Microscopic description of three-body continuum states

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Microscopic description of three-body continuum states

Damman Alix

Faculté des Sciences Université Libre de Bruxelles

Thèse de doctorat présentée en vue de l'obtention du grade de docteur en Sciences Physiques

novembre 2011

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À ma grand-mère.

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Paradoxe de la lévitation félino-tartinique:

Les lois de la tartine beurrée stipulent de manière dénitive que le beurre doit toucher le sol alors que les principes de l'aérodynamique féline réfutent strictement la possibilité pour le chat d'atterrir sur le dos. Si l'assemblage du chat et de la tartine devait atterrir, la nature n'aurait aucun moyen de résoudre ce paradoxe. C'est pour cela qu'il ne tombe pas.

Rubrique-à-Brac, Marcel Gotlib

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Remerciements

Je tiens tout d'abord à remercier le professeur Pierre Descouvemont pour m'avoir proposé ce sujet et accepté de diriger ma thèse. Tout au long de mon travail, j'ai pu proter de son expertise, à la fois dans les domaines scientique et informatique. Je le remercie également pour sa grande disponibilité et sa patience pendant la rédaction de cette thèse.

Je remercie également les diérents membres du service de Physique Nu- cléaire Théorique et Physique Mathématique (PNTPM) de l'ULB. Je pense en particulier au professeur Daniel Baye pour les discussions scientiques et non-scientiques que j'ai pu partager avec lui, ainsi qu'au professeur Jean- Marc Sparenberg pour son soutien idéologique dans l'exploitation exclusive du système Linux. Mes remerciements s'addressent également à mes jeunes collègues : Pierre Capel, Jeremy Dohet-Eraly, Thomas Druet, Veerle Helle- mans, Horacio Olivares Pilon, Edna Carolina Pinilla Bertran, et Kouhei Washiyama. Je leur souhaite bonne continuation dans leurs projets.

Je remercie ma mère et mes cinq frères pour leur soutien moral et leurs nombreux encouragements. Je pense aussi à mes anciens camarades de Licence avec qui j'ai pu partager mes moments de doute.

Je remercie tout particulièrement Rose-Marie pour tous les moments vécus pendant ces quatre dernières années.

Pour terminer, je remercie les organismes qui ont nancé le présent travail, à savoir le Pôle d'Attraction Interuniversitaire (PAI) et l'Institut Interuniversitaire des Sciences Nucléaires (IISN).

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Glossary

RGM : Resonating Group Method

GCM : Generator Coordinate Method

MRM : Microscopic R-matrix Method

CSM : Complex Scaling Method

ACCCM : Analytical Continuation of the Coupling Constant Method

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A Slater determinants in the Harmonic Oscillator model 135 A.1 Factorization of the center of mass motion . . . 135 A.2 Product of two center of mass wave functions . . . 136 B Matrix elements between 0s Harmonic Oscillator states 139 B.1 one-body matrix elements . . . 139 B.2 two-body matrix elements . . . 139 C Eective charge matrix z^{J π} for charged systems α+p+p and n+α+α 141 D Coecients of the overlap kernel for theα+N+N,N+α+αandα+α+α

systems 145

E New formulas for the projected matrix elements of the central and

Coulomb potentials 149

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CONTENTS

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List of Figures

1 Schematic representation of the⁶He halo nucleus . . . 2

1.1 Description of a three-cluster system in the context of the RGM. An internal cluster wave functionφ_i is associated with each cluster. . . 14

1.2 Description of a two-cluster system in the context of the GCM. . . 21

1.3 Description of a three-cluster system in the context of the GCM. . . 23

1.4 Schematic representation of the Microscopic R-matrix Method (MRM) applied to a two-cluster system. . . 27

2.1 Representation of the three possible congurations for the Jacobi coordinates. From left to right, we have the congurations 1(23), 2(31) and 3(12). . . 30

2.2 Description of a three-cluster system in the context of the GCM. . . 38

2.3 Schematic representation of the Microscopic R-matrix Method (MRM) applied to a three-cluster system. . . 41

3.1 Decomposition of the action of the antisymmetrizerA. . . 54

4.1 Ratioℵ^{J π}_0,γK(R) for the α+N+N and N+α+α systems . . . 78

4.2 Ratioℵ^{J π}_0,Ki(R)for the α+α+α system . . . 79

4.3 Ratios ℵ^{J π}_12,γK(R) and ℵ_2,γK^{J π} (R) for the α+N+N systems . . . 82

4.4 Ratios ℵ^{J π}_11,γK(R),ℵ^{J π}_12,γK(R), andℵ^{J π}_2,γK(R)for the N+α+α systems . . 83

4.5 Function ψ_K^`^x^`^y(αR) . . . 86

4.6 3D plots of the function N˜_11,γ,γ^{J π} . . . 87

4.7 3D plot of the functionN˜_12,γ,γ^{J π} . . . 88

4.8 Eigenvaluesµ^{J π}₁ (R)andµ^{J π}₂ (R)for theα+N+N,N+α+α, andα+α+α systems . . . 90

4.9 Eigenvalues µ˜^{J π}_i of the α+N+N. . . 91

4.10 Eigenvalues µ˜^{J π}_i of the N+α+α andα+α+α systems. . . 92

5.1 Level scheme of the ⁶He and⁶Be nuclei . . . 94

5.2 Lowest energy curves associated with the kinetic energy . . . 98

5.3 Lowest energy curves associated with the central potential . . . 99

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LIST OF FIGURES

5.4 Lowest energy curves associated with the spin-orbit potential . . . 99

5.5 Lowest energy curves associated with the Coulomb potential . . . 100

5.6 Lowest energy curves E_K,1⁰⁺(R)for the α+n+nand α+p+p systems . . . 101

5.7 Lowest energy curves E₁^{J π} for the α+n+nand α+p+psystems . . . 103

5.8 Second lowest energy curves E₂^{J π} of theα+n+nand α+p+psystems . . 104

5.9 Diagonal vs eigenphase shifts for the α+p+p system . . . 106

5.10 Convergence of 0⁺ eigenphases with respect to the cut-o value K_max . 108 5.11 Stability of 0⁺ eigenphases according to small variations of the channel radius a . . . 109

5.12 0⁺ eigenphases for the α+n+n system . . . 110

5.13 0⁺,1⁻ and2⁺ eigenphases for theα+n+nand α+p+p systems . . . 112

5.14 0⁺, 1⁻ and 2⁺ eigenphases for the α+n+n and α+p+p systems and including the spin-orbit interaction . . . 113

5.15 Diagonal phase shifts for the J^π = 2⁺ continuum of theα+n+n system (K =2,4) . . . 114

6.1 Level scheme of the ¹²C nucleus . . . 119

6.2 Schematic representation of the ¹²C nucleus . . . 121

6.3 Energy curves associated with the ⁸Be=α+α system . . . 123

6.4 0⁺ and 2⁺ resonant phase shifts for the¹²C nucleus . . . 126

6.5 1⁻,3⁻, and 4⁺ resonant phase shifts for the¹²C nucleus . . . 128

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List of Tables

1.1 Parameter values of the Minnesota potential. . . 10

3.1 Two-body matrix elementshab|P_i|abi andhab|P_i|bai . . . 53

3.2 Numbers of terms N_n,N_v, andN_c. . . 58

3.3 Comparison between numbers Nv,Nc andN˜v,N˜c. . . 64

3.4 `max for the diagonal matrix elements of the overlap kernel and kinetic energy . . . 65

3.5 `_max for the diagonal matrix elements of the central potential . . . 66

3.6 Number of nonzero coecients Λ_N . . . 69

3.7 Cumulated computation times for the projected matrix elements of the overlap kernel . . . 70

3.8 Number of nonzero coecients Λ_V . . . 71

3.9 `_max for the matrix elements of the central potential (bis) . . . 72

3.10 Cumulated computation times for the matrix elements of the central potential . . . 72

4.1 Number of channels γK . . . 75

4.2 Number of nonzero channels in theα+α+αsystem . . . 77

4.3 Convergence of the ratio ℵ^{J π}_0,γK(R) . . . 80

4.4 Convergence of the ratios ℵ^{J π}_12,γK(R)and ℵ^{J π}_2,γK(R) . . . 84

4.5 Convergence of the ratio ℵ^{J π}_11,γK(R),ℵ^{J π}_12,γK(R), andℵ^{J π}_2,γK(R) . . . 84

5.1 Experimental values of energies and widths for the states of the⁶He and 6Be nuclei. See [TCG⁺02]. . . 95

5.2 Adjustment of the exchange-mixture parameterufor the0⁺and2⁺state of theα+n+nsystem . . . 97

5.3 Theoretical and experimental properties of the bound and resonant states of theα+n+nsystem . . . 115

5.4 Theoretical and experimental properties of the resonant states of the α+p+psystem . . . 116

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LIST OF TABLES

6.1 Experimental values of energies and widths for the states of the ¹²C nucleus. Energies are given relatively to the the0⁺₁ ground state and the

8Be+α threshold. See [AS90]. . . 120 6.2 Theoretical and experimental energies for the ⁸Be nucleus . . . 124 6.3 Theoretical and experimental properties of the 0⁺ and 2⁺ states of the

12C nucleus . . . 127 6.4 Theoretical and experimental properties of the 1⁻,3⁻, and 4⁺ states of

the¹²C nucleus . . . 129 C.1 Matrix elementsz^{J π}_γK,γ0K⁰associated with the triplet (L,S,π)=(0,0,+) and

theα+p+p system. . . 142 C.2 Matrix elementsz^{J π}_γK,γ0K⁰associated with the triplet (L,S,π)=(2,0,+) and

theα+p+p system. . . 142 C.3 Matrix elementsz^{J π}_γK,γ0K⁰associated with the triplet (L,S,π)=(1,1,+) and

theα+p+p system. . . 143 C.4 Matrix elementsz^{J π}_γK,γ0K⁰associated with the triplet (L,S,π)=(1,1,−) and

theα+p+p system. . . 143 C.5 Matrix elements z^{J π}_γK,γ0K⁰ associated with the triplet (L,S,π)=(0,1/2,+)

and then+α+α system. . . 144 C.6 Matrix elements z^{J π}_γK,γ0K⁰ associated with the triplet (L,S,π)=(2,1/2,−)

and then+α+α system. . . 144 D.1 Coecients C_ij^k of the function nk for the α+N+N systems (⁶He/⁶Be)

with S = 0. The two isolated nucleons are located along the Jacobi coordinateX. . . 145 D.2 Coecients C_ij^k of the function nk for the α+N+N systems (⁶He/⁶Be)

with S = 1. The two isolated nucleons are located along the Jacobi coordinateX. . . 146 D.3 Coecients C_ij^k of the function n_k for the N+α+α systems (⁹Be/⁹B).

The twoα particles are located along the Jacobi coordinateX. . . 147 D.4 First thirteen coecients C_ij^k of the function n_k for the α+α+α system

(¹²C). The twoα particles are located along the Jacobi coordinateX. . 148

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Introduction

The atomic nuclei are complex systems made up of fermions which strongly interact.

The idea of a central nucleus containing the bulk of the atomic mass and surrounded by a cloud of orbiting electrons has been proposed a century ago by Rutherford [Rut11].

Since then, nuclear physicists have developed many theoretical and experimental tools in order to explore the structure of the nucleus. In the mid-eighties, the development of radioactive beams gave access to the study of exotic nuclei at the very limits of the nuclear stability [THH⁺85a, THH⁺85b]. Investigation of nuclei close to the neutron or proton driplines have revealed intriguing features [Jon04]. The driplines correspond to the limit of the nuclear landscape.

The separation energy of the last nucleon or pair of nucleons decreases progressively when approaching the driplines. As a result, the number of bound states also decreases.

Among very neutron-rich light nuclei, some present an unusual structure. It consists in an inert core surrounded by one or two valence neutrons forming a halo. Their size is remarkably large compared to stable nuclei with the same number of nucleons.

The terminology halo nucleus has been introduced by Hansen and Jonson [HJ87]. The valence neutrons are loosely bound to the core so that halo nuclei have one or two bound states only. This makes the study of the continuum states of particular interest.

The two-neutron halo nuclei have a particularity. They are all borromean systems.

This means that they are bound three-body structures although none of the binary subsystem is bound [ZDF⁺93]. The adjective borromean refers to a picture of three linked rings in the coat of arms of the aristocratic Borromeo family. Breaking one of the three rings leaves the other two unlinked. In analogy, all borromean systems always dissociate into three parts. The simplest two-neutron halo nucleus is the⁶He nucleus.

It can be viewed as an α+n+n system. It is schematically represented in gure 1. It possesses only one0⁺bound state situated at 0.97MeV under the three-body breakup threshold [TCG⁺02]. This energy corresponds to the separation energy of the valence neutrons. It has also a2⁺ narrow resonant state situated1.8MeV above the0⁺ ground state.

The ⁶Be nucleus is the mirror nucleus of the ⁶He. It can be seen as an α+p+p system and is the lightest true two-proton emitter following the denition of Goldansky [Gol60]. The ⁶Be nucleus is not bound but is characterized by a 0⁺ narrow resonant state (Γ ' 90 keV) located 1.37 MeV above the two-proton emission threshold. This

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INTRODUCTION

state is the isobaric analogue of the0⁺ ground state of the⁶He. The mirror2⁺ state is broad (Γ'1.16 MeV) and situated 1.7 MeV above the0⁺ state.

6He=α+n+n bound

5He=α+n unbound

2n unbound

Figure 1: Schematic representation of the ⁶He halo nucleus (left) and of the borromean rings (right).

The ⁹Be nucleus is also borromean and can be seen asn+α+α system. It possesses only one three-body3/2⁻bound state1.57MeV under the breakup threshold [TKG⁺04].

Several resonances are present in its three-body continuum. The study of this system is important for the understanding of theα(αn,γ)⁹Be process playing a signicant role in nuclear astrophysics [WH92]. The ⁹B mirror system can be seen as a p+α+α system.

Like the⁶Be nucleus, this system is unbound but presents several resonances above the three-breakup threshold.

Another last but famous example of three-body state is the Hoyle state predicted by Hoyle [Hoy54] before it was observed experimentally. The Hoyle state is situated0.29 MeV above the α+⁸Be threshold and is the rst 0⁺ excited state of the ¹²C nucleus.

This state plays a crucial role in the helium burning in stars. Helium is burned through the triple-α process. This process proceeds in two steps. First, two α particles collide to form a ⁸Be nucleus. This nucleus is unstable and spontaneously decays into two α particles. Nonetheless, the lifetime of the ⁸Be is long enough to allow the capture of a thirdαin a stellar environment. The capture of the thirdαis enhanced by the presence of the Hoyle state close to theα+⁸Be threshold.

The above examples reveal the propensity of nucleons in a nucleus to congregate into smaller subunits usually called clusters when it is energetically advantageous. Accord- ing to Horiuchi and Ikeda [HI86], a cluster is a spatially localized subsystem composed of strongly correlated nucleons. An-cluster structure appears near or above the threshold energy of the breakup into the n clusters. Moreover, the α-clustering correlation

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INTRODUCTION

is the most prominent. The α particle has one of the highest binding energies per nucleon among light nuclei [vOFKE06]. On the theoretical side, cluster models have been developed to study nuclei presenting a molecule-like structure. They can be separated into two main categories: the non-microscopic models and the microscopic models.

The rst category makes use of eective clustercluster potentials to solve the three- body Schrödinger equation. Although such models involve rather small computation times, they present some important drawbacks. Firstly, the Pauli principle is approxi- mately treated and the forbidden states must be removed from the wave function (see for instance [TDE⁺00, TBD03, FN02]). Secondly, the derivation of eective cluster cluster potentials requires experimental data and are sometimes not well dened (e.g.

the⁹Li+npotential which is required to study the ¹¹Li).

The second category is based on the construction of a fully antisymmetrized wave function and on the resolution of the A-body Schrödinger equation







A

X

i=1

Ti+

A

X

i<j

j=1

Vij







Ψ =EΨ (1)

whereT_irepresent the individual kinetic energies,V_ij the potential energies between two nucleons, and Ψthe microscopic A-body wave function. Since microscopic models rely on the resolution of a A-body problem, they represent tedious calculations. However, they oers important advantages:

(i) All nucleons are taken into account using a fully antisymmetrized A-body wave function.

(ii) Eective NN-interactions are used which provide a more predictive power to the models.

(iii) The Pauli principle is exactly treated and the forbidden states are automatically eliminated.

The rst microscopic cluster model was the Resonating Group Method (RGM) still widely used and was proposed in 1937 by Wheeler [Whe37a, Whe37b]. The former article was entitled Molecular Viewpoints in Nuclear Structure. In the Wheeler representation, neutrons and protons are divided into various groups¹ which are continually broken and reformed. The RGM wave function associated with a three-cluster system is of the form

Ψ =Aφ1φ2φ3 g, (2)

whereAis the antisymmetrization operator,φi the internal cluster wave functions, and gthe relative wave function. The latter describes the relative motion between the three clusters.

1Wheeler did not use the word clusters

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INTRODUCTION

The main disadvantage of RGM is its non-systematic aspect. The Generator Coor- dinate Method (GCM) reduces the diculty of the RGM calculations [Hor70, Won75].

The introduction of generator coordinates (used as variational parameters) in the microscopic calculations allows the use of Slater determinants which are build in a systematic way.

Besides microscopic cluster models, we can mention the existence of the ab initio models which make use of realistic interactions, tted on many properties of the nucleon- nucleon system. These interactions may include many body forces as NNN-forces. The aim of ab initio models is to solve the A-body Schrödinger equation exactly and to use realistic interactions. Although such models represent the actual trend for the study of light systems, they are extremely dicult to implement. Recent developments of ab initio models [Pie02, NQSB09] are quite successful for spectroscopic properties of low-lying states. Ab initio calculations become available with systems with more than four nucleons but they are currently limited to A≈12. Also, their application to continuum states is a very dicult task [NPW⁺07, QN08]. This is why developing simpler microscopic models based on the clusters assumption but giving the possibility to treat exactly the 3-body asymptotic conditions is of great interest.

Dealing with three-cluster systems is much more complicated than with two-cluster systems. The hyperspherical formalism has proved to be very helpful the general context of three-body problems. It is widely used in nuclear or atomic physics [ZDF⁺93, Lin95].

The three-body Schrödinger equation is replaced by a set of coupled dierential equa- tions depending on a unique one-dimensional coordinate called hyperradius. This formalism can be used to investigate three-body bound and scattering states. For instance, non-microscopic calculations have been performed by Danilin et al. [DTVZ98]

and Descouvemont et al. [DTB06] to obtain the properties of bound and resonant states of the⁶He=α+n+nsystem. The Faddeev formalism is another way to deal with three- body problems and has for example been applied to the⁶He and⁶Li systems [ZDF⁺93]

and to the ¹²C system [FMK⁺04].

Regarding the microscopic models, Korennov and Descouvemont have developed a GCM method using hyperspherical generator coordinates and applied it to the ⁶He and ⁶Li nuclei [KD04]. However, their calculations were limited to bound or quasi- bound states. The use of the GCM itself does not give access to scattering states.

Asymptotically, the GCM wave functions have a Gaussian behavior incompatible with the Coulomb behavior. For large distances, the antisymmetrization and the nuclear interaction are negligible so the wave function is deduced from the Coulomb three-body Schrödinger equation. To connect the GCM wave function to its proper asymptotic behavior, the Microscopic R-matrix Method (MRM) can be used. This method enables to compute phase shifts and provides the wave function in the whole space. The MRM is the microscopic version of the usual R-matrix method rst proposed by Lane and Thomas in 1958 [LT58]. The MRM for two-body collisions has been developed by Baye, Heenen and Libert-Heinemann [BH74, Hee76, BHLH77].

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INTRODUCTION

The challenge now is to extend the MRM to three-body collisions using the hyperspherical formalism. This constitutes the main objective of the present work. With a GCM/MRM model in the hyperspherical formalism, we can study the three-cluster bound, resonant and scattering states. Furthermore, the derivation of three-body phase shifts involves large bases and important computation times. To this end, the objective of the present work is also to develop a semi-analytic method to compute GCM matrix elements in an accurate way. We apply the model to the mirror α+n+n and α+p+p systems.

In the present work, the antisymmetrization eects in three-cluster systems are also studied. We focus on theα+N+N (⁶He/⁶Be), N+α+α (⁹Be/⁹B) and α+α+α (¹²C) systems. We rst analyze the relative contributions of the nucleon exchanges between the three clusters. Secondly, we derive the eigenvalues associated with the overlap kernel.

Finally, in the case of the α+α+α system, the continuum states are investigated through theα+⁸Be collision using the multichannel MRM. The⁸Be nucleus is described as two-αstructure and the dierent channels correspond to the0⁺,2⁺, and4⁺states of

8Be. The present MRM calculation is an extension the one performed by Descouvemont and Baye [DB87] and in which the 2⁺ and 4⁺ states of ⁸Be have not been taken into account.

The present dissertation is organized as follows:

In chapter 1, we begin with the description of the microscopic Schrödinger equation.

Then, we give an overview of the RGM. The cases of two and three-cluster structures are considered. Next, we discuss the GCM which allows a systematic construction of the total wave function with the help of Slater determinants. Finally, we conclude by giving an overview of the MRM.

In chapter 2, we present the hyperspherical formalism and its use in a GCM/MRM model to investigate three-cluster scattering states.

In chapter 3, the semi-analytical method used to compute the GCM matrix elements is explained in detail. A particular decomposition of the overlap kernel is proposed in order to study the antisymmetrization eects. In the last section, we discuss a special technique which signicantly simplies the calculation of the matrix elements and allows to save computation time.

In chapter 4, antisymmetrization eects are examined. As a rst step, we focus on the long range behavior of the norm. Secondly, we solve the eigenvalue problem of the norm. In this chapter, we consider three kinds of three-cluster systems: α+N+N (⁶He/⁶Be),N+α+α (⁹Be/⁹B), andα+α+α (¹²C).

In chapter 5, we investigate the three-body bound and continuum states of the ⁶He and ⁶Be nuclei through the hyperspherical formalism. We compare our results with others obtained from dierent models.

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INTRODUCTION

In chapter 6, bound and continuum states of the ¹²C nucleus are studied from the ground state to 9 MeV above the α+⁸Be threshold. Calculations are based on the two-body version of the MRM in which⁸Be is described as a two-α structure.

Conclusions and outlooks of this work are discussed in the nal section.

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Chapter 1

The microscopic cluster model

1.1 Microscopic Hamiltonian

1.1.1 Assumptions

Physicists have long known that nuclei are built up of protons and neutrons [Hei32, Iva32]. They constitute the fundamental components of the nuclear matter. A proper study of the nuclei should consider the indistinguishability of the nucleons and the many-body character of such systems. We also know that the internal structure of nuclei and the antisymmetrization principle may play an important role in collisions between nuclei. Therefore, quantum eects are important and an exact quantum treatment of the collision should be applied.

Microscopic models aim to describe the dynamical properties of the nuclei using a unied theory of the nucleus [WT77]. They are able to investigate both the bound states and continuum states at the same time. In order to derive the properties of a nucleus consisting of A nucleons, or of a nuclear reaction involving A nucleons, microscopic models intend to solve a Schrödinger equation of the form

HΨ(¯q₁, . . . ,q¯_A) =



− ~² 2m_N

A

X

i

∆_i+

A

X

i<j

V_ij



Ψ(¯q₁, . . . ,q¯_A) =EΨ(¯q₁, . . . ,q¯_A), (1.1) wherem_N is the nucleon mass¹ and q¯_i denotes the space, spin and isospin coordinates of the i^th nucleon. The microscopic Hamiltonian is the sum of the individual kinetic energy terms and of the interactions terms². The Hamiltonian (1.1) is non-relativistic but it does not need to be. The non-relativistic assumption remains reasonable as long

1we shall assume that both proton and neutron have the same mass.

2we shall not consider NNN-,>3N-forces

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1. THE MICROSCOPIC CLUSTER MODEL

as the kinetic energy per nucleon is much less than the nucleon rest mass of about 940 MeV. This is the case in the present work.

To proceed further, let us dene the internal Hamiltonian and the Coulomb potential. The internal Hamiltonian is dened in the center of mass (c.m.) reference frame.

Ifp_i represents the momentum associated with the nucleoni, then the c.m. momentum is dened by

Pc.m.=

A

X

i

pi, (1.2)

and the c.m. kinetic energy by

Tc.m.= P_c.m.²

2Am_N. (1.3)

The internal Hamiltonian is given by

H_int=H−T_c.m., (1.4)

The nucleon-nucleon (NN) interactionVij contains a nuclear termV_ij^N and a Coulombian term V_ij^C. The expression of the Coulombian term is

V_ij^Coul = e²

|q_i−qj| (1/2−t_z,i) (1/2−t_z,j) (1.5) whereq_i represents the space coordinates of the i^th nucleon andt_i its isospin ³. Since we use electronvolts and fermis as units of measurement for the energies and distances, the value of the constant e² equals 1.44 MeV fm. The factor(1/2−t_z,i) (1/2−t_z,j) is zero if one or both nucleons are neutrons.

In the c.m. reference frame, the microscopic Schrödinger equation becomes

H_intΨ =E_intΨ, (1.6)

in whichE_int does not include the kinetic energy of the c.m. motion. The resolution of the equation (1.6) is far from being simple. To meet the challenge, we must overcome two main diculties:

(i) We must solve, with the least possible approximations, a many-body problem.

Pursuing such an objective quickly becomes a complicated task. The diculties increase signicantly with the number of nucleons A. The ab initio models performed calculations using realistic interactions like AV18 [WSS95], Paris [LLR⁺80] or CD-Bonn [Mac01]. Realistic interactions describe all deuteron properties and the NN-phase shifts. Ab initio calculations may also include a NNN- force which can play an important role in many-body systems. However, such

3we use the conventiontz,j= 1/2for neutrons andtz,j=−1/2for protons

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1.1 Microscopic Hamiltonian

models are very complicated to implement and highly time-consuming (see for instance [NPW⁺07, NQSB09, QN08]). Therefore, ab initio models remain limited to a few nucleons systems (A.12) and cannot treat the two-body scattering problem for A > 5. The treatment of 3-body scattering states will not be accessible for a long time. Simpler models based on eective interactions involve reasonable computing capabilities while maintaining a microscopic description of the nuclei.

The eective interactions are adapted from experimental data and requiring a few parameters.

(ii) The microscopic wave function must be projected on the good quantum numbers in order to respect the symmetries of the Hamiltonian.

In the present work, we shall solve the Schrödinger equation (1.6) in the case of continuum states (E>0). To achieve this goal, we shall use a model in which the nucleons of a nucleus are grouped in sub-structures called clusters. The nucleons interact through an eective NN-interaction and the total wave function is fully antisymmetrized and projected on the good quantum numbers.

1.1.2 Choice of the eective nuclear NN-interaction

For the eective nuclear NN-interaction, we use an interaction of the form

V_ij =V_ij^Cent+V_ij^S.O., (1.7) whereV^Cent is the central part of the nuclear interaction andV^S.O.the spin-orbit part.

There is no tensor term in (1.7) but its eects are simulated by a suitable choice of the central term.

The central part is rotationally invariant and has the general expression V_ij^Cent =

Nk

X

k

V_k(|q_i−q_j|) (W_k+B_kP_σ−H_kP_τ−M_kP_σP_τ). (1.8) The spin exchange operatorPσ is dened by

P_σ = 1

2(1 +σ_iσ_j), (1.9)

with σi = 2si. The symbol si represents the spin of the i^th nucleon. The isospin exchange operatorP_τ is dened by

Pτ = 1

2(1 +τiτj), (1.10)

with τ_i = 2t_i. The central interaction contains four forces: the Wigner force, the Bartlett force, the Heisenberg force, and the Majorana force. The three latter are

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linked to the exchange of spins, isospins and spatial coordinates respectively. The weights of the four forces are determined by the four factorsW_k,B_k,H_k andM_k. In our calculations, we shall use harmonic oscillator (H.O.) functions as single particle (s.p.) wave functions. Thus, an eective potential with a Gaussian shape will simplify the calculations in our model. The functions Vk are expressed as

Vk(|q_i−qj|) =V0,k exp −(q_i−q_j)² κ²_k

!

. (1.11)

The number of Gaussians is usually ranged from one to three.

All the nuclei we shall study are composed only of α, n and p clusters. In the present work, we shall use the Minnesota potential [TLT77]. This potential is able to correctly reproduce the s-wave phase shift of both n-p and p-p systems at low energies.

It also yields good results concerning the α+N and α+α phase shifts. Finally, the ground state of the deuteron obtained from this potential is in good agreement with the experimental value. The Minnesota potential is constructed from three Gaussians and it also contains an exchange-mixture parameter u. Its value is supposed to be close to 1. The parameterucan be slightly modied to adjust the energy of the ground state or in some cases of excited states. The values for the other parameters are given in table 1.1.

k V0,k [MeV] κ_k [fm] Wk Bk Hk Mk

1 200 0.8200579 u/2 0 0 1−u/2

2 -178 1.2509777 u/4 u/4 1/2−u/4 1/2−u/4 3 -91.85 1.4664711 u/4 −u/4 −1/2 +u/4 1/2−u/4

Table 1.1: Parameter values of the Minnesota potential.

In addition, we can also mention the Volkov potential [Vol65] which is also of the form (1.8) but contains two Gaussians and no Bartlett and Heisenberg forces. This potential can also be slightly modied by changing the value of a parameterm. It was obtained from tting the bulk properties of s- and p-shell nuclei.

The spin-orbit part represents the spin-orbit coupling between two nucleons and has the form [TLT78, BP81]

V_ij^S.O.=−2S₀

~²ν⁵ exp −(qi−qj)² ν²

!

`_ij.s_ij. (1.12) where`_ij = (q_i−q_j)×(p_i−p_j)/2 ands_ij =s_i+s_j are the relative momentum and the total spin of the nucleons, respectively. The parametersS₀ and ν are the strength

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1.2 The Resonating Group Method (RGM)

and the range of this eective spin-orbit force. In the present work, we x the value of ν to 0.1 fm, which is numerically equivalent of using a zero-range force (contact force).

The value ofS0 is set to 30 MeV fm⁵ 1.1.3 Constants of motion

In classical mechanics, a constant of motion is a quantity that is conserved through- out the motion. In quantum mechanics, an observableA is a constant of motion if its eigenvalues do not vary over time or, equivalently, if it commutes with the Hamiltonian ([A, H] = 0). Those which commute amongst them can form a complete set of com- muting observables (C.S.C.O.) with the Hamiltonian. In fact, the constants of motion are linked to the symmetries of the Hamiltonian [Bay05]. The microscopic Hamiltonian (1.1) is invariant under space-inversion, translation and rotation.

The invariance under translation is related to the c.m. momentum operator P_c.m.

(1.2). The wave function solution of (1.6) is translationally invariant. The invariance under space-inversion is related to the total parity operatorΠwhich inverses the space coordinates of all nucleons. Finally, the invariance under rotation is linked to the total angular momentum operatorJand its z-componentJz. We assume that the operators Π, J and Jz are dened with respect to the c.m. of the A nucleons. The C.S.C.O associated with the internal HamiltonianH_intis given by

H_int, J², J_z,Π . (1.13)

Thus, the good quantum numbers are the total angular momentum numberJ, its projection M, and the parity number π. The Schrödinger equation (1.6) can be rewritten as

HintΨ^{J M π}=EΨ^{J M π}. (1.14)

1.2 The Resonating Group Method (RGM)

The Resonating Group Method (RGM ) [WT77, Hor77, Sai77, SLYV03, Bri08] is the oldest microscopic method for treating the clustering phenomena. It was developed by Wheeler [Whe37a, Whe37b] and dedicated to the study of nuclei which present a strong cluster structure. It allows clusters to exchange nucleons using a fully antisymmetrized trial wave function. In this way, the Pauli principle is properly taken into account.

The origin of the term Resonating Group Method is due to the way the method accounts for the structure in clusters. Because of the clustering correlations, protons and neutrons are grouped into clusters. Those are continually being broken up and reformed in various ways. The nucleus oscillates from one cluster structure to another, hence the expression resonating group.

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1.2.1 RGM wave function

Let us consider a nucleus containingAnucleons (Zprotons,N neutrons) and divided intoNcclusters, each containingAi nucleons (i= 1, . . . , Nc). The nucleons are indexed with the two indices i, j. The rst one indicates to which cluster a nucleon belongs, while the second is used to enumerate the nucleons contained in the same cluster. If the vector qi,j represents the spatial coordinates of the nucleon j of the cluster i, the center of mass coordinate r_c.m.,iof the cluster iis given by

rc.m.,i= 1 A_i

Ai

X

j

qi,j, (1.15)

From theNcc.m. coordinatesrc.m.,i, we can dene the c.m. coordinaterc.m.of the total nucleus andN_c−1relative coordinatesr_i as follows

r_c.m.= 1 A

Nc

X

i

A_i r_c.m.,i, (1.16)

r_i =r_c.m.,i− PNc

k>iA_kr_c.m.,k P_N_c

k>iA_k . (1.17)

The relative coordinatesriare also called unnormalized Jacobi coordinates. More detail about the Jacobi coordinates shall be given in the next chapter. If Nc = 2, there is only one relative coordinate r which is the dierence between the c.m. coordinates of the two clusters

r=rc.m.,1−rc.m.,2. (1.18)

In the case of a three-cluster structure, we have







r1=rc.m.,1− A₂r_c.m.,2+A₃r_c.m.,3 A2+A3

, r₂=r_c.m.,2−r_c.m.,3.

(1.19) TheAi−1 internal spatial coordinatesξ_i,j associated with the clusteriare dened as

ξ_i,j =q_i,j−r_c.m.,i (j= 1, . . . , A_i−1). (1.20) A change of coordinates using (1.17) and (1.20) allows to rewrite the internal Hamilto- nian (1.4) as

H_int=

Nc

X

i

H_i

!

+T_rel(r₁, . . . ,r_N_c−1) +

Nc

X

k>i Ai

X

j A_k

X

l

V_ij,kl(|q_i,j−q_k,l|), (1.21) whereH_iis the internal Hamiltonian of the i^thcluster,T_relthe kinetic energy related to the relative motion of the clusters, andV_ik,jl the interaction between the k^th nucleon of

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the i^th cluster and the l^thnucleon of the j^thcluster. The last three summations in (1.21) regroup all the nucleon-nucleon interactions which occurs between two dierent clusters.

Asymptotically, it represents the summation of the Coulomb interactions between all clusters. The expression of the internal HamiltonianH_i is given by

Hi =− ~² 2mN

Ai

X

k=1

∆ξ_i,k+

Ai

X

k>l=1

Vij,il(|q_i,j−qi,l|), (1.22) The internal cluster wave functionφi and the internal energyEi satisfy the Schrödinger equation

Hi φi(ξ_i) =Ei φi(ξ_i), (1.23) where ξ_i stands for all internal degrees of freedom of cluster i. Each function φi is normalized, translationally invariant and antisymmetric with respect to the nucleons.

Given the general form of the Hamiltonian (1.21) and the Schrödinger equation (1.23), we can now introduce the general expression of the RGM wave function

Ψ_RGM =Aφ₁(ξ₁)φ₂(ξ₂). . . φ_N_c(ξ_N_c) g(r₁, . . . ,r_N_c−1), (1.24) where A is an operator used to antisymmetrize the wave function, φ_i (i = 1, . . . , N_c) are the internal wave functions andg(r₁, . . . ,r_N_c−1) is the relative wave function. The latter describes the inter-cluster motion. A schematic representation of the construction of the RGM wave function in the case of a three-cluster structure is given in gure 1.1.

The internal energy of the nucleus is deduced by solving

H_intΨ_RGM =E Ψ_RGM. (1.25)

The wave function (1.24) is fully antisymmetrized thanks to the action of the projector A. Such a projector is called antisymmetrizer and is dened as

A= 1 A!

X

p

(−)^p P_p, (1.26)

where Pp corresponds to any permutation of parity (−)^p among the A! possible permutations ofA nucleons. The antisymmetrizerAplays a crucial role when the clusters are close to each other. The protons and neutrons are indistinguishable and it is im- possible to know whether a nucleon belongs to a cluster rather than to another. As a consequence, the relative coordinates ri are purely quantum quantities which have no physical existence unless the clusters are suciently distant from each other. Indeed, in the case of a two-cluster structure, there areA!/(A₁!A₂!(1 +δ_A₁_A₂δ_Z₁_Z₂))equivalent ways to dene the relative coordinater[BHLH77].

As a matter of fact, the RGM wave function (1.24) is written in a simplied form.

In practice, the RGM wave function must take into account the internal state of each cluster and have the good quantum numbersJ,M andπ. The symbolJ represents the total angular momentum number,M its projection, andπthe total parity. Each cluster

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φ₁(ξ₁)

φ2(ξ₂)

φ3(ξ₃) r1 r₂

r_c.m.,1

r_c.m.,2

r_c.m.,3

O

Ψ =Aφ1(ξ₁)φ2(ξ₂)φ3(ξ₃) g(r1,r2)

Figure 1.1: Description of a three-cluster system in the context of the RGM. An internal cluster wave functionφi is associated with each cluster.

is characterized by an intrinsic spinS_i, an intrinsic spin projection ν_i, and a parity π_i. In the present work, we consider only clusters with maximum four nucleons so that the internal orbital angular momenta are all null. To simplify the notation, we shall gather the sets of quantum numbersS_i,ν_i, andπ_iunder the symbolsλ_i. The total intrinsic spin Sof the system is thus determined by coupling all the intrinsic spinsS_i. Furthermore, a relative angular momentum operator`_i can be associated with each relative coordinate ri. It is dened by`i=ri×p_i=−i~ri×∇_r_i. The angular momentumLis obtained by coupling all the relative angular momenta`i. Each possible (L,S) doublet denes a new channel. The total angular momentumJ is given by coupling the quantum numbersL and S. Finally, the total parityπ is given by

π=π₁π₂. . . π_N_c (−)^`¹^+`²^+···+`^Nc. (1.27) Now, in order to illustrate how to construct properly the RGM wave function Ψ^{J M π}, let us rst consider the case of two-cluster systems and then three-cluster systems.

Two-cluster RGM wave function

In the case of two-cluster systems, we start from the RGM wave function

Ψ^λ¹^λ² =Aφ^λ¹(ξ₁)φ^λ²(ξ₂)g(r). (1.28)

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The relative wave functiong can be developed on the spherical harmonic basisY_`^m^`(Ω)

g(r) =

∞

X

`=0

`

X

m`=−`

Y_`^m^`(Ω)g`(r). (1.29) The partial RGM wave Ψ` is built by selecting the term which contains the spherical harmonicY_`. Thenceforward, we have

Ψ^λ_`¹^λ²^m^`=A φ^λ¹(ξ₁)φ^λ²(ξ₂)Y_`^m^`(Ω)g_`(r). (1.30) The next step consists in constructing the total intrinsic spinSby coupling the intrinsic spins S₁ and S₂. To do this, we introduce the two-cluster spin function

φ^ν_S(ξ) =X

ν1ν2

hS₁S2ν1ν2|Sνi φ^λ¹(ξ₁)φ^λ²(ξ₂) = [φ_S₁(ξ₁)⊗φ_S₂(ξ₂)]^Sν, (1.31) whereξ₁ andξ₂ have been grouped together under the symbolξ. Finally, the quantum numbersJ and M appear naturally in the channel wave function ϕ^{J M π}_`S dened as

ϕ^{J M π}_`S (ξ,Ω) =X

νm`

h`Sm_`ν|J Mi φ^ν_S(ξ)Y_`^m^`(Ω) = [φ_S(ξ)⊗Y_`(Ω)]^{J M}, (1.32) where the total parity π is equal toπ₁π₂(−)^`. Given the above denitions, we are now able to develop the RGM wave Ψ^{J M π} as a summation of partial waves Ψ^{J M π}_`S . Each partial wave is equal to the product of a channel waveϕ^{J M π}_`S and a partial relative wave g^{J π}_`S. Hence, we write

Ψ^{J M π}=X

`S

Ψ^{J M π}_`S =AX

`S

ϕ^{J M π}_`S (ξ,Ω) g^{J π}_`S(r). (1.33)

Three-cluster RGM wave function

For three-cluster systems, we generalize the above procedure. The relative wave function gis developed as

g(r1,r2) = X

`1`2LML

[Y_`₁(Ω1)⊗Y_`₂(Ω2)]^LM^L g^L_`₁_`₂(r1, r2), (1.34) where the two angular momentum `₁ and `₂ are coupled to obtain the global angular momentumL. We introduce the three-cluster spin function

φ^ν_S(ξ) =

φ^λ¹(ξ₁)⊗h

φ^λ²(ξ₂)⊗φ^λ³(ξ₃)

iS23Sν

, (1.35)

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where we dene rst the intermediate intrinsic spinS₂₃and then, the total intrinsic spin S. The symbol ξ stands for ξ₁, ξ₂, and ξ₃. The three-cluster channel wave function ϕ^{J M π}_`

1`2LS is expressed as

ϕ^{J M π}_`₁_`₂_LS(ξ,Ω₁,Ω₂) =h

φ_S(ξ)⊗[Y_`₁(Ω₁)⊗Y_`₂(Ω_r₂)]^LiJ M

. (1.36)

Here, the total parityπ is given byπ =π₁π₂π₃(−)^`¹^+`². Finally, we develop the RGM wave Ψ^{J M π} as

Ψ^{J M π}= X

`1`2LS

Ψ^{J M π}_`₁_`₂_LS =A X

`1`2LS

ϕ^{J M π}_`₁_`₂_LS(ξ,Ω1,Ω2) g^{J π}_`₁_`₂_LS(r1, r2). (1.37)

1.2.2 Asymptotic behavior of the RGM wave function

When the clusters are suciently far away from each other, the contribution of the nucleon permutations between them to the RGM wave function (1.24) becomes negligible. The antisymmetrization only occurs inside the clusters which can now be considered as separated nuclei. In the case of a two-cluster system, the relative coordinate rdened in (1.18) becomes a physical observable, i.e. the distance between the two nuclei. Besides, the nuclear NN-interaction being short-ranged, its contribution can be also neglected at large distances. Only the Coulombian contribution subsists.

To simplify the following discussion, let us consider a system made up of two clusters only. The case of three-cluster systems will be discussed in the next chapter. The asymptotic expression of the Hamiltonian (1.21) for a two-cluster system is given by

H_as−−−→

r→∞ H₁+H₂+T_rel(r) +Z₁Z₂e²

r , (1.38)

and the wave function (1.33) tends to Ψ^{J M π}_RGM,as −−−→

r→∞

X

`S

ϕ^{J M π}_`S (ξ,Ω) g_`S,as^{J π} (r). (1.39) It should be noted that the channel wave functions ϕ^{J M π}_`S satisfy the asymptotic or- thogonality relation [BHLH77]

hϕ^{J M π}_`S |ϕ^J_`0⁰S^M⁰⁰^π⁰i=δ_{J J}⁰δ_{M M}⁰δ_ππ⁰δ_``⁰δ_SS⁰, (1.40) wherertends to innity and the integration is performed overξandΩ. The asymptotic Schrödinger equation

H_as Ψ^{J M π}_RGM,as =E^{J π} Ψ^{J M π}_RGM,as (1.41)

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1.3 The Generator Coordinate Method (GCM)

can be solved exactly. For a positive energy solution (E^{J π} >0) and a xed entrance channel`⁰S⁰, the relative wave function g_`S^{J π} takes the form [DB10]

g^{J π}_`S,as(r)∝ I_`(η, kr)δ_``⁰δ_SS⁰−U_`S,`^{J π}⁰_S⁰O_`(η, kr)

, (1.42)

whereU^{J π} is the collision matrix,I` the incoming wave function,O`the outgoing wave function, η the Sommerfeld parameter, and k the wave number. The collision matrix contains all the information about the collision and its knowledge is required to derive the phase shifts.

1.3 The Generator Coordinate Method (GCM)

The Generator Coordinate Method (GCM ) is a general method used to develop a microscopic wave function able to reproduce certain collective properties of nuclei, like vibrations, rotations, or cluster structures [HW53, GW57, WT77]. The GCM is a variational method which makes use of variational parameters in the microscopic wave function to describe collective degrees of freedom. In addition to the reproduction of the nuclear collective motion of nuclei, an interesting feature of this method is the ability to easily restore the symmetries of the eective Hamiltonian (see further subsection 1.4.2).

The GCM can be applied to the study of cluster nuclei as an alternative of the RGM.

On the one hand, the microscopic wave function built from the RGM spontaneously exhibits a cluster structure and is fully antisymmetrized. But on the other hand, it is really dicult to use the RGM in numerical applications because of the action of the antisymmetrizer A on the nucleon coordinates. Thus, to avoid the unsystematic character of the RGM, Brink [Bri66] was the rst to propose to apply the GCM to cluster models. As GCM basis functions, Brink adopted the many-center cluster wave functions which has been used earlier by Margenau [Mar41]. Such functions are called Brink-Margenau wave functions or simply Brink wave functions. They depend on a generator vector which simulates the inter-cluster distance. The GCM describes the inter-cluster relative motion by a linear superposition of Brink functions. As we shall see in the subsection 1.4, the Brink functions projected on the good quantum numbers can be constructed in a systematic way. The downside is that the GCM is only valid in a small region of the space where the clusters are assumed to be close. The correct asymptotic behavior of the total wave function is recovered using the R-matrix method as described in the section 1.5.

1.3.1 GCM as a variational method

As we mentioned in the subsection 1.1.1, the A-body Schrödinger equation (1.6) cannot be solved exactly. We can at least attempt to approach the exact solution using a variational method.

Microscopic description of three-body continuum states