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## The clustering phenomenon

The formation of clusters is a fundamental aspect of nuclear many body dynamics together with the formation of mean field. Clustering aspects appear abundantly in many problems in both nuclear structure and nuclear collisions. The coexistence of the clustering aspect and the single particle aspect is a unique feature of nuclear many body systems.

The basic assumption of this model is such that nuclei can be described accurately in terms of a system of two-component nuclei; each with its free state characteristics; interacting through a deep local potential. However, many binary decompositions satisfy this minimal requirement.

Recently, Buck et al. proposed that the choice must be done with reference to the binding energies of the cluster and core.

In light stable nuclei it is well known that the clustering structure is of basic importance.I n the region of very light stable nuclei with A less than about 10, the clustering structure shows up in ground states. In heavier mass region of stable nuclei, however, the mean field is formed in ground states and the clustering structure appears in excited states.

States in nuclei that is have α-particles clusterizations with N = Z are typically not found in ground states, but are observed as excited states close to the decay thresholds into clusters, as was suggested in 1968 by Ikeda. The The Ikeda diagram is shown in Fig. 1, this links the energy required to liberate the cluster constituents to the excitation energy at which the cluster structures prevail in the host nucleus. The clear prediction, is that cluster structures are mainly found close to cluster decay thresholds.

Clustering gives rise to states in light nuclei which are not reproduced by the shell model, but the nuclear shell model does, however, play an important role in the emergence of nuclear clusters, and also in the description of special deformed nuclear shapes, which are stabilised by the quantal effects of the many-body system, namely the deformed shell gaps.

This connection is illustrated by the behaviour of the energy levels in the deformed harmonic oscillator], shown in Fig. 2. The numbers in the circles correspond to the number of nucleons, which can be placed into the crossing points of orbits. At zero deformation there is the familiar sequence of magic numbers which would be associated with spherical shell closures, and the associated degeneracies. At a deformation of the potential, where the ratios of the axes are 2:1, these same magic degeneracies reappear, but are repeated twice. This establishes an explicit link between deformed shell closures and clustering. This concept, fundamental for the understanding of the appearance of clustering within the nucleus,. the deformed magic structures with special stability are expected for particular combinations of spherical (shell-model) clusters. For example ( see table I), for super-deformed structures (2:1) the magic numbers have a decomposition into two magic numbers, of two spherical clusters, e.g. 20Ne≡ ( 16O+α). Thus, one would expect clusterisation not only to appear at a particular excitation energy, but also at a specific deformation. The Hyperdeformation (3:1) are related to cluster structures consisting of three clusters. For larger deformations longer α-chain states are produced.

Introduction

**Chapter I: Overview of nuclear structure**

I1.Introduction

I-2.The Nuclear shell model

I-2-1.Basic principles of the nuclear shell model

I-2-2.Harmonic Oscillator Potential

I-2-3.Woods-Saxon Potential

I-2-4.Shell model with pure configurations

I-2-5.Shell model with configuration mixing

I-3.Collective model

I-3-1 .Spherical nucleus

I-3-1-1.Quantisation of surface vibrations

I-3-1-2.Different vibration modes

I-3-2.Deformed nucleus

I-3-2-1.The Bohr Hamiltonian

I-3-2-2.Solution of the rotation-vibration Hamiltonian

I-3-2-2-1.The axially symmetric case

I-3-2-2-2.The Asymmetric Rotor

I-4.Algebraic collective model

I-4-1.Introduction

I-4-2.The U(6) algebra

I-4-5.Basic operators

I-4-3. Chaines of subalgebr of U(6).20

I-4-4. The IBM Hamiltonian

I-4-4-1.Vibration nuclei: the U(5) limit

I-4-4-2.Rotational nuclei: the SU(3) limit

I-4-4-3.γ-instable limit :the SO(6) limit

**Chapter II: Microscopic cluster model**

II-1.The clusteringphenomenon

II-2.Basic ideal

II-3.The appropriate core–cluster decomposition

II-4.Cluster-core potential

II-5.Hamiltonian diagonalisation

II-6.Quantum numbers and Wildermuth condition

II-7. Electromagnetic Transition

II-8.Application to Er isotopes

II-8-1.energy levels

II-8-1-1.Method of calculations

II-8-1-2.Result of calculations

II-8-2.Electric quadrupole transitions

**Chapter III: Extension of nuclear vibron model**

III-1.Introduction

III-2.The U(4) algebra

III-3.shain of subalgebra of the U(4)

III-4.Basic operators

III-5.The vibron model Hamiltonian

III-6.Dynamic symmetries

III-6-1.The U(3) limit

III-6-2.The SO(4) limit

III-7.Algebraic nuclear cluster model (6) (4)U U⊗

III-7-1.The U(6) ⊗ U(4) algebra

III-7-2.The U(6) ⊗ U(4) Hamiltonian

III-7-3.Dynamic symmetriesUU ⊗

III-7-3-2.SU(3) limit

III-7-3-1.Limit )3()5(ba

III-7-4.Transition Operators

III-8. Effect of the higher order term

III-8-1.Higher order terms in Hamiltonian

III-8-2.Higher order terms in the transition operators

General conclusion

Appendix A The eigenfunctions of the tridimensional isotropic harmonic oscillator

Appendix B

Algorithm of numerical of the calculations energy levels

Appendix C lie algebra