Dynamical approach to synthesis of superheavy elements

Y. Aritomo

Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology, Tokyo, Japan Flerov Laboratory of Nuclear Reactions, Dubna, Russia

the RIBF Nuclear Physics Seminars 169th

29th October, 2013

Fl

1. Introduction Superheavy Elements and Theoretical approaches 2. Model Dynamical model with Langevin equation Two center shell model

3. Results Evaporation residue cross section Mass distribution of Fission fragments

4. The way to synthesize new Superheavy elements by secondary beam

5. Summary

Contains

Lv Fl May 2012 IUPAC

104 105 106 107 108 109 110 111 112

flerovium livermorium

Periodic Table

Менделеев(1834-1907)

1869

Super Heavy Elements less stable

1. Introduction Nuclear Chart and Stability of Nuclei

Our Interests

・ Next magic number Z=82, N=126

・ Verification of ‘Island of Stability’

（predicted by macroscopic-microscopic

model in 1960’s）

・ Synthesis of new elements

Yu.Ts.Oganessian

0

5

10

Po

ten

tial

energ

y

(in M

eV)

Deformation

0.3.10 s.-6

1016

92U

years

Z=108

22 years later ....

0

5

10

92U

98Cf

Pote

nti

al e

ner

gy (i

n M

eV)

Deformation

Liquid DropModel

Z=108

1016

years

G. Flerov and K. Petrjzak

Leningrad 1940

N. Bohr and J.A. Wheeler (1939) Mayer and Jensen (1949) Magic numbers

Microscopic Theory

Models:

Macro-microscopic

Hartry-Fock-Bogolubov

Relativistic-mean-field

268106162

298114184

Quadrupole deformation β2

LD

Pote

ntial energ

y /

MeV

LD + shell

25098152

Fission barrier of Superheavy Elements

Shell correction energies in the

macroscopic-microscopic model

5

2

22 50.883 1 1.7826

C

S

E Z Ax

E N Z A

fissility parameter

Spherical nucleus

Surface energy

Coulomb energy

2 2 34SE r A

2 2

1 3

0

3

5C

e ZE

r A

Stability of Superheavy nuclei

Ys.Ts. Oganessian, Yu.A. Lazarev Treatise on Heavy-Ion Science vol.4 (1985)

1020

Z=107 104

102

LDM LDM

10

Experimental setup for synthesis of SHE

Lab Country City Accelerator Separator

FLNR Russia Dubna U400 U400M

DGFRS VASSILISSA

GSI Germany Darmstadt UNILAC SHIP TASCA

RIKEN Japan Wako RILAC GALIS

LBNL USA Berkeley 88-inch Cyclotron BGS

GANIL France Caen SPIRAL2's LINAC accelerator

S3 (Super Separator Spectrometer)

G.N. Flerov

(1913 -1990)

Yu.Ts. Oganessian

(1933-)

P. Armbruster

(1931-)

S. Hofmann

(1943-)

G. Muenzenberg

(1940-)

K. Morita

(1957-)

Fusion process in Superheavy mass region

FUSION

TRANSFER, QUASI-FISSION

Nuclear Molecule

Compound

Nucleus (CN)

Evaporation

Residue (ER)

FUSION-FISSION

Fission

Fragments 90~99%

„Cold“ and „Hot“ Fusion Reactions

Cold Fusion → doubly magic target nuclei: Pb, Bi;

E*(CN) = 10 – 20 MeV; evaporation of 1 – 2 neutrons;

up to now successful for Z ≤ 113

Hot Fusion → actinide targets (U, Cm, …) and 48Ca projectiles;

E*(CN) = 30 – 40 MeV; evaporation of 3 – 4 neutrons;

up to now successful for Z ≤ 118

reaction Q-value large

reaction Q-value small

Cold fusion reaction Hot fusion reaction 1994

110 Ds 62Ni + 208Pb 269110 + n (GSI)

111 Rg 64Ni + 209Bi 272111 + n (GSI)

1996

112 Cn 70Zn + 208Pb 277112 + n (GSI) named in Feb. 2010

1999

114 Fl 48Ca + 244Pu 292114 + 3n (FLNR) named in May. 2012

2000

116 Lv 48Ca + 248Cm 292116 + 4n (FLNR) named in May. 2012

2002

118 48Ca + 249Cf 294118 + 3n (FLNR)

2003

115 48Ca + 243Am 288115 + 3n 284113 + α (FLNR)

2004

113 70Zn + 209Bi 278113 + n (RIKEN)

2010

117 48Ca + 249Bk 294,293117 + 3-4n (FLNR)

Synthesis of New Elements Reports of new elements

2012

119 50Ti + 249Bk 296,295119 + 3-4n (GSI- TASCA)

20

Experimental data

Pb target

Actinide target

Evaporation residue cross sections

40

2. Model

1. 2-1. Estimation of cross sections

2. 2-2. Dynamical Equation

2* *

00

(2 1) ( , ) ( , ) ( , )2

ER cm CN

cm

T E P E W EE

10-22

10-20

10-19

~10-18

(s)

Reaction

Time

2* *

00

(2 1) ( , ) ( , ) ( , )2

ER cm CN

cm

T E P E W EE

Formation probability

Survival probability

Reaction time t < 10-22 s

10-22 < t < 10-18 s

~10-18 < t s

Quasi-fission 90~99 %

Fusion-fission

Tℓ

1st

stage

2nd

stage

3rd

stage

Touching probability

W

Recent Development of Theoretical Models

Tl PCN Wsuv

Antonenko, Admiyan,

Nasirov, Cherepanov,

Volkov, Giardina, Scheid

Simple WKB 1D di-nucleus confi.

Statistical method

Statistical model

Aritomo, Ohta

Experimental data,

Gross-Kalinovski

3D two-center shell model

3D-Langevin eq.

Langevin with Statistical

model

Bouriquet, Shen, Kosenko,

Boilley, Abe

Gross-Kalinovski 2D two-center LD model

2D-Langevin eq.

Statistical model

(KEWPIE)

Ohta Simple WKB Empirical function derived

from results with

3D-Langevin

Statistical model

Zagrebaev, Greiner Quantum

(CC or empirical model)

3D two-core model

Master eq.

Zagrebaev, Greiner,

Aritomo, Karpov,

Noumenko

Unified model

3D two-center shell model

3D-Langevin eq.

Statistical model

Statistical model

Swiatecki, Wilczynska,

Wilczynski

Empirical method 1D-Diffusion model

Analytical formula

Statistical model

Misicu, Gupta, Greiner Deformation and

Orientation

Ichikawa, Iwamoto, Moller,

Sierk

Deformation and

Quadrupole zero-point

vibrational energy

V.I. Zagrebaev, et al. Phy. Rev. C. 65. (2001) 014607

Fission width

Bohr and Wheeler (1939)

Statistical model (transition state method)

initial state and final state

Kramers (1940)

Dynamical model

friction as diffusion process

*

*

* 0

1 ( )

2 ( )

exp2

BE UBW

f B

B

dK E U KE

UT

T

Stationary solution

Fission width

～

Recent Development of Theoretical Models

Tl PCN Wsuv

Antonenko, Admiyan,

Nasirov, Cherepanov,

Volkov, Giardina, Scheid

Simple WKB 1D di-nucleus confi.

Statistical method

Statistical model

Aritomo, Ohta

Experimental data,

Gross-Kalinovski

3D two-center shell model

3D-Langevin eq.

Langevin with Statistical

model

Bouriquet, Shen, Kosenko,

Boilley, Abe

Gross-Kalinovski 2D two-center LD model

2D-Langevin eq.

Statistical model

(KEWPIE)

Ohta Simple WKB Empirical function derived

from results with

3D-Langevin

Statistical model

Zagrebaev, Greiner Quantum

(CC or empirical model)

3D two-core model

Master eq.

Zagrebaev, Greiner,

Aritomo, Karpov,

Noumenko

Unified model

3D two-center shell model

3D-Langevin eq.

Statistical model

Statistical model

Swiatecki, Wilczynska,

Wilczynski

Empirical method 1D-Diffusion model

Analytical formula

Statistical model

Misicu, Gupta, Greiner Deformation and

Orientation

Ichikawa, Iwamoto, Moller,

Sierk

Deformation and

Quadrupole zero-point

vibrational energy

Nuclear shape is described by

Two center parametrozation

2* *

00

(2 1) ( , ) ( , ) ( , )2

ER cm CN

cm

T E P E W EE

Formation probability

Survival probability

Reaction time t < 10-22 s

10-22 < t < 10-18 s

~10-18 < t s

Quasi-fission 90~99 %

Fusion-fission

Tℓ

1st

stage

2nd

stage

3rd

stage

Touching probability

σcap ? PCN

W

Tℓ

Uncertainty

of the 2nd stage

2. Model

1. 2-1. Potential

2. Two-center shell model

3.

2-2. Equation trajectory calculation

),,( z

Overview of Dynamical Process in reaction 36S+238U

10

1. Potential energy surface

2. Trajectory described by

equations

two-center parametrization

(Maruhn and Greiner,

Z. Phys. 251(1972) 431)

),,( z

Nuclear shape

(δ1=δ2)

( , , )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

δ=0

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

Fission barrier recovers at low excitation energy

298114

2. Model

1. 2-1. Potential

2. Two-center shell model

3.

2-2. Equation Taking into account the fluctuation around the mean trajectory

Thermal fluctuation of nuclear shape

thermal fluctuation of collective motion

),,( z

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 equation Friction Random force

dissipation fluctuation ordinary differential equation

Einstein relation Fluctuation-dissipation theorem

0.0

0.5

1.0

1.5

2.0

0

20

40

-0.5

0.0

0.5

z

-0.5 0.0 0.5 1.0 1.5 2.0

-0.5

0.0

0.5

z

Fission process 240U E* < 20 MeV

Trajectory on potential energy surface

Overview of Dynamical Process in reaction 36S+238U

10

3. Results

Evaporation residue cross section

Mass distribution of fission fragments

Itkis et al.

Cal.

Calculation results 48Ca + 244Pu

Calculation

4. Way to synthesize new SHE

Ti, Cr, Fe etc. beams

Transfer reaction U+Th, U+Cm

Secondary beams

Cold fusion reaction Hot fusion reaction 1994

110 Ds 62Ni + 208Pb 269110 + n (GSI)

111 Rg 64Ni + 209Bi 272111 + n (GSI)

1996

112 Cn 70Zn + 208Pb 277112 + n (GSI) named in Feb. 2010

1999

114 Fl 48Ca + 244Pu 292114 + 3n (FLNR) named in May. 2012

2000

116 Lv 48Ca + 248Cm 292116 + 4n (FLNR) named in May. 2012

2002

118 48Ca + 249Cf 294118 + 3n (FLNR)

2003

115 48Ca + 243Am 288115 + 3n 284113 + α (FLNR)

2004

113 70Zn + 209Bi 278113 + n (RIKEN)

2010

117 48Ca + 249Bk 294,293117 + 3-4n (FLNR)

Synthesis of New Elements Reports of new elements

20

Yu. Ts. Oganessian and K. Morita

190

A=304

Possibility of synthesizing 298114

Model Calculation

3. Survival Process

118

117 293117 294117

116

115 287115 288115 289115 290115

114

113 282113 283113 284113 285113 286113

283Cn

278Rg 279Rg 280Rg 281Rg 282Rg

274Mt 275Mt 276Mt 278Mt

270Bh 271Bh 272Bh 274Bh

266Db 267Db 268Db 270Db

294118

290Lv 291Lv

286Fl 287Fl

282Cn

275Hs

267Rf

CN 278113

274Rg

270

Mt

266Bh

262Db

278113

274Rg

270Mt

266Bh

262Db

258Lr

254Md

254Fm

250Cf

2* *

00

(2 1) ( , ) ( , ) ( , )2

ER cm CN

cm

T E P E W EE

Trajectory calculation Two-center parametrization

Capture

Fusion

Survival

3-dim Langevin

Statistical model

f

n

To estimate survival probability

Dynamical model

dqtqPtEW );,();,(saddle inside

*

0

W ; survival probability

One-dimensional Smoluchowski equation

tqPq

TtqP

q

tqV

qtqP

t;,;,

;,1;,

2

2

T(t) : temperature statistical code SIMDEC

Cooling curve

statistical code

q ; separation distance

We assume the particle emissions are limited to neutron emission in the neutron-rich heavy nuclei.

friction reduced ;

mass inertia ;

ondistributiy probabilit ; ;,

tqP probability distribution

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 Φ(T)=exp{-aT2/Ed}, Ed=20 MeV

VLD(q)

MeV 20

exp)(2

d

d

E

E

aTT

)( )(),(

)()()(

),()(2

1)(),,(

0

2

TqETqV

qEqEqV

TqVqI

qVTqV

shellSH

CSLD

SHLD

MeV 20

exp)( *

d

d

E

E

ET

298114

Fission barrier recovers at low excitation energy

Neutron Separation energy Shell correction energy at G.S.

3. Survival process

Rapid cooling

Approaching to the closed shell

Bf (

MeV)

A=298

A=304

Smoluchowski equation

A=304

Bf (

MeV)

4. Summary 1. The possibility of synthesizing a doubly magic superheavy nucleus, 298114184, was

investigated on the basis of fluctuation dissipation dynamics.

2. Owing to the neutron emissions, we must generate more neutron-rich compound nuclei.

3. To calculate the survival probability, we employ the dynamical model.

4. 304114 has two advantages to achieving a high survival probability.

1 ) small neutron separation energy and rapid cooling

2 ) the neutron number of the nucleus approaches that of the double closed shell

obtain a large fission barrier

5. The systematical investigation compared with the statistical model and dynamical one is necessary. We must apply the dynamical model for known systems.

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

Collaborators S. Chiba Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology K. Hagino Department of Physics, Tohoku University K. Nishio Advanced Science Research Center, Japan Atomic Energy Agency V.I. Zagrebaev, A.V. Karpov Flerov Laboratory of Nuclear Reactions W. Greiner Frankfurt Institute for Advanced Studies, J.W. Goethe University