2005)

With non-diagonal parts

γzδ γzα γδα

mzδ mzα mδα

( )

( ) ( ) )(2

1 11

1

tRgpmppmqq

V

dt

dp

pmdt

dq

jijkjkijkjjk

ii

i

jiji

+−

−

−=

=

−−

−

Key parameters on trajectory analysis

Langevin Equation

Without non-diagonal parts

γzδ γzα γδα

mzδ mzα mδα

Transport coefficients behavior change（Smolcouski Eq）

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

z

298Fl

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

z

d

298Fl

298Fl LDM

without Random force with non-diagonal

----- without non-diagonal

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

非対角成分あり 非対角成分なし

z

298Fl

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

非対角成分あり 非対角成分なし

z

d

298Fl

Neutron multiplicity

0.35

Neutron multiplicity

0.23

with non-diagonal

----- without non-diagonal

298Fl LDM

without Random force

1. Questions on the calculation for fission processFission path always passes through

bottom of potential energy surface?→ not always necessarily say

Dynamical aspects of Fission→ non-diagonal term of transport coefficients is important

2. Behaviors of trajectory are affected by non-diagonal terms of transport coefficients

mass distribution of fission fragmentstime scale of fission process

3. Eigenvalues of transport coefficients are important

Summary

Elucidation of Fission Mechanism in

Super-Heavy Element Region using

Dynamical ModelInternational molecular program "Nuclear Fission Dynamics“

- YITP2019 Work-shop Kyoto, Japan –

Yuuya Miyamoto1*, Yoshihiro Aritomo1, Shoya Tanaka1 ,Shoma Ishizaki1

Kentaro Hirose2, and Katsuhisa Nishio2

1Graduate School of Science and Engineering Research, Kindai

University Higashiosaka 577-8502, Japan2Advanced Science Research Center, Japan Atomic Energy Agency

(JAEA), Tokai, Ibaraki 319-1195, Japan

Outline

I. Background

II. Theory and Method

III. Results and Discussion

IV. Summary and conclusion

V. Future outlook on research

The Sudden Change of The FFMDs in Fermium Region

Experimental Results were Measured about 40 Years Ago

核分裂片の収率

(%)

核分裂片の質量数(u)

259Fm

Compact mass

symmetric fission

Elongated mass

asymmetric

fission like U

129Sn

110Tc 144La

142Cs114Rh

140Xe117Pd

Yie

ld (

%)

Fragment mass (u)

D.C. Hoffman et al.,

Phys. Rev. C, 21, 1980 (637).

(*FFMDs : Fission Fragment Mass Distributions )

Num

ber

of

neu

trons

Number of protons

258Fm

257Fm

256Fm

254Fm

Sudden change due

to adding one neutron

258Fm

259Md

101

258No

104

100

260Rf

101

260Md

102

TKETotal Kinetic Energy

Fission Fragment

Mass Distribution

(FFMDs)

Spontaneous fission in the mass region Mendelevium and bimodal-fission Exp. results - 260Md fission fragment -

J.F. Wild et al., Phys.

Rev. C, 41, 649 (1990)

Asymmetric

& symmetric

fission Symmetric

fission

260Md

260Md

Double hump

Distribution (TKE)

Langevin Calculation and Shape Parameterization

z : Elongation

δ : Deformation

α : Mass asymmetry

ε : Neck parameter

q(z, δ1,δ2 α, ε)

( )

( ) ( ) )(2

1 11

1

tRgpmppmqq

V

dt

dp

pmdt

dq

jijkjkijkjjk

ii

i

jiji

+−

−

−=

=

−−

−

Friction Random force

dissipation fluctuationY. Aritomo and M. Ohta, Nucl. Phys. A 744, 3-14 (2004)

δ1 = δ2

ε = const.

γij : Wall and Window dissipation (Friction)mij : Hydrodynamical mass (Inertia)

Definition of the neck parameter

r

a2a1

r R=0.75 (1+ )0 d2/3 r

z =z1 2=0r r r

z

z2a1 a2

b1

b2

a1 | |z1z2 a2

b2

|zmax1

| zmin2

| |z1

E0

E

e= /E0E

z

V V V

V0

b1

b1 b2

( )a ( )b ( )c

Neck parameter ε: ratio of smoothed potential height to the original

one where two harmonic oscillator potential cross each other.

1.00

0.00

ε val

ue

Example of the nuclear shapes in two-center parametrization and

The corresponding potentials V(Z) shown for δ1=δ2=0.5.

The mass Asymmetry α=0.0 for (a) and α=0.625 for (b) and (c)(z, δ, α) = const.

236 238 240 242 244 246 248 250 252 254 256 258 2600.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Formula Y=A*X+B

Intercept -1.93846

Slope 0.01007

Cf-

251

C

f-252

Cf-

253

C

f-254

Bk

-24

9

B

k-2

50

Bk

-25

1

B

k-2

52

Cm

-24

7

Cm

-24

8

C

m-2

49

C

m-2

50

Pu

-24

1

Pu

-24

2

P

u-2

43

Np

-239

Np

-240

N

p-2

41

N

p-2

42

U-2

37

U-2

38

U

-239

U

-240

U

Np

Pu

Cm

Bk

Cf

Mass (u)

e par

amet

er

0.0

5.0

0.0

5.0

0.0

5.0

0.0

5.0

0.0

5.0

100 1500.0

5.0

100 150 100 150 100 150

Exp. e = 0.35 e = 0.45 e = 0.55 e = 0.65

Np-242Np-240Np-239

U-237 U-238 U-239 U-240

Bk-249 Bk-250 Bk-252

Cf-254Cf-251

Np-241

Pu-241 Pu-242 Pu-243

Cm-247 Cm-248 Cm-249 Cm-250

Cf-252 Cf-253

Bk-251

Mass (u)

Yie

ld (

%)

ε neck parameter 0.0 1.0

χ2 =σ1𝑁 𝑌𝑒𝑥 − 𝑌𝑐𝑎𝑙

𝜎𝑒𝑥

2

𝑁

χ2 = 𝐶휀2 + 𝐵휀 + 𝐴

Fission Fragment Mass Distribution in the Range of Excitation Energy Ex = 10-20MeV

0.01

0.1

1

10

0.01

0.1

1

10

100 130 1600.01

0.1

1

10

100 130 160

260Fm

258Fm

256Fm250Fm

254Fm

252Fm

Yie

ld (

%)

Fragment mass (u)

Origin of dramatic changes in fission modes in the

fermium region. Ex = 7.0MeV

0.01

0.1

1

10

0.01

0.1

1

10

190 220 2500.01

0.1

1

10

190 220 250

260Fm

258Fm

256Fm

254Fm

250Fm

252Fm

Pro

babil

ity (

%)

Total kinetic energy (MeV)

-0 .5

0 .0

0.5

1.0

1.5

2.0

2.5

3.0 4060

80100

120140

160180

200220

- 4 0- 3 0- 2 0- 1 0

0

1 0

2 0

3 0

4 0

Fragment mass (u)

Ener

gy (M

eV)

Elongation Z

-0 .50 .0

0 .51 .0

1 .52 .0

2.53.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

- 1 0 0

-5 0

0

50

100

Elon

gatio

n Z

Deformation δ

Ene

rgy

(MeV

)

-0 .50 .0

0 .51 .0

1 .52 .0

2.53.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

- 1 0 0

-5 0

0

50

100

Elon

gatio

n Z

Deformation δ

Ene

rgy

(MeV

)

-0 .5

0 .0

0.5

1.0

1.5

2.0

2.5

3.0 4060

80100

120140

160180

200220

- 4 0- 3 0- 2 0- 1 0

0

1 0

2 0

3 0

4 0

Fragment mass (u)

Ener

gy (M

eV)

Elongation Z

Fm-258

α=-0.08

Fm-258

α=0.00

Scission

1st Min

2nd Min

Scission

1st Min.

2nd Min.

Fm-254

α=0.16

Scission

1st Min.

2nd Min.

Fm-254

α=0.00

Scission

1st Min

2nd Min

254Fm (Asymmetric) and 258Fm (Symmetric) fission time Ex = 7.0MeV

0123

-0.20.00.20.40.6

10-22 10-21 10-20 10-19 10-18

-0.4-0.20.00.20.4

254Fm 258FmZd

Time (s)

Scission Scission

AA’

B

C

The average fission time of 254Fm is

80 times longer than that of 258Fm.

SF half-life of neutron-rich Fm isotopes

J. Rundrup et al., PRC 13 (1976) 229.

A. Staszczak et al., Phys. Rev. C, 80, 014309 (2009).

T. Ichikawa et al. Phys. Rev. C 79, 014305 (2009).

I. Neck parameters of heavier actinide nucleus

We found that ε=0.35 cannot reproduce the FFMDs

of heavier actinide nucleus. The neck parameter is

systematically increased with the mass of the

fissioning nuclides to facilitate the neck configuration.

II. Dramatic change of fission mode in fermium

The dramatic change from an elongated asymmetric

fission to compact symmetric fission is strongly

regulated by the structure of the second fission barrier

and the dynamical motion of the nucleus.

Summery