ii. fusion and quasifission with the dinuclear system model second lecture
TRANSCRIPT
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.