Dinuclear system model                   in nuclear structure and reactions
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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

Dinuclear system model in nuclear structure and reactions


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


Dinuclear system model in nuclear structure and reactions

First lecture

I. Dinuclear effects in nuclear spectra and fission


Dinuclear system model in nuclear structure and reactions

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


Dinuclear system model in nuclear structure and reactions

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


Dinuclear system model in nuclear structure and reactions

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).


Dinuclear system model in nuclear structure and reactions

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


Dinuclear system model in nuclear structure and reactions

Mass asymmetry coordinate

A2

A1


Dinuclear system model in nuclear structure and reactions

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


Dinuclear system model in nuclear structure and reactions

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


Dinuclear system model in nuclear structure and reactions

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.


Dinuclear system model in nuclear structure and reactions

2.1 Deformation

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

Multipole moments of dinuclear system:


Dinuclear system model in nuclear structure and reactions

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


Dinuclear system model in nuclear structure and reactions

152Dy


Dinuclear system model in nuclear structure and reactions

  • 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


Dinuclear system model in nuclear structure and reactions

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


Dinuclear system model in nuclear structure and reactions

152Dy

50Ti+102Ru

26Mg+126Xe


Dinuclear system model in nuclear structure and reactions

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


Dinuclear system model in nuclear structure and reactions

 = 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.


Dinuclear system model in nuclear structure and reactions

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.


Dinuclear system model in nuclear structure and reactions

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+


Dinuclear system model in nuclear structure and reactions

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

a

a


Dinuclear system model in nuclear structure and reactions

Ba


Dinuclear system model in nuclear structure and reactions

_

+

splitting

oscillations in h

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


Dinuclear system model in nuclear structure and reactions

potential

wavefunctions

Positive parity

Negative parity

x


Dinuclear system model in nuclear structure and reactions

238U

236U

234U

232U


Dinuclear system model in nuclear structure and reactions

-

223Ra

-

-

-

+

-

+

-

+

+

+

3/2

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

3/2


Dinuclear system model in nuclear structure and reactions

225Ra


Dinuclear system model in nuclear structure and reactions

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.


Dinuclear system model in nuclear structure and reactions

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).


Dinuclear system model in nuclear structure and reactions

60Zn

I=0

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


Dinuclear system model in nuclear structure and reactions

60Zn

8Be

I=0

a


Dinuclear system model in nuclear structure and reactions

I=8


Dinuclear system model in nuclear structure and reactions

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.


Dinuclear system model in nuclear structure and reactions

60Zn


Dinuclear system model in nuclear structure and reactions

60Zn


Dinuclear system model in nuclear structure and reactions

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.


Dinuclear system model in nuclear structure and reactions

Hyperdeformed states can be quasibound states of the dinuclear system.

V(R)

quasibound states

Rm

R


Dinuclear system model in nuclear structure and reactions

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.


Dinuclear system model in nuclear structure and reactions

80

L=0

80

L=0


Dinuclear system model in nuclear structure and reactions

Optimumconditions:

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.


Dinuclear system model in nuclear structure and reactions

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


Dinuclear system model in nuclear structure and reactions

Coordinates:

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:


Dinuclear system model in nuclear structure and reactions

space-fixed axis

z

sym. axis of heavy cluster

z1‘‘

q1, 1

q2, 2

e

z‘

mol. axis


Dinuclear system model in nuclear structure and reactions

Hamiltonian:

Moments of inertia:

Potential:


Dinuclear system model in nuclear structure and reactions

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


Dinuclear system model in nuclear structure and reactions

238U

First excited state of mass asymmetry motion


Dinuclear system model in nuclear structure and reactions

Bending oscillations of heavy nucleus around the molecular(R) axis with small angle e

Moment of inertia of bending motion


Dinuclear system model in nuclear structure and reactions

Approximate eigenenergies

Oscillator energy of bending mode


Dinuclear system model in nuclear structure and reactions

K=2

n=1 bending mode

K=1

238U (=234Th+a)


Dinuclear system model in nuclear structure and reactions

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)


Dinuclear system model in nuclear structure and reactions

b

U

b = a/b

a

scission point at

<3MeV

Rmin Rb R

potential energy:


Dinuclear system model in nuclear structure and reactions

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


Dinuclear system model in nuclear structure and reactions

Relative primary (before evaporation of neutrons) yields of fission fragments:

with .

Examples:

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


Dinuclear system model in nuclear structure and reactions

104Mo + 132Sn

104Zr + 132Te

bimodal fission


Dinuclear system model in nuclear structure and reactions

258Fm

258No


Dinuclear system model in nuclear structure and reactions

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.


Dinuclear system model in nuclear structure and reactions

  • Calculation procedure:

  • Relative probabilities for the formation of different binary systems

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

Examples:

ternary fission of 252Cf,

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


Dinuclear system model in nuclear structure and reactions

252Cf


Dinuclear system model in nuclear structure and reactions

56Ni

12C

8Be


Dinuclear system model in nuclear structure and reactions

  • 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.


Dinuclear system model in nuclear structure and reactions

  • 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.


Dinuclear system model in nuclear structure and reactions

  • 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.

D.G.


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