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 238 U 7. Binary and ternary fission 8. Summary Contents

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

 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. 8 Be  + .  First evidence for nuclear molecules in scattering of 12 C on 12 C and 16 O on 16 O 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). 1. Introduction

1.Relative motion of nuclei: formation of dinuclear system in heavy ion collisions, molecular resonances, decay of dinuclear system: fission, quasifission, emission of clusters 2.Transfer of nucleons between nuclei: change of mass and charge asymmetries between the clusters The dinuclear system has two main degrees of freedom:

Mass asymmetry coordinate A2A2 A1A1

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 Let us first consider some selected aspects of the dinuclear system model. The degrees of freedom of this model are  internuclear motion ( R )  mass asymmetry motion (  )  deformations (vibrations) of clusters  rotation (rotation-oscillations) of clusters  single-particle motion

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:

152 Dy

Dinuclear system model is used in various ranges of  :  = : large quadrupole deformation, hyperdeformed states  = : quadrupole and octupole deformations are similar, superdeformed states  ~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: B 1, B 2, B 0 are negative binding energies of the clusters and the united (|  |=1) nucleus. V(R, ,I) is the nucleus-nucleus potential. Example: 152 Dy

 152 Dy 50 Ti+ 102 Ru 26 Mg+ 126 Xe

: 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. Moment of inertia of DNS: Example: 152 Dy

 = 0.34: 50 Ti+ 102 Ru, Hyperdeformed properties: U=20 MeV above g.s., about estimated energy of L=0 HD-state of 152 Dy, (calc)=131 MeV -1, (est)=130 MeV -1,  2 (calc)=1.3,  2 (est)0.9. (calc)=104 MeV -1, (exp)=85±3 MeV -1, Q 2 (calc)=24 eb ( 2 =0.9), Q 2 (exp)= 18±3 eb Similar:  = 0.71: 22 Ne+ 130 Ba 26 Mg+ 126 Xe and 22 Ne+ 130 Ba have SD properties.  = 0.66: 26 Mg+ 126 Xe, Superdeformed properties:

For nuclear structure studies we assume  as a continuous coordinate and solve a Schrödinger equation in mass asymmetry. Wave function    contains different cluster configurations. At higher excitation energies: statistical treatment of mass transfer. Diffusion in  is calculated with Fokker-Planck or master equations. 2.4 Mass asymmetry motion

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. A Z (A-4) (Z-2) +  - particle 

Ba

splitting _+_+ oscillations in  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

238 U 236 U 234 U 232 U

223 Ra 3/ (I,K - ) (I,K + ) 3/2

225 Ra

Here: application of dinuclear model to structure of 60 Zn, 194 Hg and 194 Pb a) Cluster structure of 60 Zn Zn 56 Ni+tresh. 2.7 MeV above g.s. Assumption: g.s. band contains -component Zn 52 Fe+ 8 Be, tresh MeV above g.s. / 48 Cr+ 12 C, tresh MeV above g.s. Extrapolated head of superdef. band: 7.5 MeV Assumption: superdeformed band contains 8 Be- component. 4. Normal- and superdeformed iiiiibands

Unified description of g.s. and sd bands by dynamics in mass asymmetry coordinate. b) Potential U(I) for 60 Zn mono-nucleus (U(I=0) = 0 MeV 56 Ni+4.5 MeV 52 Fe+ 8 Be 5.1 MeV 48 Cr+ 12 C 9.0 MeV Stepwise potential because of large scale in  Barrier width is fixed by 3 - state (3.504 MeV).

60 Zn I=0 x=-1 for  x1 for  

60 Zn 8 Be  I=0

I=8

c) Spectra and E2(I=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.

60 Zn

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 V(R) RRmRm

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

Optimum conditions: Decay of the dinuclear system by - transitions to lower L-values in coincidence with quasifission of dinuclear system (lifetime against quasifission s). Estimated cross section for formation of HD- system is about b. Heavy ion experiments with coincidences of -rays and quasifission could verify the cluster interpretation of HD-states.

6. Rotational structure of 238 U Description of nuclear structure with dinuclear model for large mass asymmetries Heavy cluster with quadrupole deformation + light spherical cluster, e.g.  - particle z 1 ‘‘ R z‘ A2A2 A1A1

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  is the angle between R and the body- fixed symmetry axis of heavy cluster. b) Mass asymmetry coordinate with positive x values only:

z z‘ z 1 ‘‘           mol. axis sym. axis of heavy cluster space- fixed axis

Hamiltonian: Moments of inertia: Potential:

If C 0 is small: approximately two x independent rotators If C 0 is large: restriction to small , x bending oscillations Wave function: Heavy cluster is rotationally symmetric: J 1 =0,2,4... Parity of states: (-1) J 2 Example: 238 U

238 U First excited state of mass asymmetry motion

Bending oscillations of heavy nucleus around the molecular(R) axis with small angle  Moment of inertia of bending motion

Approximate eigenenergies Oscillator energy of bending mode

238 U (= 234 Th+) K=1 n=1 bending mode K=2

7. Binary and ternary fission a) 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. mass and charge numbers: deform. parameters: (ratios of axes) Characteristics of DNS:

b a a/b R min R b R <3MeV U scission point at potential energy:

S=S n ~8 MeV is excitation energy in neutron induced fission S=0 in spontaneous fission deformation energy E def, difference to ground state Total kinetic energy (TKE): excitation energy:

Relative primary (before evaporation of neutrons) yields of fission fragments: with. Examples: Potential for neutron-induced fission of 235 U leading to 104 Mo Sn and 104 Zr Te Kinetic energy and mass distributions of spontaneous fission of 258 Fm and 258 No

104 Mo Sn 104 Zr Te bimodal fission

258 Fm 258 No

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: 1.Relative probabilities for the formation of different binary systems 2.Relative probabilities of ternary system, normalized to unity for each binary system Examples: ternary fission of 252 Cf, induced ternary fission of 56 Ni ( 32 S + 24 Mg).

252 Cf

56 Ni 12 C 8 Be

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  -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 238 U.  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.