Reaction mechanism of fusion-fission process in superheavy mass region

Y. Aritomo1, K. Hagino2, K. Nishio1, and S. Chiba1

1Japan Atomic Energy Agency, Tokai, Japan2 Department of Physics, Tohoku University, Sendai, Japan

「微視的核反応理論による物理」YITP, Kyoto, Japan, 1-3 August, 2011

1. ModelCoupled-channels method + Dynamical Langevin calculation

Trajectory analysis Langevin equationTwo center shell model

2. Results36S+238U and 30Si+238U

Mass distribution of Fission fragmentsCapture Cross-sectionFusion Cross-section

3. Mechanism of Dynamical processPotential energy surface on scission lineAnalysis of trajectory behaviorAnalysis of probability distribution

4. Summary

36S + 238U30Si + 238U

Exp. by

K. Nishio

et al.

Zcn=106 Zcn=108

Phenomenalism

Dynamical Model based on Fluctuation-dissipation theory

(Langevin eq, Fokker-Plank eq, etc) Classical trajectory analysis

We can obtain…. Fission, Synthesis of SHE

Mass and TKE distribution of fission fragments ACN : 200~300

Neutron multiplicity

Charge distribution

Cross section (capture, mass symmetric fission, fusion)

Angle of ejected particle, Kinetic energy loss ( two body)

Conditions

Nuclear shape parameter

Potential energy surface (LDM, shell correction energy, LS force)

Transport coefficients (friction, inertia mass) Liner Response Theory

Dynamical equation (memory effect, Einstein relation)

What we can obtain under the conditions

1. Model

1. 1-1. Potential

2. 1-2. Dynamical Equation

1-3. Simulation of Experiment and Cross Sections

Estimation of cross sections

Capture Cross Section

Fusion Cross Section

Coupled-channel method

Dynamical calculation

Langevin eq.

CN

Formation probability PCN

計算処方の概観

Time-evolution of nuclear shape

in fusion-fission process

1. Potential energy surface

2. Trajectory described by

equations

Model: Outlook of calculation methods

two-center parametrization

(Maruhn and Greiner,

Z. Phys. 251(1972) 431)

),,( z

Nuclear shape

δ=0

(δ1=δ2)

Trajectory which enters

into the spherical region

= fusion trajectory

( , , )q z

2

0

( 1)( , , ) ( ) ( , )

2 ( )

( ) ( ) ( )

( , ) ( ) ( )

DM SH

DM S C

SH shell

V q T V q V q TI q

V q E q E q

V q T E q T

T : nuclear temperature

E* =aT2 a : level density parameter

Toke and Swiatecki

ES : Generalized surface energy (finite range effect)

EC : Coulomb repulsion for diffused surface

E0shell : Shell correction energy at T=0

I : Moment of inertia for rigid body

Φ(T) : Temperature dependent factor

MeV 20

exp)(2

d

d

E

E

aTT

Potential Energy

qi : deformation coordinate （nuclear shape）

two-center parametrization (Maruhn and Greiner, Z. Phys. 251(1972) 431)

pi : momentum

mij : Hydrodynamical mass (inertia mass）

γij: Wall and Window (one-body) dissipation （friction ）

)(2

1 11

1

tRgpmppmqq

V

dt

dp

pmdt

dq

jijkjkijkjjk

ii

i

jiji

ijjk

k

ik

ijjii

Tgg

tttRtRtR

process) (Markovian noise white:)(2)()( ,0)( 2121

),,( z

)(2

1 1*

int qVppmEE jiij

energy excitation : energy, intrinsic : *

int EE

多次元ランジュバン方程式Multi-dimensional Langevin Equation

Newton equationFriction Random force

dissipation fluctuation

Two Center Shell Model

2

22

LS L0

ˆ ( ) ( ) ( )2

H V V Vm

r r p s r p

r

a2a1

r R=0.75 (1+ )0 2/3 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

J. Maruhn and W. Greiner, Z. Phys, 1972

2 2 2 2

1 1 1

2 2 2 2 2 2

1 1 1 1 1

1

2 2 2 2 2 20 2 2 2 2 2

2

2 2 2 2

2 2 2

1 1

01

( ) 1 12

0

z

z

z

z

z z z

z c z d z g z

z z

V z m z c z d z g z

z z

z z z

r

r

r

r

r

r

r r

r

1

2

0

0

z z zz

z z z

Neck parameter is the ratio of smoothed potential

height to the original one where two harmonic

oscillator potential cross each other

ε = 1.0

0.35

0.0

224Th

Time dependent adiabatic fusion-fission potential

Time-dependent

weight function

V. Zagrebaev, A. Karpov, Y. Aritomo, M. Naumenko and W. Greiner, Phys. Part. Nucl. 38 (2007) 469

0.4

0.8 3.31.8 2.81.3 2.3 3.8

0.2

0.6

0.8

1.0

r/R0n

eck p

ara

me

ter

()e

2. Results

2-2.

1. Capture and Fusion Cross sections

2. Mass distribution of Fission Fragments

34, 36S+238U and 30Si+238U 2-2.

Touching probability CC method

Perform a trajectory calculation

starting from the touching distance between target and projectile

to the end each process.

Fusion box and Sample trajectory 36S+238U

E*= 40 MeV

L = 0

θ = 0

z-α

δ=0

z-δ

α=0

QF

FFFF?

QF

FF

DQF

Results 36S+238U MDFF and Cross sections

FF

process

Results 30Si+238U MDFF and Cross sections

3. Mechanism of Dynamical process

30Si+238U 36S+238U

200u74u

137u

178u90u

134u

Clarification of the mechanism of Fusion-fission process

MDFF at Low incident energy

0.4

0.8 3.31.8 2.81.3 2.3 3.8

0.2

0.6

0.8

1.0

r/R0

neck p

ara

me

ter

()e

(a) 1-dim Potential energy on scission line

30Si+238U

36S+238U

(b) Trajectory Analysis on Potential Energy Surface z-A plane

E* = 35.5 MeV

L=0, θ=0

30Si+238U

δ=0.22, ε=1.0

δ=0.22, ε=0.35

δ=0.22, ε=1.0

E* = 39.5 MeV

L=0, θ=0

36S+238U

Trajectory Analysis on Potential Energy Surface 30Si+238U

E* =

35.5 MeV

L=0, θ=0

Trajectory Analysis on Potential Energy Surface 36S+238U

E* =

39.5 MeV

L=0, θ=0

0.8 1.0 1.2 1.4 1.60.48

0.60

0.72

0.84

B

z0.8 1.0 1.2 1.4 1.6

0.48

0.60

0.72

0.84

B

z0.8 1.0 1.2 1.4 1.6

0.48

0.60

0.72

0.84

B

z

Trajectory Analysis using ALL trajectories

~40,000 points

(c) Trajectory Analysis “Probability Distribution”

E* =

35.5 MeV

L=0, θ=0

E* =

39.5 MeV

L=0, θ=0

3. Summary

1. In order to analyze the fusion-fission process in superheavy mass region, we apply the Couple channels method + Langevin calculation.

2. Incident energy dependence of mass distribution of fission fragments (MDFF) is reproduced in reaction 36S+238U and 30Si+238U.

3. The shape of the MDFF is analyzed using (a) 1-dim potential energy surface on the scission line (b) sample trajectory on the potential energy surface (c) probability distribution

4. The relation between the touching point and the ridge line is very importantto decide the process fusion hindrance

leading to synthesize SHE

4. Summary