II. Fusion and quasifission with the dinuclear system model

Second lecture

1. Introduction

2. Models for fusion with adiabatic and diabatic potentials

3. Dynamics of fusion in the dinuclear system model

4. Quasifission and incomplete fusion

5. Summary and conclusions

Contents

Work of

G. G. Adamian, N. V. Antonenko, R. V. Jolos, S. P. Ivanova, A. Zubov

Joint Institute for Nuclear Research, Dubna

Collaboration with

Junqing Li, Wei Zuo, Institute of Modern Physics, Lanzhou

Enguang Zhao, Ning Wang, Shangui Zhou Institute of Theoretical Physics, Beijing

1. Introduction

Superheavy elements are presently produced in heavy ion reactions up to element 118.

Experiments done at GSI (Darmstadt), JINR (Dubna), GANIL (France), LBNL (Berkeley), RIKEN (Japan), IMP(Lanzhou)

Two types of reactions are successful:

Cold and hot complete fusion reactions with excitation energies of the compound nucleus of E*CN of 12-15 and 32-36 MeV, respectively.

a) Lead-based cold fusion reactions

70Zn+208Pb278112277112+n, =1 pb

E*CN 12 MeV for Z108

b) Hot fusion reactions with 48Ca projectiles

48Ca+244Pu292114289114+3n, =1 pb

E*CN 32 MeV

Small cross sections:

Competition between complete fusion and quasifission

Idea of Volkov (Dubna) to describe fusion reactions with the dinuclear system concept:

Fusion is assumed as a transfer of nucleons (or clusters) from the lighter nucleus to the heavier one in a dinuclear configuration.

This process is describable with the mass asymmetry coordinate =(A1-A2)/(A1+ A2).

A1 A2

If A1 or A2 get small, then ||1 and the system fuses.

1. Relative motion of nuclei,

capture of target and projectile into dinuclear system, decay of the dinuclear system: quasifission

2. Transfer of nucleons between nuclei, change of mass and charge asymmetries leading to fusion and quasifission

The dinuclear system model uses two main degrees of freedom to describe the fusion and quasifission processes:

Aim of lecture:

Consideration of the fusion dynamics,

comparison of models in the calculation of residue cross sections for superheavy nuclei,

discussion of quasifission with master equations,

incomplete fusion as a signature for the dinuclear model.

2. Models for fusion with adiabatic and diabatic potentials

Heavy and superheavy nuclei can be produced by fusion reactions with heavy ions.

Two main collective coordinates are used for the description of the fusion process:

1. Relative internuclear distance R

2. Mass asymmetry coordinate for transfer

Description of fusion dynamics depends strongly whether adiabatic or diabatic potential energy surfaces are assumed.

diabatic

adiabatic

touching configuration

V

R

Diabatic potentials are repulsive at smaller internuclear distances R<Rt.

Explanation with two-center shell model:i i

R R1 1

2 2

adiabatic model diabatic model

Velocity between nuclei leads to diabatic occupation of single-particle levels, Pauli principle between nuclei

diabatic

adiabatic

~R

a) Models using adiabatic potentials

Minimization of potential energy, essentially adiabatic potential in the internuclear distance, nuclei melt together.

Large probabilities of fusion for producing nuclei with similar projectile and target nuclei.

R 0

entrance

quasifissionfusion

touching configuration

R

48Ca + 246Cm (from Zagrebaev)

b) Dinuclear system (DNS) concept

Fusion by transfer of nucleons between the nuclei (idea of V. Volkov, also von Oertzen), mainly dynamics in mass asymmetry degree of freedom, use of diabatic potentials, e.g. calculated with the diabatic two-center shell model.

entrance quasifission

totouching configuration

fusion

R

58Fe+244Pu 302120

110Pd+ 110Pd

~R

double folding

TCSM

2R0

Calculation of diabatic potential:

= single particle energies

= occupation numbers

Diabatic occupation numbers depend on time:

De-excitation of diab. levels with relaxation time, depending on single particle width.

diabatic

adiabatic

3. Dynamics of fusion in the dinuclear system model

Evaporation residue cross section for the production of superheavy nuclei:

a) Partial capture cross section cap

Dinuclear system is formed at the initial stage of the reaction, kinetic energy is transferred into potential and excitation energy.

Bqf=barrier for quasifission

V(R)

touching point R

b) Probability for complete fusion PCN

DNS evolves in mass asymmetry coordinate by diffusion processes toward fusion and in the relative coordinate toward the decay of the dinuclear system which is quasifission.

B*fus=inner fusion barrier

V()

i-1 1

B*fus

222Th

Calculation of PCN and mass and charge distributions in and R:

Fokker-Planck equation, master equations, Kramers approximation.

Competition between fusion and quasifission, both processes are treated simultaneously.

c) Kramers formula for PCN:

Rate for fusion: fus

Rate for quasifission: qf = R + sym,

i.e. decay in R and diffusion in to more symmetric DNS.

Cold fusion (Pb-based reactions):

Hot fusion (48Ca projectiles):

d) Importance of minima in the driving potential

Shell and deformation effects lead to local minima in V(). In liquid drop model the mean value of runs to symmetric fragmentation: , then quasifission.

Minima in V() hinder motion to , then larger probability for fusion.

Idea of A. Sandulescu and W. Greiner (1976) for optimum selection of target and projectile: choice of fragmentations in minima of V().

A. Sandulescu and W. Greiner (1976)

114110106

102

100104 108 112

Z=114

Z=108

Example: 54Cr + 208Pb 262Sg

V()

B*fusBsym

54Cr + 208Pb

B*fus=5.7 MeV, Bsym=3.6 MeV, Bqf=2.7 MeV

Dependence of fusion probability PCN

on barrier height Bsym.

B*fus Bsym

54Cr + 208Pb 262Sg

e) Optimum excitation energy for Pb-based fusion reactions

B*fus

V()

-1 1i

E*CN

E*CN = Ecm + Q

V(i) + B*fus

Cold fusion

f) Survival probability Wsur

De-excitation of excited compound nucleus by neutron emission in competition with fission. Other decays (decays) can be neglected.

For Pb-based reactions with emission of one neutron:

n =width for neutron emission f =fission width, f>>n P1n =probability of realisation of 1n channel

g) Results for cold fusion reactions

Cold reactions: AZ+208Pb superheavy + n

React. Super-heavy

E*CN (MeV)

PCN cap (mb)

Wsur ERth ER

ex

50Ti+ 208Pb

257104+n

16.1 3x10-2 5.3 9x10-5 14.3 nb

10 nb

70Zn+ 208Pb

272112+n

9.8 1x10-6 3.0 6x10-4 1.8 pb 1 pb

86Kr+ 208Pb

293118+n

13.3 1.5x 10-10

1.7 2x10-2 5.1 fb < 0.5pb

nb

pb

fb

76 73

70

6768

68

110

114

112

76Ni+208Pb

64Ni+208Pb

ANi+208Pb110

e.g. 48Ca+244Pu288,289114 + 4,3 n, x 3n , 4n ~ 1 pb

Initial dinuclear system is more asymmetric than in Pb-based reactions: B*fus is smaller.

Larger fusion probability, but smaller survival probability because of higher excitation energy E*CN.

h) Hot fusion reactions with 48Ca projectiles and actinide nuclei as target

Reaction Super-heavy

E*CN (MeV)

xnth

(pb)

xnexp

(pb)48Ca + 238U

283112 + 3n

33 1.7 1 - 5

48Ca + 244Pu

288114 + 4n

35.5 1.0 ~ 1

48Ca + 242Pu

286114 + 4n

32 1.0 2.5

48Ca + 248Cm

292116 + 4n

35 0.2 ~ 0.5

capture

fusion

PCN

Z=112

Z=114

Z=116

Z=118

4. Quasifission and incomplete iiiiifusion

a) Master equation for mass and charge transfer

Probability PZN(t) to find the dinuclear system in fragmentation:

Z1=Z, N1=N, Z2=Ztot-Z1, N2=Ntot-N1

Z1+N1=A, Z1+Z2+N1+N2=Atot

Starting point: shell model Hamiltonian

in second quantization:

Rates depend on single-particle energies and temperature related to excitation energy.

Only one-nucleon transitions are assumed.

qfZ,N : rate for quasifission

fisZ,N : rate for fission of heavy nucleus

Transfer rates

for example

t = 10-22 s : transition time of nucleon,

temperature T(Z,N),

nA1(T), nA2(T): Fermi occupation numbers

A1, A2: single particle energies.

b) Fusion probability and mass and charge yields

Fusion probability:

t0 = reaction time: (3-5)x10-20 s yield of quasifission:

Reaction time determined by

mass yield: charge yield:

c) Total kinetic energy

Total kinetic energy distribution and its dispersion depend on deformation of fragments.

Strong polarisation effects for symmetric DNS: for A=Atot/2 20 3-4 times larger deformations than in ground state.

Distribution of clusters in charge, mass and deformation in calculation of TKE:

with frequency vib and stiffness parameter Cvib of quadrupole vibrations.

d) Hot fusion reactions with 48Ca beam

Quasifission measurements by Itkis et al.

e.g. 48 Ca + 248Cm 296116

Quasifission events near initial mass overlap with deep inelastic collision events and are taken out in the experimental analysis.

In our calculations only quasifission is shown. Calculated primary fragment distributions.

Maxima in yields arise from minima in driving potential U().

E*CN= 33.4 MeV

286112

E*CN=42 MeV

Pb ZrSn

E*CN= 37 MeV

J=70

J=0

Minima in variance 2TKE are related to stiff

nuclei: Zr, Sn, Pb.

Importance of fluctuations of deformations for variance 2

TKE .

deformation

nucleon exchange

E*CN= 33 MeV

58Fe+248Cm306122

e) Competition between fusion-fission and quasifission processes

Ratio of fusion-fission to quasifission events:

E*CN(MeV) r40Ar + 165Ho 89

1201.10.7

48Ca + 244Pu 34.850

1.4 x 10-2

1.1 x 10-1

86Kr + 208Pb 1730

2.1 x 10-7

2.0 x 10-5

f) Incomplete fusion

Transfer cross sections to more asymmetric systems

Cross section of the production of primary heavy nucleus:

Here: 48Ca + 244,246,248Cm, 76Ge + 208Pb

Cross section with evaporation of x neutrons:

48Ca+244Cm

48Ca+246Cm

48Ca+248Cm

Charge number of heavy nucleus

48Ca+244Cm Ecm=207 MeV

48Ca+246Cm Ecm=205.5 MeV

48Ca+248Cm

Ecm=204 MeV

nb

66,68,70Zn + 208Pb

5. Summary and conclusions The concept of the nuclear molecule or

dinuclear system describes nuclear structure phenomena connected with cluster structures, the fusion of heavy nuclei to superheavy nuclei, the competing quasifission and fission.

The dynamics of the dinuclear system has two main degrees of freedom: the relative motion of the nuclei and the mass asymmetry degree of freedom.

Superheavy nuclei are produced in heavy ion collisons in a process of three stages:

(1) Capture of nuclei in a dinuclear system,

(2) Fusion to compound nucleus by nucleon transfer between touching nuclei in competition with quasifission, which is the decay of the dinuclear system,

(3) De-excitation of compound nucleus by neutron emission into the ground state of superheavy nucleus.

The dinuclear system has a repulsive, diabatic potential energy surface towards smaller internuclear distances. This potential hinders the nuclei to amalgamate directly.

The transfer of nucleons and the quasifission are statistical diffusion processes in the excited dinuclear system. They are described by Fokker-Planck or master equations or simply by the Kramers approximation. Fusion and quasifission are simultaneously treated.

A precise prediction of optimum production cross sections for superheavy nuclei can be obtained from the calculation of isotopic series of reactions. The cross sections depend sensitively on the interplay between fusion and survival probabilities.

For elements Z > 118 the production cross sections seem to be lower than 0.1 pb. The border of 0.1 pb is the limit of measurements in the near future.

D.G.