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Dinuclear system model in nuclear structure and reactions. The two lectures are divided up into. I. Dinuclear effects in nuclear spectra and fission II. Fusion and quasifission with the dinuclear system model. First lecture.
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Dinuclear system model in nuclear structure and reactions
The two lectures are divided up into
I. Dinuclear effects in nuclear spectra and fission
II. Fusion and quasifission with the dinuclear system model
First lecture
I. Dinuclear effects in nuclear spectra and fission
Contents
1. Introduction
2. The dinuclear system model
3. Alternating parity bands
4. Normal and superdeformed bands
5. Hyperdeformation in heavy ion collisions
6. Rotational structure of 238U
7. Binary and ternary fission
8. Summary
Work of
G. G. Adamian, N. V. Antonenko, R. V. Jolos, Yu. V. Palchikov, T. M. Shneidman
Joint Institute for Nuclear Research, Dubna
Collaboration with
N. Minkov
Institute for Nuclear Research and Energy, Sofia
1. Introduction
The dinuclear system has two main degrees of freedom:
Relative motion of nuclei: formation of dinuclear system in heavy ion collisions, molecular resonances, decay of dinuclear system: fission, quasifission, emission of clusters
Transfer of nucleonsbetween nuclei: change of mass and charge asymmetries between the clusters
Mass asymmetry coordinate
A2
A1
Applications of dinuclear system model
Aim of lecture:
Consideration of nuclear structure effects and fission due to the dynamics in the relative motion, mass and charge transfer and rotation of deformed clusters in a dinuclear configuration
2. The dinuclear system model
Let us first consider some selected aspects of the dinuclear system model.
2.1 Deformation
Dinuclear configuration describes quadrupole and octupolelike deformations and extreme deformations as super and hyperdeformations.
Multipole moments of dinuclear system:
Comparison with deformation of axially deformed nucleus described by shape parameters:
152Dy
2.2 Potential and moments of inertia
Clusterisation is most stable in minima of potential U as a function of . Minima by shell effects, e.g. magic clusters.
Potential energy of dinuclear system:
B1, B2, B0are negative binding energies of the clusters and the united (=1) nucleus. V(R,,I) is the nucleusnucleus potential.
Example: 152Dy
152Dy
50Ti+102Ru
26Mg+126Xe
Moment of inertia of DNS:
: moments of inertia of DNS clusters
For small angular momenta:
For large angular momenta and large deformations:
Exp.: Moments of inertia of superdeformed states are about 85% of rigid body limit.
Example: 152Dy
= 0.34: 50Ti+102Ru,
Hyperdeformed properties: U=20 MeV above g.s., about estimated energy of L=0 HDstate of 152Dy, (calc)=131 MeV1, (est)=130 MeV1, 2(calc)=1.3, 2(est)0.9.
= 0.66:26Mg+126Xe,
Superdeformed properties:
(calc)=104 MeV1, (exp)=85±3 MeV1, Q2(calc)=24 eb (2=0.9), Q2(exp)= 18±3 eb
Similar: = 0.71:22Ne+130Ba
26Mg+126Xe and 22Ne+130Ba have SD properties.
2.4 Mass asymmetry motion
For nuclear structure studies we assume h as a continuous coordinate and solve a Schrödinger equation in mass asymmetry.
Wave function yI(h) contains different cluster configurations.
At higher excitation energies: statistical treatment of mass transfer. Diffusion in hiscalculated with FokkerPlanck or master equations.
3. Alternating parity bands
Ra, Th and U have positive and negative parity states which do not form an undisturbed rotational band. Negative parity states are shifted up. This is named parity splitting.
5
6+
3
4+
1
2+
0+
Parity splitting is explained by reflectionasymmetric shapes and is describable with octupole deformations.
Here we show that it can be described by an asymmetric mass clusterization.
Configuration with alphaclustering can have the largest binding energy.
AZ (A4)(Z2) + a  particle
a
a
Ba
_
+
splitting
oscillations in h
Lower state has positive parity, higher state negative parity. Energy difference depending on nuclear spin is parity splitting.
potential
wavefunctions
Positive parity
Negative parity
x
238U
236U
234U
232U

223Ra



+

+

+
+
+
3/2
(I,K) (I,K+)
3/2
225Ra
4. Normal and superdeformed iiiiibands
Here: application of dinuclear model to structure of 60Zn,194Hg and 194Pb
a) Cluster structure of 60Zn
1. 60Zn56Ni+a, tresh. 2.7 MeV above g.s. Assumption: g.s. band contains acomponent.
2. 60Zn52Fe+8Be, tresh. 10.8 MeV above g.s. / 48Cr+12C, tresh. 11.2 MeV above g.s.
Extrapolated head of superdef. band: 7.5 MeV
Assumption: superdeformed band contains 8Becomponent.
Unified description of g.s. and sd bands by dynamics in mass asymmetry coordinate.
b) Potential U(h , I) for 60Zn
mononucleus (h=1,1) U(I=0) = 0 MeV 56Ni+a  4.5 MeV 52Fe+8Be 5.1 MeV 48Cr+12C 9.0 MeV
Stepwise potential because of large scale in h. Barrier width is fixed by 3 state (3.504 MeV).
60Zn
I=0
x=h1 for h>0 x= h+1 for h<0
60Zn
8Be
I=0
a
I=8
c) Spectra and E2(DI=2)transitions
Experimentally observed lowest level of sd band: 8+
I(12+sd 10+gs)/I(12+sd 10+sd) = 0.42 calc. aa = 0.54 exp.
I(10+sd 8+gs)/I(10+sd 8+sd) = 0.63 calc. aa = 0.60 exp.
60Zn
60Zn
5.Hyperdeformed states
in heavy ion collisions
Dinuclear states can be excited in heavy ion collisions.
The question arises whether these states are hyperdeformed states.
Shell model calculations of Cwiok et al. show that hyperdeformed states correspond to touching nuclei.
Possibility to form hyperdeformed states in heavy ion collisions.
Hyperdeformed states can be quasibound states of the dinuclear system.
V(R)
quasibound states
Rm
R
Investigation of the systems:
One to three quasibound states with
Energy values at L=0, quadrupole moments and moments of inertia of quasibound configurations are close to those estimated for hyperdeformed states.
80
L=0
80
L=0
Optimumconditions:
Decay of the dinuclear system by gtransitions to lower Lvalues in coincidence with quasifission of dinuclear system (lifetime against quasifission 1016 s).
Estimated cross section for formation of HDsystem is about 1 mb.
Heavy ion experiments with coincidences of grays and quasifission could verify the cluster interpretation of HDstates.
6. Rotational structure of 238U
Description of nuclear structure with dinuclear model for large mass asymmetries
Heavy cluster with quadrupole deformation + light spherical cluster, e.g. a particle
z1‘‘
z‘
A1
A2
R
Coordinates:
a) Polar angles from the spacefixed zaxis
: defining the bodyfixed symmetry x axis of heavy cluster x : defining the direction of R
eis the angle between R and the bodyfixed symmetry axis of heavy cluster.
b) Mass asymmetry coordinate with positive x values only:
spacefixed axis
z
sym. axis of heavy cluster
z1‘‘
q1, 1
q2, 2
e
z‘
mol. axis
Hamiltonian:
Moments of inertia:
Potential:
If C0is small: approximately two x independent rotators
If C0 is large: restriction to small e, x bending oscillations
Wave function:
Heavy cluster is rotationally symmetric: J1=0,2,4...
Parity of states: (1)J2
Example: 238U
238U
First excited state of mass asymmetry motion
Bending oscillations of heavy nucleus around the molecular(R) axis with small angle e
Moment of inertia of bending motion
Approximate eigenenergies
Oscillator energy of bending mode
K=2
n=1 bending mode
K=1
238U (=234Th+a)
7. Binary and ternary fission
Characteristics of DNS:
mass and charge numbers:
deform. parameters: (ratios of axes)
b
U
b = a/b
a
scission point at
<3MeV
Rmin Rb R
potential energy:
Total kinetic energy (TKE):
excitation energy:
S=Sn~8 MeV is excitation energy in neutron induced fission
S=0 in spontaneous fission
deformation energy Edef , difference to ground state
Relative primary (before evaporation of neutrons) yields of fission fragments:
with .
Examples:
Potential for neutroninduced fission of 235U leading to 104Mo + 132Sn and 104Zr + 132Te
Kinetic energy and mass distributions of spontaneous fission of 258Fm and 258No
104Mo + 132Sn
104Zr + 132Te
bimodal fission
258Fm
258No
b) Ternary fission
Ternary system consists of two prolate ellipsoidal fragments and a light charged particle (LCP) in between.
LCP has one or several alphaparticles and neutrons from one or both binary fragments.
Ternary system can not directly formed from the compound nucleus because of a potential barrier between binary and ternary fission valleys.
Examples:
ternary fission of 252Cf,
induced ternary fission of 56Ni (32S + 24Mg).
252Cf
56Ni
12C
8Be
D.G.