Fusion-Fission Dynamics for Super-Heavy Elements

Bülent Yılmaz1,2 and David Boilley1,3

• Fission of Atomic Nuclei• Super-Heavy Elements (SHE)• Measurement of fission time of SHE @ GANIL• Kewpie2: A cascade code• Fission time of SHE: isomeric structure effects• Conclusions

Advances in Nuclear Physics, Istanbul, 01/07/2008

1 GANIL,Caen,France 2 Ankara U.,Ankara,Turkey 3 Caen U.,Caen,France

Symmetric fission process

Surface tension vs. Coulomb repulsion

q

V(q)

scission

B f

saddle

initial state

LDM

fission time

E ?

Advances in Nuclear Physics, Istanbul, 01/07/2008

q

V(q)

scission

B f

saddle

initial state

LDM

fission time

E ?

Symmetric fission process

Advances in Nuclear Physics, Istanbul, 01/07/2008

1- Statistical Models:

2- Dynamical Models:

¡ f =! gs

2¼

0

@

s

1+µ

¯2! sd

¶2

¡¯

2! sd

1

A e¡ B f =T

diffusion over the fission barrier by Langevin or Fokker-Planck equations

(q,p) space

Symmetric fission process

q

V(q)

scission

B f

saddleinitial state

E ?

Advances in Nuclear Physics, Istanbul, 01/07/2008

sample stochastic events

neutron

gamma

proton

alpha

Deexcitation Scheme of Hot Compound Nucleus

...

Evaporation Residue

Fissio

n Fra

gmen

ts Fission Fragments

Advances in Nuclear Physics, Istanbul, 01/07/2008

Super-Heavy Elements

• Heaviest nuclei

• Synthesis by heavy-ion fusion reactions

• Existance of critical initial center of mass energy

• Low production cross section (picobarn)

• No liquid drop potential barrier

• Stability by shell correction energies

n

1 1,A Z

2 2,A Z

1A

2A 1 2A A

1 2Z Z

2A 2A

1A 1A

Reaction mechanism leading to SHE

Advances in Nuclear Physics, Istanbul, 01/07/2008

Island of stability predicted due to shell closure

Nuclear Chart

Z=114

N=

184

Z=120Z=124

?

Advances in Nuclear Physics, Istanbul, 01/07/2008

long fission times measured with crystal blocking technique

Measurement of fission time of SHE @ GANIL

M. Morjean et al, EPJD 45, 27 (2007).

238U + Ge @ 6.1 MeV/u

%10 of the capture events has tf > 10¡ 18s

A. Drouart et al, AIP Conf. Proc. 1005, 215 (2008).

238U + Ni @ 6.6 MeV/u

with Z = 124 MeVE ? = 70

208Pb + Ge @ 6.2 MeV/u

(> 10¡ 18s)

D. Jacquet et al, AIP Conf. Proc. 853, 239 (2006).

with E ? = 67 MeVZ = 120

long fission times observed

no hint of long fission time

Z = 114

New probe into SHE stability: M. Morjean et al, accepted to PRL Advances in Nuclear Physics, Istanbul, 01/07/2008

Kewpie2: A Cascade Code

dP0

dt = ¡ ¡ t;0P0

dP1

dt = ¡ n;0P0 ¡ ¡ t;1P1

.

.

.dPk

dt = ¡ n;k¡ 1Pk¡ 1 ¡ ¡ t;kPk

Pk ! A ¡ kZXPopulations:

+

Bohr-Wheeler fission rate with Strutinsky and Kramers corrections

Weisskopf neutron rate

Mean Fission Time

Kewpie2: A. Marchix, PhD Thesis (2007). Kewpie: B. Bouriquet, Comp. Phys. Com. 159, 1 (2004).

Fission Time Distribution

Bateman equations

¡ t;k = ¡ f ;k +¡ n;k

tf =1

Ptot(0) ¡ Ptot(1 )

Z 1

0tµ

¡dPtot

dt

¶dt

=km a xX

k=0

Z 1

0t¡ f ;kPk(t)dt

Advances in Nuclear Physics, Istanbul, 01/07/2008

Kewpie2: A Cascade Code

How can we reach 10% of long-fission events?

with a constant potential

Bf

P (tf > 10¡ 18s) = 0:3% atat

atatP (tf > 10¡ 18s) = 84%

Bf = Bn=2= 3

Bf = Bn = 6

MeV

MeV

Bf ' j¢ Eshell j

Advances in Nuclear Physics, Istanbul, 01/07/2008

Using Moller and Nix shell correction energies, the statistics of the long fission times calculated by Kewpie2 are far smaller than what is observed experimentally for Z=120 and Z=124 nuclei.

Kewpie2: A Cascade Code

How can we reach 10% of long-fission events?

with a damped potential according to Ignatyuk’s presription

Ed = 18:5 MeV

Bf (E ?) ' j¢ Eshell j expµ

¡E ?

Ed

¶ ¢ Eshell = 12 MeV

MeV

MeV

up to 5 isotopes

with arbitrarily fixed shell correction energies along the deexitation chain

up to next 2 isotopes¢ Eshell = 10

¢ Eshell = 7 for the others

P (tf > 10¡ 18s) = 9%

Advances in Nuclear Physics, Istanbul, 01/07/2008

Isomeric Structure

Are the observed long fission times due to high fission barriers or/and isomeric potential structures?

some possible double-bump (isomeric) shapes

single-bump shape

Advances in Nuclear Physics, Istanbul, 01/07/2008

B f

In order to understand the consequences of the potential structure beyond the saddle point two potential shapes has been considered.

optimizes fission time

Isomeric Structure

some possible double-bump (isomeric) shapes

single-bump shape

t(2)f ¼3£ t(1)

fno evaporation Advances in Nuclear Physics, Istanbul, 01/07/2008

Langevin Equations for the deformation coordinate

_q(t) = p=M

h²(t)²(t0)i = 2M ¯T±(t ¡ t0)

_p(t) = ¡ V0(q) ¡ ¯p+²(t)

h²(t)i = 0 +

Dynamical Model

Monte-Carlo neutron evaporation scheme using Weisskopf rate formula

qBq0 qS qB qSq0

scission

saddle saddle

scission

initial position initial

position

double-bump potentialsingle-bump potential

¯ = 2£ 1021s-1 E ? = 70MeV M = m0A=4 ~! = 1MeV Bn = 6MeV

Advances in Nuclear Physics, Istanbul, 01/07/2008

fission time for Z=124-like nuclei

PreliminaryPreliminary

Advances in Nuclear Physics, Istanbul, 01/07/2008

fission time for Z=124-like nuclei

E ? = 0

E ? = 5MeV

E ? = 20MeV

E ? = 2MeV

¢ Eshell = 10MeV

double-bump potential single-bump potential

¢ E shell

B f (E ?) = V1

·1¡ exp

µ¡

E ?

Ed

¶¸+ V2 exp

µ¡

E ?

Ed

¶

V2

V1

Advances in Nuclear Physics, Istanbul, 01/07/2008

PreliminaryPreliminary

fission time for Z=124-like nuclei

Advances in Nuclear Physics, Istanbul, 01/07/2008

Conclusions

• Very large barriers are still necessary to explain the long fission time

Thank You

• In future, if we have more precise measurement of fission time and distribution we cannot forget about the isomeric states.• This study will be continued with more realistic model...

Advances in Nuclear Physics, Istanbul, 01/07/2008