Study on Sub-barrier Fusion Reactions and Synthesis of Superheavy Elements Based on Transport The

ory

Zhao-Qing Feng

Institute of Modern Physics, CAS

Contents• Introduction• Improved isospin dependent quantum mo

lecular dynamics model• Study on dynamics of fusion reactions ne

ar Coulomb barrier• Production cross sections of the superhe

avy nuclei based on dinuclear system model

• Summary

1. Introduction• 60s, 20 century, Shell model prediction:

“stability island” around Z=114,N=184

• Experiments GSI: 110-112 Dubna: 113-116 Riken: 113 IMP: 105, 107 (new nuclei)

• Theoretical models for the description of superheavy nuclei:

Dinuclear system model (Adamian et al. NPA 633 (1998)

409, Li et al. EPL 64(2003)750, Feng et al. CPL 22 (2005) 846)

Fluctuation-dissipation model (Aritomo et al. PRC 59 (1999) 796)

Nucleon collectivization model (Zagrebaev et al. PRC 65 (2001) 014607)

Macroscopic dynamical model (S. Bjornholm and W.J. Swiatecki, NPA 391(1982) 471)

Improved isospin dependent quantum molecular dynamics model (Wang et al. PRC69 (2004) 034608), Feng et al. NPA 750 (2005) 232

2. Improved isospin dependent quantum molecular dynamics model• Purpose: to study fusion mechanism near Co

ulomb barrier• Improved aspects including: 1. Yukawa term is replaced by introducing de

nsity dependent surface term derived from self-consistently Skyrme interaction. (Wang et al. PRC 65 (2002) 064608)

2. Introducing surface symmetry term. (Wang et al. PRC 69 (2004) 034608)

3. Nucleon’s fermionic nature is improved by using phase space constraint method. (M. Papa et al. PRC 64 (2001) 024612)

4. Coulomb exchange term is included in the model. (Wang et al. PRC 67 (2003) 024604)

5. Shell effect is considered in the model. (Feng et al. NPA, 750 (2005) 232 )

6. Switch function method is introduced in the model, which can effectively prevent some unphysical nucleus emissions in the process of projectile and target appoarchng. (Feng et al. HEP&NP,2005,29(1) 41 )

2.1 Introduction on the improved isospin dependent quantum molecular dynamics model• In the improved model, the effective interaction

potential energy is denoted as

shelleffsurfsymvolcoul UUUUUUU

Rde

Lrerfttr

eU

p

iijjziz

ij jicoul

3/43/1

2

2

343

)4/()1)(1(14

i ij i ij

ijijvolU

00 12

i

ij

ij

jisurfsurf L

rrL

gU

0

2

223

2

i ij

ijeff gU

0

Lrr

Lji

ij 4)(

exp4

1 2

2/3

rdrrksym

)()(2

C-U

2

0

symsym-surf

i ij

jisym

ijjziz

symsym L

rrL

kttC

U2

0 2231

2

• Switch function method is introduced, which can prevent some unphysical nucleons emission. So the surface interaction energy of the system is written as

).1(arg SUSUUU surfcomp

surft

surfproj

surfsyst

S is called as switch function

Taking coefficients must satisfy the continuity of the surface energy and its first derivative!

55

44

33

2210

)()(

)()(

lowup

low

lowup

low

lowup

low

lowup

low

lowup

low

RRRRC

RRRRC

RRRRC

RRRRC

RRRRCCS

C0 C1 C2 C3 C4 C5

0 0 0 10 -15 6

Parameter set in the model

/MeV/MeVCsym/MeVsym/fm2 gsurf/MeVfm2 g /Me

V 0/fm-3-356.0 303.0 7/6 32.0 0.08 8.0 10.0 0.165

Parameter set taken by Wang et al.

The ground state properties, static (dynamical) barriers fusion (capture) excitation function as well as neck dynamical behaviour et al. can be described very well using the improved model.

2.2 Consideration of shell effect in ImIQMD• As we know that shell effect is the diversity of shell

model (shell structure) and macroscopic model (bulk property). Thus, the shell correction energy can be obtained from the variance of shell levels and uniformed levels, which is written by

.~EEEshell

]2/[

1

)(22~ N

iishell deegeeEEE

Using Strutinsky method (NPA 95 (1967) 420), the shell correction energy is written as

i

ii eefeeeg 2

2)(exp1)(

• The smoothed level density is usually given by.2.1

.)(2

deegN

Gaussian distribution widthIn the calculation, 3rd-order Laguerre polynomial is used. The Fermi energy is obtained by

The shell levels are calculated by using deformed two center shell model. (R.A. Gherghescu, Phys. Rev. C 67 (2003) 014309)

.

/exp1 2 rdaRraEU shellIQMD

shell

• In ImIQMD, the Shell correction energy is denoted by

Using canonical equation, the force can be obtained as

,/exp1

/exp02 raRr

aRraEF shellIQMD

shell

02/exp1

/exp raRraRr

aeeF

i

iiiishell

One can obtain the force of each nucleon derived from the shell correction energy as

From energy density functional, we can also know thatshell effect mainly embodies the surface of the nucleus!(M. Brack, C. Guet, H.B. Hakansson, Phys. Rep. 123 (1985) 276)

• Considering the Woods-Saxon distribution form of the nuclear density , it is more self-consistently by denoting the shell correction energy as

.0

rdEU shellIQMDshell

,0

shellIQMD

shellEF

i

iiishell

eeF

0

10

20

30

40

50

60

70

80

DTCSM levels smoothed levels

proton levels

E/M

eV

E Ferm

i=42.

54M

eV

E Ferm

i=41.

77M

eV

s1/2

p3/2

p1/2

d5/2

s1/2/ d3/2

f7/2

f5/2

48Cag(e)

00

2

2,)( iii

iziizi m

peprL

It is very important to fill these levels in ImIQMD. In our calculation, we label each nucleon according to angular momentum and single particle energies, which are obtained respectively by

3. Dynamical study on fusion reactions near Coulomb barrier • Based on improved isospin dependent quantum mol

ecular dynamics model, the static and dynamical Coulomb barrier, fusion/capture cross sections, neck dynamical behaviour et al. are studied systematically.

tppt EERERV )()(3.1 Dynamical barrier

Here, Ept, Ep and Et are the total, projectile and target energy respectively, the kinetic energy part is approximated by using Thomas-Feimi model as

i

ikin mE

3/222

23

253

The static nucleus-nucleus interaction potential

Static barriers, prox. (W.D. Myers et al., PRC 62 (2000) 044610)

5 10 15 20 2550

100

150

200

250

300

5 10 15 20 2550

100

150

200

250

300

ImIQMD Prox.

V b/MeV

16O+208Pb

16O+238U

5 10 15 20 2520

40

60

80

100

120

V b/MeV

R/fm

48Ca+208Pb

5 10 15 20 2520

40

60

80

100

120

R/fm

48Ca+238U

Dependence on the projectile-target combinations leading to the same compound nucleus formation 258Rf

5 10 15 20 25100

150

200

250

300

350

ImIQMD Prox.

V b/MeV

124Sn+134Xe

5 10 15 20 25100

150

200

250

300

350

86Kr+172Er

5 10 15 20 2550

100

150

200

250

300

V b/MeV

R/fm

50Ti+208Pb

5 10 15 20 2560

80

100

120

140

R/fm

24Mg+234U

The static and dynamical interaction potentials calculated by using the ImIQMD for the reaction sytems40,48Ca+40,48Ca.

Dependence of dynamical barriers on incident energy

Dependence of fusion barrier on projectile neutron number leading to the same element formation

3.2 Neck dynamical behaviour

0 50 100 150 200 2500.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

40Ca+40Ca 40Ca+48Ca 48Ca+48Ca

N/Z

t/(fmc-1)

Ec.m.=50MeV below the barriers for b=0fm

0 50 100 150 200 2500.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

40Ca+40Ca 40Ca+48Ca 48Ca+48Ca

N/Z

t/(fmc-1)

Ec.m.=60MeV above the barriers for b=0fm

Time evolution of N/Z at neck region for 40,48Ca+ 40,48Ca

Nucleon transfer in neck region for reaction system 48Ca+238U

N/Z dependence on incident energy at neck

region for system 48Ca+208Pb

Neck radius development in the process of neck formation

3.3 The calculation of fusion/capture cross sections

max

0

),(2)(b

fusfus bdbbEpE

Experimental data taken in Refs M. Trotta et al., Phys. Rev. C 65 (2001) R011601H.A Aljuwair et al., Phys. Rev. C 30 (1984) 1223

Fusion excitation functions for 40,48Ca+40,48Ca

Positive Q value will lead to the enhancement of sub-barrier fusion cross sections

40,48Ca+124,116Sn, 16,18O+42,40Ca, 9,11Li+208,206Pb suggested by V.I. Zagrebaev PRC 67 (2003) 061601R.

Capture cross sections for heavy systems

Experimental data taken from E.V Prokhorova et al., nucl-exp/0309021 and W.Q Shen et al., Phys. Rev C 36 (1987) 115

M. Dasgupta et al., Nucl. Phys A 734 (2004) 148K. Nishio et al., Phys. Rev. Lett 93 (2004) 162701

M. G. Itkis, Yu. Ts. Oganessian, E. M. Kozulin et al., Proceedings on Fusion Dynamics at the Extremes, Dubna, 2000, edited by Yu. Ts. Oganessian and V. I. Zagrebaev page 93.

Preliminary consideration on the calculation of the evaporation cross sections based on ImIQMD

Formation probability at excitation energy E* is written as

*exp1

*)(0

0

EEPEPCN

Where E0 is the critical excitation energy depending on the reaction system, is the barrier distribution width, we can take it as 2/)( ds BB

So the evaporation cross section can be denoted by

dbbEWEPbEbp mcsurmcCN

b

fusevap ),()(),(2 ....0

max

4. Production cross sections of the superheavy nuclei based on dinuclear system model• In dinuclear system mdoel, evaporation cross

section is denoted by),(),(),()12()( ......

2.. JEWJEPJETJE mcsurmcCN

Jmcmcc

Schematic illustration of the fusion process

Cap. Q-fission Fission

• , T(Ec.m.,J) is usually taken 0.5.• Fusion probability • The mass distribution probability P(A1,E1,t) is given by master equation

which is solved numerically in the model.If only considering the competition of neutron emission and fission, the survival probability Wsur with emitting X neutrons can be written as

)2/( ..2

mcE

BGA

CN dAJJEAPJP0

1int11 ))(),(,()(

'1

'11

'11

)],,(),,([),,(11

'1

'1,

11

AAAAA

tEAPdtEAPdWdt

tEAdP

iifin

inx

iCNCNsur JEJE

JEJxEPJxEW

),(),(

),(),,(),,( **

*

1

**

Energy and angular-momentum dissipation are described by Fokker-Planck equation

fDl

fvl

fDp

fvp

flpf

ru

rfp

tf

llpprel

2

2

2

2

Based on dinuclear system model, the production cross sections of superheavy nuclei in cold fusion reactions are studied systematically.

• Height of the pocket are 6.39 MeV (0.61) 4.80 MeV (0.56) 1.70 MeV (0.04) 1.71MeV (0.02)

• In the DNS model, the compound nucleus formation is governed by the driving potential.

),(),()()()(),( 11211 RAURAUAUAUAURAU NCLDLDLD

Production cross sections for asymmetric and nearly symmetric reaction systems, comparisonwith coupled channel model which has includednucleon transfer and surface vibration is also shown.(V.Yu. Denisov Prog. Part. Nucl. Phys. 46 (2001) 303)

Feng, Jin, Fu et al., Chin. Phys. Lett., 22(4), 2005, 846

Improvement of dinuclear system model• In order to describe correctly the capture process, ba

rrier distribution function method is included in the model. (P.H. Stelson, PLB 205 (1988) 190, V.I. Zagrebaev et al., PRC 65 (2001) 014607) The transmission coefficient is denoted as

dBEJJ

JRB

J

BfJET mc

)1()(2)(

2exp1

1)(),(

2

2..

The barrier distribution function satisfies the normalization condition, which is usually taken as a asymmetric Gaussian distribution form. 1)( dBBf

mm

mm

BBBB

BBBB

NBf

,exp

,exp

)(2

2

2

1

• Here Bm=(B0+Bs)/2, B0 and Bs are the height of the Coulomb barrier and the saddle point respectively. Gaussian distribution function 2= (B0-Bs)/2, 1 is less than the value of 2 (usually 2 MeV). V. I. Zagrebaev PRC64 (2001)034606

Capture cross section can be reproduced very well by introducing the barrier distribution function method

Comparison of calculated evaporation residue cross sections with experimental data for 1,2,3,4 neutron emission

Experimental data taken from E.V Prokhorova et al., nucl-exp/0309021Yu. Ts. Oganessian et al., Phys. Rev. C 64, 054606 (2001).

Production cross sections of superheavy nuclei 286-xn112, 29

2-xn114, 296-xn116 in 48Ca induced reactions and comparison

with Dubna data (Yu.Ts. Oganessian et al., PRC 70 (2004) 064

609)

Extension to multi-dimension degree of freedom for the driving potential• In the DNS model, we only consider the mass asymmetry degree

of freedom, so there is some difficulties for describing the mass distribution of quasi-fission or fission, as well as reasonably showing the formation process of the compound nucleus. We need to consider the center of mass distance R and deformation degree of freedom et al. in the process of the superheavy compound nucleus formation.

Y. Aritomo, M. Ohta, NPA 744 (2004) 3

5. Summary• The isospin dependent quantum molecular dynamics

model is improved by introducing switch function method for the surface term and considering shell effect. Experimental fusion/capture cross sections can be reproduced very well using the improved model. Fusion barrier and neck dynamical behaviour in fusion process are studied systematically.

• The dinuclear system model is improved by introducing the barrier distribution function method, dynamical deformation is considered in the capture process. Evaporation residue cross sections can be regenerated well for 1n,2n,3n 4n evaporation. Further studies are in progress!