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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


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

  • A dinuclear system or nuclear molecule is a cluster configuration of two (or more) nuclei which touch each other and keep their individuality, e.g. 8Be a + a.

  • First evidence for nuclear molecules in scattering of 12C on 12C and 16O on 16O by Bromley, Kuehner and Almqvist (Phys. Rev. Lett. 4 (1960) 365); importance for element synthesis in astrophysics.

  • Dinuclear system concept was introduced by V. V. Volkov (Dubna).

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



Applications of dinuclear system model

  • Nuclear structure phenomena: normal-, super- and hyperdeformed bands, alternating parity bands

  • Fusion to superheavy nuclei, incomplete fusion

  • Quasifission, no compound nucleus is formed

  • Fission

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

  • The degrees of freedom of this model are

  • internuclear motion ( R )

  • mass asymmetry motion ( h)

  • deformations (vibrations) of clusters

  • rotation (rotation-oscillations) of clusters

  • single-particle motion

Let us first consider some selected aspects of the dinuclear system model.

2.1 Deformation

Dinuclear configuration describes quadrupole- and octupole-like deformations and extreme deformations as super- and hyperdeformations.

Multipole moments of dinuclear system:

Comparison with deformation of axially deformed nucleus described by shape parameters:


  • Dinuclear system model is used in various ranges of h:

  • h=0 - 0.3: large quadrupole deformation, hyperdeformed states

  • h=0.6 - 0.8: quadrupole and octupole deformations are similar, superdeformed states

  • h~1: linear increase of deformations, parity splitting

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 nucleus-nucleus potential.

Example: 152Dy




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 HD-state of 152Dy, (calc)=131 MeV-1, (est)=130 MeV-1, 2(calc)=1.3, 2(est)0.9.

 = 0.66:26Mg+126Xe,

Superdeformed properties:

(calc)=104 MeV-1, (exp)=85±3 MeV-1, 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 Fokker-Planck 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.








Parity splitting is explained by reflection-asymmetric shapes and is describable with octupole deformations.

Here we show that it can be described by an asymmetric mass clusterization.

Configuration with alpha-clustering can have the largest binding energy.

AZ (A-4)(Z-2) + a - particle







oscillations in h

Lower state has positive parity, higher state negative parity. Energy difference depending on nuclear spin is parity splitting.



Positive parity

Negative parity



















(I,K-) (I,K+)



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 a-component.

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 8Be-component.

Unified description of g.s. and sd bands by dynamics in mass asymmetry coordinate.

b) Potential U(h , I) for 60Zn

mono-nucleus (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).



x=h-1 for h>0 x= h+1 for h<0






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.



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.


quasibound states



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.






Decay of the dinuclear system by g-transitions to lower L-values in coincidence with quasifission of dinuclear system (lifetime against quasifission 10-16 s).

Estimated cross section for formation of HD-system is about 1 mb.

Heavy ion experiments with coincidences of g-rays and quasifission could verify the cluster interpretation of HD-states.

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







a) Polar angles from the space-fixed z-axis

: defining the body-fixed symmetry x axis of heavy cluster x : defining the direction of R

eis the angle between R and the body-fixed symmetry axis of heavy cluster.

b) Mass asymmetry coordinate with positive x values only:

space-fixed axis


sym. axis of heavy cluster


q1, 1

q2, 2



mol. axis


Moments of inertia:


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


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


n=1 bending mode


238U (=234Th+a)

7. Binary and ternary fission

  • Binary fission

  • The fissioning nucleus with A and Z is described at the scission point as a dinuclear system with two fission fragments in contact.

Characteristics of DNS:

mass and charge numbers:

deform. parameters: (ratios of axes)



b = a/b


scission point at


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 .


Potential for neutron-induced 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



b) Ternary fission

Ternary system consists of two prolate ellipsoidal fragments and a light charged particle (LCP) in between.

LCP has one or several alpha-particles 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.

  • Calculation procedure:

  • Relative probabilities for the formation of different binary systems

  • Relative probabilities of ternary system, normalized to unity for each binary system


ternary fission of 252Cf,

induced ternary fission of 56Ni (32S + 24Mg).





  • 8. Summary

  • The concept of the dinuclear system describes nuclear structure phenomena connected with cluster structures, the fusion of heavy nuclei to superheavy nuclei, the 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.

  • Parity splitting is interpreted by oscillations with even and odd parities in a potential with minima at the a-cluster fragmentation.

  • Normal- and superdeformed bands can be explained by the dynamics in the mass (or charge) asymmetry coordinate.

  • Hyperdeformed states can be seen as quasibound, molecular states in the internuclear potential.

  • Mass asymmetry motion and bending oscillations of the heavy cluster in very mass asymmetric dinuclear systems are used to interpret the structure of 238U.

  • Relative probabilities for binary and ternary fission can be statistically calculated with the potential depending on mass asymmetry and deformation.

  • Further studies on mass asymmetry motion and rotation in the dinuclear system model are necessary.


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