electromagnetic processes in few-body systems...final-state interactions in the quasi-elastic...

ELECTROMAGNETIC PROCESSES

IN FEW-BODY SYSTEMS

by

GAOTSIWE JOEL RAMPHO

ELECTROMAGNETIC PROCESSES IN FEW-BODY SYSTEMS

by

GAOTSIWE JOEL RAMPHO

submitted in accordance with the requirements

for the degree of

DOCTOR OF PHILOSOPHY

in the subject

PHYSICS

at the

UNIVERSITY OF SOUTH AFRICA

PROMOTER : PROF. S. A. SOFIANOS

JOINT PROMOTER : PROF. M. BRAUN

NOVEMBER 2010

I declare that “ ELECTROMAGNETIC PROCESSES IN FEW-BODY SYSTEMS “ is my own

work and that all the sources that I have used or quoted have been indicated and acknowledged

by means of complete references.

————————————- ————————————-SIGNATURE DATE

(Mr. G. J. Rampho)

Acknowledgments

I would like to thank the following individuals for providing support in terms of expert advice

and informative discussions relating to the work in this thesis :

• Prof. S. A. Sofianos : University of South Africa, SOUTH AFRICA.

For his expert guidance, support and inspiration during the research related to this thesis.

• Prof. M. Braun : University of South Africa, SOUTH AFRICA.

For providing expert guidance and assistance in compiling this thesis.

• Prof. L. M. Lekala : University of South Africa, SOUTH AFRICA.

For his support and encouragement towards finalising and submitting this thesis.

• Prof. S. Oryo : Tokyo University of Science, JAPAN

For informative discussions and for providing Japanese-to-English translations of some

useful material related to this thesis.

• Dr. S. S. Shimanskiy : Joint Institute for Nuclear Research, RUSSIA.

For enlightening discussions relating to the relevance of some of the physics questions

addressed in this thesis.

I am also thankful to

• the University of South Africa for the financial support and motivation,

• members of the Physics Department (University of South Africa) for their support, and

• my family for their patience, understanding and encouragement

during the compilation of this thesis.

i

Summary

Electromagnetic processes induced by electron scattering off few-nucleon systems are theoret-

ically investigated in the non-relativistic formalism. Non-relativistic one-body nuclear current

operators are used with a parametrization of nucleon electromagnetic form factors based on

recent experimental nucleon scattering data. Electromagnetic form factors of three-nucleon

and four-nucleon systems are calculated from elastic electron-nucleus scattering information.

Nuclear response functions used in the determination of differential cross sections for inclusive

and exclusive quasi-elastic electron-nucleon scattering from the 4He nucleus are also calculated.

Final-state interactions in the quasi-elastic nucleon knockout process are explicitly taken into

account using the Glauber approximation. The sensitivity of the response functions to the

final-state interactions is investigated.

The Antisymmetrized Molecular Dynamics approach with angular momentum and parity pro-

jection is employed to construct ground state wave functions for the nuclei. A reduced form of

the realistic Argonne V18 nucleon-nucleon potential is used to describe nuclear Hamiltonian.

A convenient numerical technique of approximating expectation values of nuclear Hamiltonian

operators is employed. The constructed wave functions are used to calculate ground-state ener-

gies, root-mean-square radii and magnetic dipole moments of selected light nuclei. The theoret-

ical predictions of the nuclear properties for the selected nuclei give a satisfactory description

of experimental values. The Glauber approximation is combined with the Antisymmetrized

Molecular Dynamics to generate wave functions for scattering states in quasi-elastic scattering

processes. The wave functions are then used to study proton knockout reactions in the 4He

nucleus. The theoretical predictions of the model reproduce experimental observation quite well.

Keywords : Antisymmetrized Molecular Dynamics; Angular Momentum Projection;

Glauber Approximation; Current Operators; Few-Body Systems; Electron-Nucleus Scattering;

Electromagnetic Transitions; Differential Cross Section.

ii

Contents

1 Introduction 1

1.1 Electron-Nucleus Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Few-Body Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Antisymmetrized Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Antisymmetrized Molecular Dynamics 11

2.1 The Quantum Many-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 The Two-Body Nucleon Potential . . . . . . . . . . . . . . . . . . . . . . 14

2.2 The Ground-State Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Spin and Parity Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 The Variational Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 The Scattering Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Electromagnetic Transitions 37

3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 The Transition Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 The Transition Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 The Differential Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.5 Nuclear Current Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5.1 One-Body Current Operators . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5.2 Effects of Two-Body Currents . . . . . . . . . . . . . . . . . . . . . . . . 59

iii

CONTENTS

4 Transitions in Few-Nucleon Systems 63

4.1 Ground-State Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.1.1 Energy, Radius and Magnetic Moment . . . . . . . . . . . . . . . . . . . 65

4.1.2 Density Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 Elastic Electron-Nucleus Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.1 Charge Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.2 Magnetic Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3 Quasi-elastic Electron-Nucleon Scattering . . . . . . . . . . . . . . . . . . . . . . 87

4.3.1 Inclusive Electron-Proton Scattering . . . . . . . . . . . . . . . . . . . . 90

5 Summary and Conclusions 95

A Jacobi Coordinates 100

B Energy Expectation Values 107

B.1 Overlap Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

B.2 The Variational Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

C Transition Matrix Elements 116

C.1 Nuclear Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

C.2 Nuclear Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

C.3 Electromagnetic Form Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

C.4 Nuclear Fragmentation Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . 123

iv

Chapter 1

Introduction

One of the aims of nuclear physics studies is to establish a complete theoretical description

of the structure of nuclear systems. The correct theoretical description of nuclear structure is

expected to help explain and accurately predict different electromagnetic process in nuclei [1].

Fundamental to a complete description of nuclear structure are the wave function describing

nuclear systems, the Hamiltonian describing interactions in the nucleus and electromagnetic

form-factors describing the distributions of charge and currents in the nucleus. None of these

three components is completely understood and, therefore, none can be completely determined

for a given nuclear system. As a result, theoretical models of these components, based on

different approximations that are guided by experimental observations, are usually employed

in the description of nuclear systems. The quality of such models is often judged by their

ability to explain existing experimental observations. Parallel to the theoretical developments,

the developments in experimental technologies has led not only to the availability of more pre-

cise experimental data, but also to data in kinematical regions previously not accessible. The

availability of precise experimental data in a wide kinematical region allow for more accurate

quantitative testing and, therefore, development of realistic theoretical models that, in turn,

generate more accurate predictions of experimental outcomes [2].

In this thesis elastic and inelastic electron-nucleus scattering are used to investigate some elec-

tromagnetic processes in few-nucleon systems. The nuclear systems are modeled with the

antisymmetrized molecular dynamics wave functions. In Section 1.1 some of the advantages

of using electron scattering in probing nuclei are indicated. Some of the reasons why many

investigations in nuclear physics involve few-body systems are given in Section 1.2. Section 1.3

is devoted to summering the historical development of the antisymmetrized molecular dynamics

approach. The organisation of the thesis is outlined in Section 1.4.

1

Introduction

1.1 Electron-Nucleus Scattering

Most theoretical and experimental investigations into the properties and structure of atomic

nuclei are based on scattering techniques. The formulation of a standard scattering problem

involves (i) a projectile particle, preferably of known properties and structure, (ii) a target sys-

tem, of unknown properties and structure, and (iii) the interaction Hamiltonian, of known form,

between the projectile and the target. In the study of few-nucleon systems considerations of

prescription (i) identify hadrons (simple internal structure), leptons (“no” internal structure),

and electromagnetic radiation (no charge) as the most suitable choices as projectile probes.

Some of the information obtainable from the scattering process may, to some extent, be unique

to the type of projectile used. Therefore, the choice of the projectile may be informed by the

objectives of the investigation. The use of a given projectile has associated advantages and

disadvantages. For example, in probing properties of nucleon-nucleon interactions in nuclei,

hadrons appear to be the more relevant projectile candidates because the interaction indicated

in (iii) is fundamentally the same as the nucleon-nucleon interactions in the probed nucleus.

However, the internal structure of the projectile hadron and the lack of a complete theory

explaining hadronic interactions introduce additional complexities that limit the amount of

information, about the target, obtainable from the scattering process. Because of the strong

coupling between the hadron and target nucleus the structure information of the two systems

become difficult to separate. For the same reason, the hadron will also tend to experience a

very short mean-free-path inside the nuclear target, which limits the volume of the target that

is accessible. However, considerable progress has been made in accurately separating projectile

properties from those of the nuclear target in hadron-nucleus scattering [3, 4]. This has im-

proved the amount and quality of information obtainable from the use of hadrons in probing

nuclei.

Electrons are charged particles with very small rest-mass and interact with the probed nucleus

via the electromagnetic and weak forces. This form of interaction is based on the theory of

quantum electrodynamics, a theory that is very well developed and understood. Therefore,

prescriptions (i) and (iii) suggest that electrons will form a more favourable projectile in nu-

2

Introduction

clear structure investigations. There are additional advantages associated with using electrons

for probing the structure of nuclei and nucleons. Interactions among constituents in a nucleus,

and in a nucleon, are of strong nature while the electron-nucleus interaction is mainly electro-

magnetic, which is comparatively very weak. For a nucleus of charge Ze the electromagnetic

interaction with an electron is of order Zα where α is the fine structure constant. The inter-

action of the electron with the nucleus is, therefore, not expected to distort the initial nuclear

state significantly. As a result, perturbation theories, where the electron-nucleus interactions

are treated as small perturbations to the internal nuclear interactions, provide reliable theo-

retical tools for the analysis of the electron-nucleus scattering process. In such analyses, the

scattering cross section of the process is very well approximated by a product of two independent

factors, one depending on the structural information of the electron, and the other depending

on the structural information of the nucleus. Therefore, information about the electromagnetic

structure of the target is easily extracted. A similar factorisable form of the cross section arise

when electromagnetic radiation (real photons) are used as the probing tool. Investigations with

polarised electrons or photons provide additional more detailed information about the target.

Real photons can only be polarised along the direction transverse to the direction of propaga-

tion and the photon energy is always equal to the magnitude of the momentum transferred to

the target. However, electrons can be polarised along longitudinal and transverse directions.

Also, the energy and momentum transferred to the target by the electron can be independently

adjusted to carefully select desired kinematical conditions for specific nuclear structure studies.

Therefore, electron scattering provide a simple, clean and reliable mechanism of probing nuclear

and nucleon structure.

Depending on the magnitude of the energy and momentum transfer to the target nucleus differ-

ent processes can be induced in the nucleus [5, 6]. Generally, at low energy transfers the virtual

photon tend to interact with the nucleus as a whole. Below fragmentation energy threshold of

the system the dominant processes include excitations to discrete energy and resonant states of

the system. For intermediate energy transfers, above the fragmentation threshold, the photon

tend to interact with correlated groups of constituent nucleons or individual nucleons in the

target nucleus. This can result in a nucleon or a group of nucleons being removed from the

3

Introduction

nucleus. At even higher energies the photon can interact with single nucleons thereby exciting

resonant states of the nucleon. These different kinematical energy regions are usually identified

with elastic, inelastic, quasi-elastic and deep-inelastic electron scattering. These energy regions

can also be identified using the magnitude of the transferred momentum as reference. An infor-

mative review of the historical development of the experimental and theoretical electron-nucleus

scattering techniques is given in reference [6].

1.2 Few-Body Systems

General nuclear, and nucleon, structure as well as properties of the nucleon-nucleon interac-

tions are the focus of most investigation in nuclear physics. Besides the understanding of static

and dynamical properties of nuclear matter, such studies are aimed at constructing a compre-

hensive description of properties of nucleon-nucleon interactions. Few-nucleon systems provide

unique favourable environment for such investigations. The simplest bound nuclear system is

a proton-neutron system, and is the least dense light nucleus. This nucleus provides a better

environment for the study of the least distorted nucleon structure, as well as intermediate and

long-range behaviour of the nucleon-nucleon force. The bound three-nucleon system provides

the simplest nuclear system for investigating, in addition to the two-body nucleon-nucleon force,

the three-body nucleon-nucleon-nucleon force, and the effects of interchanging a proton and a

neutron (isospin-dependence) on the properties of nuclei. The bound four-nucleon system, two

neutrons and two protons, is the most dense light nucleus. Therefore, this nucleus provides a

good environment for studying, among other concepts, nucleon-nucleon correlation effects, and

the short-range behaviour of the nucleon-nucleon force. Systems of five and six bound nucleons

can be used to investigate instabilities and clustering effects in nuclei. The properties of nuclear

matter indicated can be investigated using many-body nuclear systems, however, most of these

concepts may be diluted or masked by many-body effects. In theoretical investigations, the

interaction models in few-nucleon systems can be treated realistically and the resulting dynam-

ical equations can be solved exactly. The formulation and solution of dynamical equations for

many-body systems is quite challenging.

4

Introduction

Progress towards a better understanding of the nuclear force has been made over the years.

Based on the accumulated experimental nucleon-nucleon scattering data different phenomeno-

logical nucleon-nucleon interaction models have been suggested. The models are constructed

by fitting the models to existing nucleon-nucleon scattering data as well as some properties of

the 2H nucleus [7, 8, 9, 10, 11]. These interaction models are known as modern or realistic

nucleon-nucleon potentials and are able to explain most of the static properties of light nuclei

[12]. Since the exact form of the short-range behavior of the nucleon-nucleon interaction is not

completely determined, yet, the short-range part of many of these potential models is often

musked by introducing short-distance cut-off factors. Knowledge of the short-range behavior of

the nucleon-nucleon force may reveal some information about the limits and boundaries between

hadronic degrees of freedom and quark degrees of freedom. The suggestion that the short-range

behavior of the strong nuclear interaction could manifest itself through short-range nucleon-

nucleon correlations and momentum distributions in a nucleus [13] refocused the need for a bet-

ter understanding of, and therefore more intensive investigations into, these concepts. Hence,

there are continued experimental [14, 15, 16, 17, 18, 19, 20] and theoretical [21, 22, 23, 24, 25]

investigations of these concepts.

The structure of few-nucleon systems and light nuclei is widely and continuously investigated,

both theoretically and experimentally. Therefore, reviews on a variety of aspects concerning

nuclear structure, including the electromagnetic properties, can be readily found in the lit-

erature [2, 26, 27]. Over the years a variety of methods have been developed and refined in

the study of electromagnetic processes in nuclei. Very accurate wave functions for bound and

scattering states in few-nucleon systems can now be constructed using realistic Hamiltonian for

the systems. A demonstration of the level of accuracy that can now be achieved in describing

ground state properties of the four-nucleon system using seven different state-of-the-art meth-

ods is given in reference [28]. The use of some of these methods in the study of electromagnetic

processes in three-nucleon and four-nucleon systems is shown in references [2, 26]. Although the

construction of bound-state wave functions in many theoretical approaches is almost standard,

the construction of scattering-state wave functions still pose a challenge in most. Scattering

wave functions for two-body and three-body scattering reactions can be treated accurately in

5

Introduction

the Faddeev formalism [2, 29]. In the hyperspherical harmonics approach some progress in

being made towards the construction of accurate wave functions for two-body scattering pro-

cesses [30]. The Lorentz Integral Transform approach treats bound states and scattering states

in the same way [31, 32, 33]. In this approach the traditional solutions to the Schrodinger

equation are not determined. Instead, the integral transforms of the nuclear response functions

from Schrodinger-like equations are generated. Some microscopic simulation methods have also

been used in the study of static and dynamical properties of nuclei [34, 35, 36, 37, 38, 39, 40].

These methods are continuously developing.

A realistic final state wave function of a fragmentation scattering process is required to satisfy

a number of conditions. Because of current conservation requirements, and gauge invariance

in the analysis, the wave functions for the scattering state must be orthogonal to the wave

function for the initial state of the system. The function must also take into account final

state interactions of the fragments in the reaction products. When these considerations are

ignored the scattering state can be approximated by a plane wave. For very high energy (ultra-

relativistic) scattering the factors contributing to the distortion of the plane wave become very

small and the plane wave approximation become very good. In practice, not many methods can

incorporate all the contributing factors in a consistent way. Therefore, for high energy scattering

the final state is usually approximated with a plane wave and the distortion factors included

as corrections. A simple but relatively accurate technique for constructing scattering wave

functions that takes into account final state interactions is a semi-classical approximation. The

familiar versions of this approach are the Glauber Approximation (GA) [41] and the Generalized

Eikonal Approximation (GEA) [42]. These approximations are widely used in nuclear scattering

investigations [27, 43, 44, 45]. The GA is applicable to processes involving relativistic energies

and is valid only at small scattering angles. The energy-range of applicability and the scattering-

angle validity of the GEA are less restricted, and therefore, the GEA can generate more accurate

results than the GA [46].

6

Introduction

1.3 Antisymmetrized Molecular Dynamics

Following the Time-Dependent Cluster Model [47, 48], a number of microscopic simulation

models were developed [49, 50] for application to fermionic systems. These models combine

Fermi-Dirac statistics with elementary quantum mechanics to treat the motion of particles in

a system [49]. However, the models are not fully quantum mechanical and do not assume a

shell structure for the system. The Fermionic Molecular Dynamics (FMD) model [49] is more

general than the Antisymmetrized Molecular Dynamics (AMD) model [50]. As a results the

AMD model has received much more attention than FMD. The AMD total wave function is

constructed as Slater determinant of single-particle shifted Gaussian wave functions. The shift

parameters of the Gaussian functions are complex variational parameters which are treated as

generalised coordinates of the system. The width parameters are taken as free real parameters

and are chosen to be the same for all the Gaussian. The equations of motion for the variational

parameters are determined from the quantum variational principle. The equations are then

solved by using the frictional cooling technique [51] to determine the parameters.

The AMD approach was used to study the dynamics of heavy-ion collisions [51] and elastic

proton-nucleus scattering [52, 53]. Clustering in nuclei as well as angular distributions of scat-

tered protons in proton-nucleus scattering can be well explained with the AMD model [52].

Some properties and processes in physical systems are governed by conservation laws related to

parity and total angular momentum of the system. The AMD wave function does not have def-

inite parity nor does it possess definite total angular momentum. As a result, for applications

in realistic investigations of properties of physical systems the AMD wave function requires

some improvement. A number of modifications have since been introduced in the AMD for-

malism. In reference [54] numerical technique are used to project the AMD wave function

onto the eigenstates of parity and total angular momentum. The resulting wave functions have

definite parity and total angular momentum. In addition, a number of different techniques

were considered for constructing a more flexible total wave function. One way was to use linear

combinations of variational spatial, spin and isospin functions to represent single-particle wave

functions [55, 56]. The resulting wave functions are suitable for investigating systems with

7

Introduction

tensor forces. The other approach is to use a linear combination of several Slater determinants

[39, 54, 57] to represent the total wave function of the system. In the approach of reference [58]

orthogonal single-particle wave functions characteristic of the Hartree-Fock orbitals are con-

structed from the AMD wave function. The AMD+Hartree-Fock wave function is also more

flexible than the original AMD wave function. These modifications to the AMD wave function

introduced significant improvements in the description of ground-state properties, mean-field

and cluster structure of nuclear systems. It should be noted that most of the studies indicated

employed phenomenological nucleon-nucleon potentials of Gaussian radial form. The main rea-

son for this is that the expectation values of this type of potentials can be evaluated analytically.

To extend the application of the AMD approach to realistic potentials further proposals were

made. In references [59, 60] the G-matrix approach was used to incorporate two-body corre-

lations in the wave function. The resulting Bruckner-AMD wave function can be constructed

with realistic nuclear potentials. In reference [61, 62] Jacobi coordinates are employed to con-

struct the wave function. Many-body correlations are included, more variational parameters

are considered and the effects of the center-of-mass are completely and explicitly removed.

Most realistic potentials are of non-Gaussian form and are, in many cases, expanded as the

sum-of-Gaussian for application in the AMD approach.

In this thesis a technique for evaluating expectation values of non-Gaussian potentials suggested

in reference [63] is employed for the first time in the AMD approach. The technique approxi-

mates the expectation value of the potential operator with a rapidly converging series of Talmi

integrals. Ground-state properties of selected few-nucleon systems are determined with the sim-

pler version of the reduced Argonne nucleon-nucleon potential. This thesis also discusses the

use of AMD in constructing wave functions that give a good approximation of nucleon-nucleus

scattering processes. Such wave functions are used in the analysis of electron-induced nuclear

fragmentation processes. The wave functions are constructed by combining the AMD approach

with the GA. In this thesis such wave functions are used to investigate inclusive and exclusive

one-nucleon fragmentation processes in few-nucleon systems. Wave functions for both bound

and scattering states are constructed with the parity and angular momentum projected AMD

8

Introduction

approach. The theoretical result are compared with experimental data for the same kinematics.

1.4 Thesis Outline

This thesis is made up of five chapters and three appendices. In Chapter 2 the basic for-

mulation of the AMD formalism are given. The formalism is centered on the construction of

bound-state wave functions of nuclear systems using a semi-realistic non-relativistic Hamilto-

nian. The evaluation of the relevant expectation values is explained in Appendix B. The parity

and total angular momentum projection of the fundamental AMD wave function is given. A

description of the projection operators is continued in Appendix C. The chapter is concluded

with an outline of a model combining the AMD formalism with the GA to construct scattering

wave functions. The model is based on nucleon-nucleus scattering and is adapted for nucleon

knock out processes. In Chapter 3 the general formalism of electron scattering off nuclei in the

one-photon-exchange approximation is recalled. The Hamiltonian describing the interaction

between external electron and non-relativistic nuclei, in the impulse approximation, is indi-

cated. Non-relativistic one-body nuclear charge and current operators are introduced. The

construction of the nuclear transition operators and their multipole expansion is explained.A

brief description of the derivation of the electron-nucleus scattering differential cross section is

also given. The differential cross sections are applicable to inclusive and exclusive one-nucleon

knock-out processes. A phenomenological parametrisation of nucleon electromagnetic form fac-

tors constructed from recent nucleon-nucleon scattering experimental data is used.

In Chapter 4 the quality of the AMD model wave functions for bound and scattering states

for nuclei is tested. The ground-state energies, root-mean-square radii and magnetic dipole

moments of selected few-nucleon systems are calculated. The ground-state wave functions are

further tested by calculating charge and magnetic form factors of three-nucleon and four-nucleon

systems. The results are compared with results obtained from other theoretical methods as

well as selected experimental data. The two-body fragmentation process of the 4He nucleus

is investigated. The focus is on the electron-induced proton knockout reaction 4He(e, e′ p)3H

which is widely investigated, both experimentally and theoretically. The AMD results are also

9

Introduction

compared with results obtained from other theoretical methods as well as selected experimental

data. The neutron knockout is not as widely investigated as the proton knockout. Therefore, not

many results for the process are readily available for comparison. In Chapter 5 the properties

of the AMD model wave functions are summarised, a reflection on the ability of the AMD

model wave functions to explain experimental observations is given and some conclusions on the

competency of the AMD model as compared to conventional methods are drawn. In Appendix A

Jacobi vectors for the position and momentum vectors in a many-body system are introduced.

10

Chapter 2

Antisymmetrized Molecular Dynamics

Based on the Time-Dependent Cluster Model [47, 48] a special case of the time-dependent

variational theory that is applicable to fermionic systems was developed [51]. This special case,

called antisymmetrized molecular dynamics (AMD), is characterized by two features. One fea-

ture is that the spatial component of the wave function describing individual constituent par-

ticles of the system is represented by a Gaussian wave packet with a complex time-dependent

centroid. The variational parameters are determined from the quantum variational principle.

The width parameter of the Gaussian packet is treated as a real constant. The other feature

is that the total wave function of the system is constructed as a Slater determinant of single-

particle wave functions. The spins of the particles are also considered fixed. This formulation

of the wave function is referred to as the original version AMD. The AMD formalism does

not fully incorporate quantum effects. As a result, there have been numerous improvements

introduced in the AMD formalism. These include representing single-particle wave functions

with either deformed Gaussian [57] or linear combinations of Gaussian [58], superposing several

Slater determinants [39] to represent the total wave function of the system. Also, in addition

to projecting the AMD wave function onto the eigenstates of parity and total angular momen-

tum of the system [54], AMD was coupled with other methods like the generator coordinate

method and the Hartree-Fock approach as well as the incorporation of the Vlasov equation [64]

to study various aspects of numerous nuclear systems. The most recent improvement is the

use of Jacobi coordinates to describe the internal relative motions of constituent particles of

a system [65]. These modifications generated the so-called advanced or extended versions of

AMD. A review and summary of the different improvements of the AMD formalism are found

in references [39, 40].

11


In this chapter the basic formulation of the AMD approach is highlighted. In Section 2.1 the

Hamiltonian for a general non-relativistic quantum many-body system is outlined. The Hamil-

tonian involves the treatment of realistic two-body interaction potentials. In Section 2.2 the

construction of the AMD wave function for the ground state of a system is outlined. A descrip-

tion of the construction of wave functions with definite parity and total angular momentum

from the AMD wave function is given in Section 2.3. In Section 2.4 the determination of the

ground state energy of the system using the AMD wave function is discussed. In Section 2.5

an outline of the derivation of the equations of motion for the complex variational parameters

is given. A variational technique used in determining the parameters is also explained. The

technique requires a slight modification to the equations of motion for the parameters. The

resulting equations resemble Hamilton’s classical equations of motion for canonical variables.

The matrix elements related to the evaluation of expectation values of the relevant operators

are given in Appendix B. In Section 2.6 a technique of combining the AMD formalism with

the Glauber Approximation to construct scattering wave functions is outlined. The technique

is specialised to nucleon-nucleus scattering.

2.1 The Quantum Many-Body Problem

Consider a system of A particles with individual mass mi and position vector ri, where i =

1, 2, 3, . . . , A. In the non-relativistic approximation dynamical properties of the system are

described by the non-relativistic Schrodinger equation

H Ψ(r1, r2, r3, . . . , rA) = EΨ(r1, r2, r3, . . . , rA) (2.1)

where H is the Hamiltonian of the system, E total energy and Ψ the wave function of the

system, respectively. The general form of the Hamiltonian in this formalism is

H = −∑

i

~2

2mi

∇2i + U(r1, r2, r3, . . . , rA) (2.2)

where the sum determines the total kinetic energy operator of the system developed from the

single-particle kinetic energy operators, and U the potential energy function describing all the

12


interactions in the system. The total potential energy of the system is considered as composed

of contributions from two-body (V (2)), three-body (V (3)), four-body (V (4)) through A-body

(V (A)) interaction potentials. As a result, the total potential energy of the system may be

determined by the sum

U =∑

ij

[

V (2)(rij) +∑

k

[

V (3)(rij,k) +∑

l

[

V (4)(rijk,l) + V (4)(rij,kl) + · · ·]

]]

(2.3)

where i 6= j 6= k 6= l and the vectors rij , rij,k , rijk,l , rij,kl are relative position vectors

between the interacting particles, and are explained in Appendix A. The many-body potentials

(V (n)) (n ≥ 4) are not well understood and are often omitted in studies of many-body systems.

The contributions of such interactions to properties of a system are expected to be small com-

pared to contributions from two-body interactions [66, 67]. Hence, in this thesis only two-body

interactions are considered.

In relative coordinates the kinetic energy operator of the system can be separated into the sum

of the kinetic energy operator for the internal motions in and the kinetic energy operator for

the center-of-mass of the system. Hence the total kinetic energy operator may be expressed in

the form

T = −A∑

i=1

~2

2mi∇2

i = −A−1∑

i=1

~2

2µi∇2

i −~

2

2M∇2 (2.4)

where 1/µi =∑i

n=1 1/mn and −i~∇i are the reduced mass and relative momentum operator,

respectively, corresponding to the i-th relative coordinate. M =∑A

n=1 mn is the total mass

and −i~∇ the total momentum operator of the system. Adopting the notation,

T0 = −A−1∑

i=1

~2

2µi

∇2

i and Tcm = − ~2

2M∇2 , (2.5)

then equation (2.4) may be expressed in the form T = T0 + Tcm. If the internal properties

of the system, like ground state energies, are studied the center-of-mass kinetic energy Tcm is

easily eliminated. Recall that the potentials V (n) depend only on the relative positions of the

interacting particle. Considering only the internal motions and interactions, the Schrodinger

13


equation for the system has the form

[

T0 + U(R)]

Ψ(R) = EΨ(R) (2.6)

where R represents the set of the individual particle position vectors r1 , r2 , r3 , . . . , rA .When the interactions in the system are dominated by two-body interactions then the potential

may be approximated by

U(R) =∑

i<j

V (rij) (2.7)

where V (rij) are the two-body interaction potentials. This thesis focuses only on nuclear

systems. However, interaction potentials in other systems of interest have a similar structure.

2.1.1 The Two-Body Nucleon Potential

A realistic description of a nucleon-nucleon interaction potential incorporates as many nucleon

degrees of freedom as possible. Such an interaction potential should also be able to explain

the experimentally observed properties of nuclear systems. The Paris [7, 68], the Nijmegen

[8, 69], the Bonn [9, 70], the Urbanna [10, 71] and the Argonne [11, 72] nucleon-nucleon po-

tentials were constructed based on this approach considering phenomenological descriptions

to different degrees. The Argonne potential is a local two-body potential and it incorporates

most of the attributes of most realistic models. Therefore, in this work a reduced version of

the Argonne potential is used to describe the nuclear Hamiltonian. The Argonne potential is

characterised by an elaborate treatment of the electromagnetic properties of and long-range

one-pion-exchange by nucleons. The short-range and intermediate-range behavior of the inter-

action is treated phenomenologically.

The different components of the Argonne nucleon-nucleon potential are the electromagnetic

component vγ(NN), the one-pion-exchange component vπ(NN) and the short-range and intermediate-

range components vR(NN). As a result, the potential model is written in the form

v(NN) = vγ(NN) + vπ(NN) + vR(NN) . (2.8)

14


where each component receives contributions from proton-proton (pp), neutron-proton (np)

and neutron-neutron (nn) interactions. A comprehensive construction of the components of the

potential is given in references [11, 72]. For convenience the expressions for these components

are given. The general structure of the electromagnetic component can be expressed in the

form

vγ(NN) = vγ(pp) + vγ(np) + vγ(nn) (2.9)

=

[

3 πme r

4 ~ c+ ln

( me r

~ c

)

− 5

6− γ

]

δpj − αβnFnp(r)

r( 1 − δij ) +

[

α′ FC(r)

r− αα′

mj

F 2C(r)

r2− α

4m2j

Fδ(r)

]

δpj −α

6

µi µj

mimjFδ(r) σi · σj −

α

4

µi µj

mimj

Ft(r)

r3Sij −

α

2

[ 4µj − 1

m2j

δpj +µn

mnmr

( 1 − δij )] FLS(r)

r3L · S (2.10)

where ~, c, γ, α, α′ and βn are constants, δ the usual delta function, me the electron mass, mi

and µi are, respectively, the mass and magnetic moment of the i-th nucleon with mr as the

reduced mass for a proton-neutron combination. The operators

L = ( ri − rj ) × ( pi − pj ) and S = ~ (σi + σj ) /2 (2.11)

are, respectively, the relative orbital angular momentum and total spin angular momentum

operators of the interacting nucleons, where r = ri − rj is the relative position, pi and σi the

linear momentum and spin angular momentum operators, respectively, of the i-th nucleon and

Sij =1

r2ij

[

3 ( σi · rij ) ( σj · rij ) − ( σi · σj ) r2ij

]

(2.12)

the tensor operator.

The functions Fi(r) are different nucleon form factors and are explicitly given in reference [72].

The one-pion-exchange component has the form

vπij =

f 2πNN

4 π

mπ

3Xπ

ij τ i · τ j (2.13)

15


where mπ is the pion mass, fπNN the pion-nucleon coupling constant, τ the nucleon isospin

operator and

Xπij = Y (mπ rij) σi · σj + T (mπ rij)Sij . (2.14)

The radial components Y (x) and T (x) are the Yukawa and tensor functions given by

Y (x) =[

1 − e−cπ x2

] e−x

x(2.15)

T (x) =

(

1 +3

x+

3

x2

)

[

1 − e−cπ x2

]

Y (x) (2.16)

where x = mπ r and 1− e−cπ r2

is a cut-off factor with cπ as the cut-off parameter that depends

on the pion mass. The short-range and intermediate-range component have the form

vR(r) = VC(r) + VT (r)Sij + VLS(r) L · S (2.17)

where each of the radial components is expressed as

Vi(r) = Ai T2(mπ r) +

Bi + Ci (mπ r) +Di (mπ r)2

1 + exp [ (r − ro)/ao ](2.18)

and ao , ro , Ai , Bi , Ci , Di are fitting parameters. The values of these parameters are de-

termined by fitting the model potential to experimental nucleon-nucleon scattering data.

In the investigations of nuclear dynamics it is sufficient to consider only the dominant compo-

nents of the potential in describing the nuclear Hamiltonian. The Argonne potential can be

reduced to different forms labeled as AV n (n = 1, 2, 4, 6, 8, 14, 18) convenient for nuclear dy-

namics investigations [12]. The reduced versions of the Argonne nucleon-nucleon potential are

required to reproduce selected nucleon-nucleon scattering data. In operator form the general

Argonne nucleon-nucleon potential can be written in the form

V (rij) =

28∑

n

vn(rij)Onij (2.19)

where Vn(rij) are radial components and Onij are two-body nucleon operators. The Argonne

16


στστc

r [fm]

V(r

)[M

eV]

32.521.510.50

100

50

0

-50

-100

Figure 2.1: The central (c), spin (σ), isospin (τ) and spin-isospin (στ) components of the AV4’nucleon-nucleon potential. The central component Vc(r) has a maximum value of almost 2417.92MeV at r = 0 fm.

V4’ version of the Argonne V18 potential used in this work consists of the first four operators

O1→4ij ≡

1 , σi · σj

⊗

1 , τ i · τ j

(2.20)

with the related form factors v(r) shown in Figure 2.1. The Argonne V4’ potential model can

also be cast in the form

V (rij) =1∑

T=0

1∑

S=0

V cST (rij) Ω

(S)ij Ω

(T )ij (2.21)

where V cST (rij) are the radial form factors of the potential. The subscripts S (T ) indicate the

17


total spin (isospin) of the interacting nucleon pair. Using the definitions

Ω(S)ij =

1

2

[

1 + (−1)S+1 P σij

]

and Ω(T )ij =

1

2

[

1 + (−1)T+1 P τij

]

(2.22)

of the spin and isospin projection operators, where P σij (P τ

ij) is the spin (isospin) exchange

operator, one obtains

V (rij) = Vc(rij) + Vσ(rij)Pτij + Vτ (rij)P

σij + Vστ (rij)P

σij P

τij . (2.23)

The relation between the form factors in equations (2.19) and (2.23) for the Argonne V4’

potential is

Vc = vc − vτ − vσ + vστ (2.24a)

Vτ = 2[

vτ − vστ

]

(2.24b)

Vσ = 2[

vσ − vστ

]

(2.24c)

Vστ = 4 vστ . (2.24d)

The evaluation of the expectation values of the potential is greatly simplified when the these ra-

dial components are given in terms of Gaussian functions. As a result, in many calculations the

non-Gaussian radial forms of the potentials are often expanded in terms of Gaussian functions.

2.2 The Ground-State Wave Function

Consider a nuclear system consisting of A particles. The wave function Ψ describing the system

depends on the position ri, spin σi and isospin τ i (i = 1, 2, 3, . . . , A), variables of the constituent

particles. In what follows the collective variable, υi, is used to represent the set ri σi τ i .For a systems of fermions the total wave function is required to be antisymmetric with respect

to the interchange of any two particles in the system. One way of constructing such a wave

18


function is by the use of Slater determinants of single-particle wave functions ψi(υj),

Ψt(υ1,υ2,υ3, . . . ,υA) =1√A!

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ψ1(υ1) ψ2(υ1) ψ3(υ1) · · · ψA(υ1)

ψ1(υ2) ψ2(υ2) ψ3(υ2) · · · ψA(υ2)

ψ1(υ3) ψ2(υ3) ψ3(υ3) · · · ψA(υ3)

......

.... . .

...

ψ1(υA) ψ2(υA) ψ3(υA) · · · ψA(υA)

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

=1√A!

det[

ψi(υj)]

. (2.25)

where ψi(υj) depends on the degrees of freedom of the i-th particle only. The functions ψi(υj)

used in this thesis do not form an orthogonal basis. However, an orthonrmal basis can be

constructed with the wave functions ψi(υj) [58, 59, 60]. The ground state wave function of the

system is represented by Ψt(S) where S is a set of variational parameters, yet to be introduced.

The single-particle wave function ψ(υ) is assumed to be separable in the form

ψ(υ) = φ(r)χ(σ) ξ(τ ) (2.26)

where φ(r), χ(σ), and ξ(τ ) are the spatial, spin and isospin wave functions, respectively. The

intrinsic spin state χ(σ) of a fermion, with spin s and third component of the spin ms, is

denoted by

χ(σ) = | s , ms 〉 =

∣

∣

12, +1

2

⟩

= | ↑ 〉∣

∣

12, −1

2

⟩

= | ↓ 〉(2.27)

where | ↑ 〉 is referred to as the spin-up and | ↓ 〉 the spin-down state. These two spin states are

orthogonal to each other. The spin of fermions can be treated as fixed, in the | ↑ 〉 and | ↓ 〉states [51]. However, to improve the quality of the wave function, the spins of the fermions may

be allowed to vary. This is done by representing the fermion spin as a general spinor. Such a

spinor can be expressed as a linear combination of the spin-up and spin-down states so that

19


the fermion spin wave function has the form [39]

χ(u) =

(

1

2+ u

)

| ↑ 〉 +

(

1

2− u

)

| ↓ 〉 (2.28)

where u is a complex time-dependent variational parameter. In this thesis the time-independent

spin states given by equation (2.27) are employed. The isospin state, ξ(τ ), in the case of a

nucleon, with isospin t and third component of the isospin mt, is given by

ξ(τ ) = | t , mt 〉 =

∣

∣

12, +1

2

⟩

= | p 〉∣

∣

12, −1

2

⟩

= |n 〉(2.29)

where | p 〉 refers to a proton while |n 〉 refers to a neutron state. These states are time-

independent.

All the constituent particles are described by the same form of the spatial wave function. The

spatial component, φ(r), is parametrized as a normalized Gaussian wave packet [51]

φ(r) =

(

2α

π

)3/4

exp

[

−α

(

r − s√α

)2

+1

2s2

]

. (2.30)

where the centroid of the wave packet, s, is a complex time–dependent three-dimensional vector.

The phase s2/2 is included mainly to simplify the structure of the elements of the resulting

overlap matrix, 〈φi |φj 〉. The width parameter α is treated as a real constant and assumed to be

the same for both protons and neutrons in the nucleus. It can be argued that the assumption is

still valid if the masses of the constituents are not significantly different, like in nuclear systems

and quark systems. When the momentum operator of a fermion is denoted by p, then the real

component Re[s] and the imaginary component Im[s] of the parameter s are given by

Re[s] =√α〈φ(r) | r |φ(r) 〉〈φ(r) |φ(r) 〉 and Im[s] =

1

2 ~√α

〈φ(r) |p |φ(r) 〉〈φ(r) |φ(r) 〉 (2.31)

where r is the position operator of the nucleon. The single-particle wave function in equation

(2.30) satisfies the minimum uncertainty ∆r ∆p = ~/2 [40] which helps inform the choice of

20


the value for the width parameter α. The value of α is chosen so as to generate reasonable

description of the ground state properties of light nuclei. The expectation values of relevant

operators calculated with this form of the wave function can be conveniently determined. Some

of the expectation values are given in Appendix B.

2.3 Spin and Parity Projections

Consider single-particle wave functions ψ ℓ mℓ

s ms with definite orbital angular momentum ℓ, spin

angular momentum s and their respective projections along the quantization axis mℓ and ms.

A wave function for the system with definite total orbital angular momentum LML and spin

angular momentum SMS,∣

∣LML, SMS⟩

, can be constructed from the functions ψ ℓ mℓ

s ms through

the coupling

∣

∣LML, SMS⟩

∝∑

[CG]

[

[

[

ψℓ1mℓ

1

s1ms1

⊗ ψℓ2mℓ

2

s2ms2

]L1ML1

S1MS1

⊗ ψℓ3mℓ

3

s3ms3

]L2ML2

S2MS2

⊗ · · · ⊗ ψℓAmℓ

A

sAmsA

]LML

SMS

(2.32)

where the single-particle wave functions with orbital angular momentum ℓ1 and ℓ2 couple to an

intermediate function with composite orbital angular momentum L1, single-particle wave func-

tions with spin angular momentum s1 and s2 couple to an intermediate function with composite

spin angular momentum S1, and so on. Then LA−1MLA−1 = LML and SA−1M

SA−1 = SMS are

respectively the total orbital and spin angular momenta, with the respective projections, of the

resulting wave function. The summation is over those Clebsch-Gordan coefficients CL ML

S MS that

have the combinations of the spin and orbital angular momentum projections that couple to

MLi and MS

i . These states can be used to generate the total wave function

∣

∣ J MJ⟩

=∑

n

CJ MJ

n

∣

∣LMLn SM

Sn

⟩

(2.33)

that has definite total angular momentum J and projection M on the quantisation axis. This

procedure cannot be applied to the AMD wave function since the single-particle wave functions

in this approach do not have definite orbital and spin angular momenta. However, wave func-

tions with definite J and M can be generated from the AMD wave function using standard

21


numerical projection techniques [63, 73].

The effects of coordinate inversion on the wave function of a system defines the parity (π) of

the system. For a more realistic description of the system, a wave function with definite parity

is required. The AMD wave function, as constructed in the previous section, does not possess

definite parity. This short-coming in the AMD wave function is corrected by the use of the

coordinate inversion operator P π. This operator is used to project the AMD wave function

onto states with definite parity. The operator generates the transformation [63]

P π ΨAMD(S) = ΨAMD(−S) (2.34)

where S ≡ s1 , s2 , s3 , . . . , sA . Wave functions with even parity, Ψ+(S), or odd parity,

Ψ−(S), are then constructed by the linear combinations

Ψ±(S) =1√2

[

ΨAMD(S) ± P π ΨAMD(S)]

. (2.35)

of the parity projected AMD wave functions. The wave function Ψπ(S), where π ≡ ±, gener-

ated as in equation (2.35) from the AMD wave function has definite parity.

To improve on the realistic description of a system with definite total angular momentum, the

wave function of the system must be an eigenfunction of total angular momentum. To generate

a wave function with definite total angular momentum J from Ψπ(S) the Peierls-Yoccoz angular

momentum projection technique [73] is used. The technique employs the projection operator

defined by [73]

P JMK =

2 J + 1

8 π2

∫

dΩDJ∗MK(Ω) R(Ω) (2.36)

where DJ∗MK(Ω) are the complex conjugates of the Wigner D-functions that depend on the Euler

angles Ω = α, β,γ. R(Ω) is the rotation operator

R(Ω) = exp(

−iα Jz

)

exp(

−i β Jy

)

exp(

−i γ Jz

)

(2.37)

where Jτ (τ = y, z) are the components of the total angular momentum operator. The proper-

22


ties of the projection operator P JMK are further explained in Appendix C. The wave function

ΨJπM (S) with the desired properties

J2ΨJπ

M (S) = J (J + 1) ΨJπM (S) (2.38)

Jz ΨJπM (S) = M ΨJπ

M (S) (2.39)

is generated from Ψπ(S) through the transformation

ΨJπM (S) =

∑

K

cJπMK P J

MK Ψπ(S) (2.40)

where Ψπ(S) is given by equation (2.35) and −J ≤ K ≤ J . The expansion coefficients cJπMK are

determined by solving the generalized eigenvalue problem

∑

N

[

HJπKN − EJπ

K N JπKN

]

cJπKN = 0 (2.41)

where HJπKN and N Jπ

KN are the matrix elements of the Hamiltonian and norm of the system,

respectively. The commutator relations

[

H , P π]

= 0 (2.42)

[

H , P JMK

]

= 0 (2.43)

are used to construct the matrix elements

HJπKN =

⟨

ΨJπMK(S) |H | ΨJπ

MN(S)⟩

(2.44)

=⟨

Ψπ(S)∣

∣

∣P J†

MK H P JMN

∣

∣

∣Ψπ(S)

⟩

(2.45)

=⟨

ΨAMD(S)∣

∣H P JKN

[

1 ± P π]∣

∣ ΨAMD(S)⟩

(2.46)

and

N JπKN =

⟨

ΨAMD(S)∣

∣P JKN

[

1 ± P π]∣

∣ ΨAMD(S)⟩

(2.47)

23


where use was made of equation (2.35) and the standard properties

P π† = P π (2.48a)

P π†P π = 1 (2.48b)

P J†MK = P J

KM (2.48c)

P JKMP

JMN = P J

KN (2.48d)

of the projection operators, where the operator P † represents the Hermitian conjugate of the

operator P . The eigenvalue problem in equation (2.41) can be evaluated before or after the

determination of the variational parameters S. Since the implementation of the numerical pro-

jection given by equation (2.36) involve a three-fold integration that is computed numerically,

projecting the wave function before the variation of parameters becomes computationally de-

manding. However, the resulting variational parameters of the wave function are more unique

than in the case of projection after variation [74, 75].

2.4 The Variational Energy

The variational energy of the system is obtained as the expectation value of the Hamiltonian

〈H 〉 whence the ground state energy of the system E0 can be obtained by subtracting the

contribution of the center-of-mass motion Tcm to give

E0 = 〈H 〉 − 〈 Tcm 〉 . (2.49)

When there are internal clusters formed in the system, then E0 can be calculated from the

number of fragments NF , the binding energies of the fragments E0F and the inter-fragment

relative motion kinetic energies T0 as

E0 =

NF∑

i

E0i + γ 〈 T0 〉 (2.50)

24


where γ is either a constant or a linear function of NF , depending on the form of clusterization

considered. In the AMD formalism γ = NF , hence the ground state energy of the system is

given by

E0 = 〈H 〉 − 〈 Tcm 〉 −NF 〈 T0 〉 . (2.51)

Because of the Gaussian form of the spatial component of the single-particle wave function the

expectation value of the center-of-mass kinetic energy has the simple form

〈 Tcm 〉 =1

2M

(

3 ~2 αA+ K2

cm

)

(2.52)

for a system of A particles where M is the total mass and

Kcm = Im

(

A∑

i

si

)

(2.53)

the momentum of the center-of-mass of the system. The first term in the right-hand-side of

equation (2.52) is due to the zero-point oscillation of the center-of-mass. The expression for

the ground state energy of the system with corrections for the center-of-mass zero-point motion

and relative motion of fragments has the form [51]

E0 = 〈H 〉 − 3 ~2 α

2MA+ T0 (A−NF ) (2.54)

where α and T0 are free parameters chosen to satisfactorily reproduce the experimental binding

energies of the three-nucleon systems. The values α = 0.12 and T0 = 10.0 MeV are used with α

taken as the same for protons and neutrons. The parameter T0 is treated as a free parameter.

The number of fragments NF is determined as a function of βij = |Re(si − sj) |. This number

is characterized by a clusterization function

f0(βij) =

1 if βij ≤ r0

exp[

−α0 ( βij − r0 )2] if βij > r0

(2.55)

where r0 is a constant. The constant r0 defines the cut-off distance for a particle to be included

25


or excluded from a cluster. The function f0(βij) is determined by the two parameters α0, r0 .The counting of internal clusters is done with the help of [51]

NF =A∑

i=1

(nimi )−1 (2.56)

where

mi =A∑

j=1

f0(βij)

nj

. (2.57)

The weight number ni is given by

ni =

A∑

j=1

f1(βij) (2.58)

where f1(βij) is defined by equation (2.55) but with a different parameter set α1, r1 . The

values used of the parameters for the clusterization function are α0 = α, α1 = 2α, r0 = 0.5 fm

and r1 = r0/2 [51]. Other forms of the function f(βij) are used in reference [40].

To determine the upper bound for the ground state energy of the system in equation (2.49) the

variational energy functional is set up using the expectation value for the Hamiltonian operator

and the norm of the projected wave function. The expectation values of operators calculated

with the projected wave function ΨJπM (S) have the same form as those calculated with the

unprojected wave function except for the replacement S∗ −→ S∗ and S −→ −S. where

S = Rr(Ω)S (2.59)

are the rotated parameters. Equations (2.46) and (2.47) can be written in a compact form

OJ±KN = O(0),J

KN ±O(1),JKN (2.60)

where the two components are given by

O(0),JKN =

⟨

ΨAMD

∣

∣

∣O P J

KN

∣

∣

∣ΨAMD

⟩

(2.61)

O(1),JKN =

⟨

ΨAMD

∣

∣

∣O P J

KN Pπ∣

∣

∣ΨAMD

⟩

. (2.62)

26


Consideration of the normalization of the wave functions transforms the expectation values into

the form

〈Ψ | O |Ψ 〉 = 〈Ψ |Ψ 〉〈Ψ | O |Ψ 〉〈Ψ |Ψ 〉 , (2.63)

so that the variational energy functional is expressed as

EJ±KN(S,S∗) =

N (0),JKN H(0),J

KN ±N (1),JKN H(1),J

KN

N (0),JKN ±N (1),J

KN

(2.64)

where N (i),JKN and H(i),J

KN are, respectively, the norm and expectation value of the Hamiltonian.

To determine the parameter set S, their dynamical equations are solved so as to minimise the

energy functional EJπ(S, (S∗). The dynamical equations are derived in the next section. The

variational energy is the upper bound to the ground state energy of the system. This energy

may be different from the lowest energy obtained from the solution of the eigenvalue problem

in equation (2.41).

2.5 Equations of Motion

The wave function of the system Ψ(S) depends on the set of complex time-dependent variational

parameters S. To establish the time evolution of such a wave function, the form of the time

dependence of the variational parameters, and therefore the equations of motion, is determined.

The equations of motion of these parameters are derived from the time-dependent variational

principle [76]

δ

∫ t2

t1

L (Ψ,Ψ∗) dt = 0 (2.65)

with the constraints

δΨ(t1) = δΨ(t2) = δΨ∗(t1) = δΨ∗(t2) = 0 (2.66)

where L (Ψ,Ψ∗) is the Lagrangian of the system and Ψ∗ the complex conjugate of the wave

function. This time-dependent variational principle requires the time development of the La-

grangian to be determined entirely by the time evolution of the wave function. That is, L (Ψ,Ψ∗)

should depend on time only through Ψ(t). A Lagrangian, for the system, with such a property

27


has the form [76]

L (Ψ,Ψ∗) =i~

2

⟨

Ψ∣

∣

dd t

Ψ⟩

−⟨

dd t

Ψ∣

∣ Ψ⟩

〈Ψ |Ψ 〉 − 〈Ψ |H | Ψ 〉〈Ψ |Ψ 〉 . (2.67)

In this definition the wave function Ψ is not normalized. Therefore the equations of motion,

for the variational parameters, resulting from this Lagrangian will not be depending on either

the normalization or the phase of the wave function. In this formulation of the Lagrangian the

wave function and its complex conjugate are treated as independent.

Because of the reason stated above the variation of the wave function in equation (2.67) with

respect to time can be cast in the form

d Ψ

d t=∑

i

[

d si

dt

∂Ψ

∂ si+

d si∗

dt

∂Ψ

∂ si∗

]

. (2.68)

Therefore, the equations of motion of the variational parameters can be obtained from the

Lagrangian by expressing equation (2.67) in terms of the parameters as

L (S,S∗) =i ~

2

∑

j

(

d sj

dt

∂

∂ sj− d sj

∗

dt

∂

∂ sj∗

)

ln N −E (2.69)

with the constraints

δs(t1) = δs(t2) = δs∗(t1) = δs∗(t2) = 0 (2.70)

which correspond to equation (2.66). In this case

N (S,S∗) = 〈Ψ(S) |Ψ(S) 〉 and E (S,S∗) =〈Ψ(S) |H |Ψ(S) 〉〈Ψ(S) |Ψ(S) 〉 (2.71)

are the norm of the wave function and energy functional of the system, respectively. Note

that s and s∗ are treated as independent variables. Minimizing the action in equation (2.65)

with the Lagrangian given by equation (2.69) and constraints (2.70) results in the equations of

28


motion

i ~

2

∑

i

∂2 ln N∂ si ∂ s∗

j

d si

dt=

∂2 E

∂ s∗j

(2.72)

− i ~

2

∑

j

∂2 ln N∂ si ∂ s∗

j

d s∗j

dt=

∂2 E

∂ si(2.73)

of the variational parameters. Introducing the matrix C defined by the elements

Cij =∂2 ln N∂ si ∂ s∗

j

, (2.74)

the equations of motion can be compactly expressed in the form [76]

i ~

2

0 C

−C∗ 0

dS ∗

d t

dS

d t

=

∂ E

∂ S ∗

∂ E

∂ S

. (2.75)

The matrix C is Hermitian and positive definite. A numerical solution of the equations (2.75)

can be determined. The solution to these equations provide the time evolution of the varia-

tional parameters and therefore of the wave function of the system.

The time-dependent variational parameters are determined using variational technique called

frictional cooling [50]. The technique involves modifying and then introducing a “friction”

coefficient to the equation of motion for the variational parameters, equation (2.75). Solving

the modified equation of motion, or cooling equations, minimizes the energy of the system. To

prove this statement and set up the cooling equations, the matrix C is replaced by a simpler

positive definite matrix, the unit matrix I. Then the equations of motion for the variational

parameters takes the form

i ~

2

0 1

−1 0

dS ∗

d t

dS

d t

= µ

∂ E

∂ S ∗

∂ E

∂ S

. (2.76)

29


It can be observed that these modified equations have the general form

d ρ

d t=

µ

i ~

∂ E

∂ ρ∗and

d ρ∗

d t= − µ

i ~

∂ E

∂ ρ(2.77)

where µ is the friction coefficient and ρ ≡ sx, sy, sz. These equations have a form similar to

that of Hamilton’s classical equations of motion for canonically conjugate variables. When µ

is a complex constant, say µ = a + i b, where a and b are arbitrary, then the variation of the

energy functional with time is given by

d E

d t=

∑

i

[

∂ E

∂ si

d si

d t+∂ E

∂ s∗i

d s∗i

d t

]

(2.78)

=1

i~

∑

i

[

( a+ i b )∂ E

∂ si

∂ E

∂ s∗i

− ( a− i b )∂ E

∂ s∗i

∂ E

∂ si

]

(2.79)

=2 b

~

∑

i

∂ E

∂ si

∂ E

∂ s∗i

, (2.80)

which is valid for the Cartesian components of the vector parameters s and s∗. Note that

equation (2.80) is obtained from equation (2.76) which result from the use of the unit matrix.

However, the same result can be obtained from equation (2.75) which involve the positive

definite matrix C [40]. Since the two terms in equation (2.80) are positive at all times, it is

evident that

d E

d t> 0 when b > 0 (2.81)

d E

d t= 0 when b = 0 (2.82)

d E

d t< 0 when b < 0 , (2.83)

so that choosing b < 0 results in the energy functional decreasing with time during the variation

of the parameters. Hence the parameter b is referred to as the coefficient of friction and this

technique of lowering the energy of a system frictional cooling. The equations are solved with

the constraint∑

i si = 0 so that only the zero-point oscillation of the center-of-mass need

be subtracted from the expectation value of the Hamiltonian of the system. Note that the

30


variation of the energy functional given by equation (2.64), with respect to the parameters S,

has the form

∂ E

∂ S∗ =N (0) ∂H(0) ±N (1) ∂H(1)

N (0) ±N (1)±(

N (1) ∂N (0) −N (0) ∂N (1)) (

H(0) −H(1))

(N (0) ±N (1) )2 (2.84)

where ∂O(i) represents ∂O(i)/∂ S∗ and the dependence on J and K is suppressed. The cooling

equations (2.76) are solved numerically using the modified second order Runge-Kutta method.

2.6 The Scattering Wave Function

In this section the construction of wave functions for the nucleon-nucleus scattering process

at high energies is explained. The model wave function is derived from the AMD formalism

and the Glauber approximation [41]. To understand the basis of the model wave function for

scattering processes, consider a nucleon scattering off a nucleus of atomic number A. In the

cluster approach the wave function describing the scattering process is well described by the

ansatz

Ψsc(R, rn, rAn) = QAn(rAn) ΦA(R)ψn(rn) (2.85)

where ΦA(R) is the wave function of the target nucleus, ψn(rn) the wave function of the projec-

tile nucleon and QAn(rAn) the function describing the relative motion of the projectile-target

system. In this approach the functions ΨA(R) and ψn(rn) are known whereas the function

QAn(rAn) is to be determined. The function QAn(rAn) is generally determined by solving the

Schrodinger equation with an effective potential describing the interaction between the nucleon

and the target. The accuracy of this ansatz depends on how close the effective potential is to

the actual interaction potential. This procedure is quite involved because many partial-wave

are often required to attained convergence to a stable solution [77, 78, 79]. In the case of high

energy scattering the details of the nucleon-nucleus interactions become less significant in the

construction of the scattering wave function. Therefore, semi-classical treatments of the process

can yield reasonably accurate wave functions.

To construct the AMD wave function for scattering states in break-up reactions, the wave

31


function of the initial state is expressed in the form

ΨJAπA

MA=

1√A

det

[

A∏

m=1

ψn(m)

]

(2.86)

=1√A− 1

A

ψn(A) det

[

A−1∏

m=1

ψn(m)

]

(2.87)

= A

ψn(A) ΨJA−1πA−1

MA−1(n′)

(2.88)

where n′ represents all the A − 1 nucleons excluding n [34]. The nucleon radial function the

form

φ(s, r) =

(

2α

π

)3/4

exp

[

−α(

r − s√α

)2

+s2

2

]

(2.89)

=

(

2α

π

)3/4

exp

[

−α r2 + 2√α s · r − s2

2

]

(2.90)

≈(

2α

π

)3/4

exp

[

2√α s · r − s2

2

]

for r → ∞ (2.91)

where r → ∞ means large r. Since s is complex, the large r approximation reveals that

φ(s, r) =

(

2α

π

)3/4

exp

[

2√α sR · r + i 〈p〉 · r − s2

2

]

(2.92)

where 〈p〉 = 2√α sI is the average momentum of the nucleon with sR and sI as the real and

imaginary components of s. Based on this approximation the wave function of a free nucleon

is expressed as

ψA(s, r) =1

(

2 π)3/2

exp [ i pA · rA ] χA ξA (2.93)

where χA ξA is the spin-isospin of the nucleon.

Consider a nucleon knockout reaction AX(e,e′ x)A−1Y. When the ejected nucleon is still in close

proximity to the recoil nucleus it tends to interacts with single nucleons in the nucleus. However,

when the free nucleon is further away from the recoil nucleus it tends to interact with clusters

two nucleons, three nucleons, or more, in the nucleus. These interactions are explained by the

Glauber nucleon-nucleus scattering series [41], and are illustrated in Figure 2.2. Denote the

32


(a)

~p1

~p′1

AX

~q

A−1Y

~px

~PA~PA−1

(b)

~p2

~p′1

~p′2~p1

AX

~q

A−1Y

~px

~PA~PA−1

(c) ~q

~p2

~p3

~p′1 ~p′′1

~p′2~p′3

~p1

AXA−1Y

~px

~PA~PA−1

Figure 2.2: The Feynman diagram vertex for a nucleon removal from the AX nucleus. Thefree nucleon (a) does not interact with the nucleons (b) interacts with single nucleons and(c) interacts with pairs of nucleons, in the recoil nucleus A−1Y.

33


position vectors of nucleons by R ≡ r1 , r2 , r3 , . . . , rA and their initial linear momenta by

P ≡ p1 , p2 , p3 , . . . , pA . The position vector for the ejected nucleon can be expressed in

the form rA = bA + zA pA where bA is the impact parameter. Also, a subset of R comprising

of the components of nucleon position vectors transverse to pA, B, can be identified. The wave

function for the nucleon-nucleus system is then expressed in the form

Ψf(R) = ϕ(A)G(B) ΨA−1(R) (2.94)

where G(B) is the Glauber multiple scattering operator. The Glauber operator has the form

G(B) = G(0)(B) +G(1)(B) +G(2)(B) +G(3)(B) + · · · (2.95)

= 1 −A−1∑

j=1

θ(

zj − zA

)

Γj

(

bA − bj

)

+

A−1∑

j=1

A−1∑

l 6=j=1

θ(

zj − zA

)

Γj

(

bA − bj

)

θ(

zl − zA

)

Γl

(

bA − bl

)

− · · · (2.96)

where Γ(

b)

is the nucleon profile function and θ(

zi − zA

)

the Heaviside step-function. The

step-function restricts the motion of the ejected nucleon in the forward direction. The first

term in equation (2.96), corresponding to G(0)(B) represents the absence of interactions be-

tween the free nucleon and the recoil nucleus. The second term, involving the summation of

single profile functions in equation (2.96), corresponding to G(1)(B), represents the interaction

of the scattered nucleon with single nucleons in the nucleus. The term involving double sum-

mations of products of profile functions corresponds to G(2)(B) and represents the interaction

of the scattered nucleon with pairs of nucleons in the nucleus, and so on (see Figure 2.2). The

contributions of each term in the series (2.96) follow a diffraction pattern [80, 81, 82]. That is,

in a given energy and angle range one of the terms generally dominates. Therefore, the series

is convergent for a given energy-angle range. The contributions of single scattering terms is

dominant mainly in the forward scattering (small angles) with small contributions from double

and even smaller contributions from triple scattering processes. Therefore, the GA is applica-

ble in processes involving relativistic energies (∼ 1 GeV or more) and is valid only at small

scattering angles. The inclusion of transverse contributions extends the energy-angle range of

34


applicability of the approximation [42, 46].

The Glauber multiple-scattering operator is determined by the profile function Γ(b). The

form of this function is deduced from the nucleon-nucleon elastic scattering amplitude F (s, q)

through the Fourier transformation

Γ(b) =1

2 π i pn

∫

e−iq·b F (s, q) d2q (2.97)

where s is the invariant total energy of the interacting pair of nucleons. A realistic description

of the nucleon-nucleon scattering amplitude has the general form [80]

F (s, q) =∑

m=1

fm(s, q)Om (2.98)

where Om are operators involving different degrees-of-freedom of the interacting nucleons as

well as the momentum transferred. The components fm(s, q) are parametrized as Gaussian

functions

fm(s, q) = σm(s) e−βm(s)q2

(2.99)

where σm and βm are complex parameters. The values of these parameters are determined from

analysis of experimental nucleon-nucleon elastic scattering phase-shift data. In this work only

the scalar component of the amplitude,

F (q) = f1(q) =i pn

4 πσ(

ρ+ i)

e−β q2

(2.100)

is considered, where σ is the total nucleon-nucleon elastic scattering cross section, ρ the ratio

of the real and imaginary components of the nucleon-nucleon elastic scattering amplitude for

the forward scattering kinematics and β the range parameter. For this parametrization of the

scattering amplitude the profile function has the form

Γ(b) =σ( 1 − i ρ )

4 π a2e−b

2

/2 a2

(2.101)

where a is a constant related to β [43]. Since the amplitude f(s,k) is experimentally deter-

35


mined the profile function can be derived independent of the knowledge of nucleon-nucleon

interactions.

36

Chapter 3

Electromagnetic Transitions

Most of the accumulated information about nuclear structure is derived from electron-nucleus

scattering and photonuclear reactions. The theory underlying the interactions between elec-

tromagnetic radiation and charged particles with matter, quantum electrodynamics, is a well

developed theory. Therefore the type of interaction involved is well understood. Interactions

in nucleons and nuclear systems involve strong forces which imply that the interaction between

electromagnetic radiation and charged particles with nuclei is comparatively very small. As a

result, in the theoretical investigations of nucleon and nuclear systems the interaction between

electromagnetic radiation and matter can be described accurately with perturbation theories.

In the perturbation treatment the interactions of electromagnetic radiation with nuclear matter

are described by processes of the order αn where α ≈ 1/137 is the fine structure constant and n

a positive integer. Therefore the interactions are dominated by processes of order 1/137, involv-

ing the exchange of a single-photon. Contributions from processes involving the exchange of

two or more photon are not significant and can, therefore, be neglected. In the one-photon ap-

proximation the Hamiltonian describing the interaction between the photon and the target can

be expressed in a form that makes it convenient to extract information about the structure of

the target. Therefore, electrons and photons provide the most reliable and clean probe into the

structural attributes of matter, to the level of nuclei and nucleons. Informative reviews of the

historical development of the electron-nucleus scattering techniques and theory can be found

in references [2, 6, 26, 27, 88]. In this chapter a summary of the electron-nucleus interactions

and the mechanism of extracting information about the target nucleus in such interactions is

given.

This chapter is started with the introduction the general kinematics of electron scattering off

37


atomic nuclei in the one-photon-exchange approximation. In Section 3.1 the notation and

convention employed for the parameters used to explain the scattering processes are given. To

simplify most of the discussions the units are set such that Planck’s constant ~ and the speed of

light c are unity. The construction of the nuclear transition operator is outlined in Section 3.2.

The Hamiltonian describing the interaction between external electron and non-relativistic nuclei

is indicated. The multipole expansion of the transition operator is given in Section 3.3. In

Section 3.4 key factors involved in the derivation of the differential cross section for electrons

scattering off atomic nuclei are highlighted. The cross sections are specialised to exclusive one-

nucleon knock-out and inclusive scattering processes. The nuclear charge and current operators

are explained in Section 3.5. The non-relativistic one-body current operators employed, and

the relativistic corrections, are given. The parametrisation of nucleon electromagnetic form

factors is also briefly discussed.

3.1 Kinematics

Consider the electron-nucleus scattering processes where an energetic electron e interacts with

a nucleus of atomic number A by transferring some energy and momentum to the nucleus.

Depending on the magnitude of the energy and momentum of the electron, the electron can be

considered to interact with the nucleus as a whole, the individual nucleons in the nucleus, or the

constituents of the nucleons. As a result, the interaction of the electron with the nucleus may

generate a number of possible processes in the nucleus. Processes that may be induced range

from excitations to discrete energy levels of and resonances in the nucleus, to fragmentation of

the nucleus and excitations of resonances in the nucleons. Therefore, a careful choice of the

initial energy and momentum of the electron may isolate a desired process in the nucleus for

investigation. Consider the electron-induced fragmentation process

e+ A → e′ + x+B (3.1)

where a nucleon x is knocked out of the nucleus leaving a nucleus B with A − 1 nucleons. In

an experimental set up of such a process the scattered electron is observed in coincidence with

the ejected nucleon. This process is used to extract information about energy and momentum

38


density distributions in the nucleus. Specific information about nuclear electromagnetic struc-

ture can be obtained from this process. A related process of interest is the inclusive scattering

process

e+ A → e′ +B (3.2)

where only the scattered electron is observed. Information about nuclear electromagnetic struc-

ture is deduced from the properties of the scattered electron. As will be explained later only lim-

ited information can be derived from the corresponding photo-nuclear processes γ+A → x+B

and γ+A → B′. For high-energy electrons these processes can be treated as one-step reactions

so that the dynamics of the reaction mechanisms can be neglected [5].

In the Born approximation the electron-nucleus scattering processes indicated in equations (3.1)

and (3.2) are well described by an exchange of a single photon between the electron and the

nucleus. Figure 3.1 is a Feynman diagram for the process in equation (3.1) which illustrates the

interaction of a free electron of energy Ei and momentum ki with a nucleus of energy Ei and

momentum P i. During the interaction the electron transfers some energy ω and momentum q,

in the form of a virtual photon, to the nucleus. In the case of reaction (3.1) the electron knocks

out a nucleon with energy Ex and momentum px out of the nucleus. After the interaction

the electron scatters with energy Ef and momentum kf while the daughter nucleus exits with

energy Ef and momentum P f . The energy and momentum of a free relativistic electron are

related by

E =[

k2 +m2e

]1/2(3.3)

where me is the electron rest-mass and k = |k | the magnitude of the electron momentum. In

the scattering process the corresponding relations for the electron energies modify to Ei = ki

and Ef = kf in the case of ultra-relativistic free electrons since me is then negligibly small

compared to the magnitude of the momenta. In terms of the electron energies and momenta

the energy and momentum transferred are given by

ω = Ei − Ef (3.4a)

q = ki − kf (3.4b)

39


( q, ω )

θe

( P i, Ei )

( ki, Ei )

( P f , Ef )

( px, Ex )

( kf , Ef )

Ψi Ψf

ψx

ψi ψf

X X ′

x

e− e−

Figure 3.1: A Feynman diagram for an electron e− interacting with a nucleus by exchanginga virtual photon of momentum q and energy ω. See text for the explanation of notation.

which define the energy and momentum loss by the electron. The energy and momentum

conservation in the process requires that

Ei + Ei = Ef + Ex + Ef (3.5a)

P i + ki = P f + px + kf (3.5b)

40


θe

θqθx

pm

qpx

qkf

ki

Figure 3.2: A diagrammatic representation of the relative directions of different momentainvolved in the process explained in Figure 3.1.

for the electron-nucleus system. The variables

Em = Ex − ω (3.6a)

pm = px − q (3.6b)

called the missing energy Em and missing momentum pm, are often used in the analysis of

some electron-nucleus scattering processes like nucleon knockout reactions, for example. The

direction of the missing momentum relative to the transferred momentum is illustrated in

Figure 3.2. The energy and momentum of the photon can be combined into a momentum

41


four-vector qµ = (ω, q). The magnitude of this four-vector has the form

q2µ = −qµqµ = q2 − ω2 (3.7)

=∣

∣ki − kf

∣

∣

2 −(

Ei − Ef

)2(3.8)

= 2[

Ei Ef − ki kf cos θe −m2e

]

(3.9)

where θe is the angle into which the electron is scattered. In the ultra-relativistic limit equation

(3.9) reduces to

q2µ ≈ 4 ki kf sin2 θe

2(3.10)

since the contributions resulting from me are negligibly small.

As a result of the interaction with the electron, the nucleus make a transition from an initial

state described by the wave function Ψi to a final state described by the wave function Ψf .

These wave functions are required to be eigenfunctions of the same nuclear Hamiltonian H ,

H Ψi = Ei Ψi (3.11a)

H Ψf = Ef Ψf (3.11b)

with Ei as the energy of the target nucleus and Ef the energy of the final nuclear system. If

the interaction generates only excitations to discrete energy levels or resonances in the nucleus

then Ef corresponds to the binding energy of the resulting systems. However, if the interaction

results in the fragmentation of the nucleus then Ef represents a combination of the kinetic en-

ergies of the relative motions of the fragments and bound state energies of composite fragments,

if any. The two wave functions must be orthogonal to each other. As will be shown later, a

realistic description of the transition process rely on the realistic representation of the nuclear

Hamiltonian H and the accurate determination of the wave functions Ψi and Ψf .

42


3.2 The Transition Operator

The probability for an electromagnetic transition in a nucleus, from a state described by the

wave function Ψi to another state described by the wave function Ψf , is given by the expectation

value 〈Ψf | T |Ψi〉. T is the electromagnetic transition operator. For the electron-nucleus

interaction this operator is given by

T = 〈ψf | H |ψi〉 (3.12)

where ψi (ψf ) is a wave function describing the initial (final) state of the electron and H the

Hamiltonian describing the interaction between the electron and the nucleus. The rate of the

transition Wfi is determined by the golden rule of Fermi [89]

Wfi =2π

~

∣

∣

∣

⟨

Ψf

∣

∣ T∣

∣Ψi

⟩

∣

∣

∣

2

dρf (3.13)

where dρf is the density of final states of the system. Since the rate of the transition is propor-

tional to the transition amplitude, a transition with a large amplitude has a higher probability

of occurring. To determine the form of the transition operator the wave functions for the elec-

tron and the interaction Hamiltonian are required.

In the initial state energetic electrons are free particles described by the relativistic Dirac wave

equation [90]. The wave functions describing such electron are, therefore, plane waves of the

form

ψkν (xν) = us(k) ei kν xν

(3.14)

where us(k) are Dirac spinors of spin s. The interaction of an electron with a nucleus of charge

Z e generates Coulomb distortions in the electron-nucleus wave function. Since such distortions

will be of the order Z e2, they are very small in the case of light nuclei. For high-energy

electrons scattering off light nuclei the effects of the Coulomb distortions are not significant

and may, as a result, be ignored. In this approximation the final state of the electron is quite

accurately approximated by a plane wave of the form given by equation (3.14). In that case

the electromagnetic Møller four-potential Aµ(qν) ≡ φ(qν) , A(qν) generated by the electrons

43


has the form [90]

Aµ(qν) =4 π e

q2ν

[

usf(kf) γµ usi

(ki)]

ei qν xν

=4 π e

q2ν

aµ ei qν xν

(3.15)

where aµ = usf(kf) γµ usi

(ki) and γµ are Dirac matrices. The interaction between the electron

and the nucleus is governed by the electron potentials given by equation (3.15) and those

generated by the target nucleus, at the nuclear site. The transition operator of the target

nucleus determined by the interaction with the electron can be written in the form [90]

T =

∫

jµ(x)Aµ(x) dx = − i4 π e2

q2ν

∫

jµ(x) aµ eiq·x dx (3.16)

where jµ ≡ ( ρ , j ) the nuclear four-current operator composed of the nuclear charge operator

ρ(x) and nuclear current operator j(x). The time-dependence of the electron and nuclear

operators in equation (3.16) council out. Therefore the final form of the nuclear transition

operator is given by

T = − i4 π e2

q2ν

∫

[

a0 ρ(x) − a · j(x)]

eiq·x dx (3.17)

which is the familiar form of the operator commonly found in the literature [90]. Note that

slight modifications to equation (3.17) generate an operator corresponding to the interaction

of the electromagnetic radiation (real photons) with nuclear matter.

The total nuclear current operator j(x) is composed of different components originating from

different dynamical processes in the nucleus. The main components are the convection current

jc(x) generated by the orbital motions of the nucleons and the magnetisation current jm(x)

involving the motions of the spin of the nucleons. In terms of these two components the nuclear

current operator is expressed as

j(x) = jc(x) + jm(x) . (3.18)

To maintain the generality of the discussion the definitions of the charge and current operators

44


will be given later. The Fourier transform of the current four-operator is given by

jµ(q) =

∫

jµ(x) eiq·x dx (3.19)

where jµ(q) is the current operator in momentum space.

3.3 The Transition Multipoles

Consider a nucleus undergoing a transition between two states that both have definite total

angular momentum J . It is then convenient to express the transition operator as a linear

combination of operators, each capable of inducing a transition of a definite angular momentum

ki

kf

θe

θx

px

q

φ

zh−

h+

Scattering Plane

Reaction Plane

Figure 3.3: A convenient orientation of the system for evaluating the nuclear multipoles.The z-axis is directed along the transferred momentum q. The electron momenta ki and kf

as well as q define the scattering plane. h±1 is the polarisation of the electron.

45


in the target nucleus. The analysis of electron-nucleus scattering processes can be carried

out in a number of different frames of reference. However, the results are projected onto

invariant forms using rotational symmetries. A coordinate system in which the projection

of vectors onto spherical bases is simplified will be convenient. The analysis of multipole

transitions is conveniently explained in a coordinate system defined by the z-axis directed

along the momentum transfer vector q, as shown in Figure 3.3. The coordinate system is then

oriented so that the vectors ki and kf lay on the xz-plane, the scattering plane. Then the unit

basis vectors in this coordinate system are given by

uz = q , uy = ki × kf and ux = uz × ux (3.20)

where q = q/|q|. The corresponding set of unit spherical vectors is defined by

e0 = uz and e±1 = ∓ 1√2

[

ux ± i uy

]

. (3.21)

Other orientations of the system may proof more convenient for evaluating expectation values

of certain operators, as shown in the next chapter.

The transition multipoles are constructed by expanding the exponential in equation (3.17) in

terms of spherical Bessel functions and spherical harmonics. The standard transformation of

the scalar plane wave to a spherical wave in an arbitrary coordinate system has the form [91]

eiq·x = 4 π

∞∑

ℓm

iℓ jℓ(q x) Ymℓ ∗ℓ (q) Y mℓ

ℓ (x) (3.22)

where jℓ(q x) are the spherical Bessel functions and Y mℓ

ℓ (x) the scalar spherical harmonics. In

the chosen coordinate system Y mℓ ∗ℓ (q) → δmℓ0

√

( 2 ℓ+ 1 )/4 π. The multipole expansion of a

general vector plane wave, in the chosen coordinate system, has the form [91]

em eiq·x =√

4 π

∞∑

ℓJ

iℓ√

( 2 ℓ+ 1 ) jℓ(q x)⟨

ℓ 0 1m∣

∣J m⟩

YmJℓ(x) (3.23)

46


where

YMJℓ(x) =

∑

mℓm

⟨

ℓmℓ 1m∣

∣J M⟩

Y mℓ

ℓ (x) em (3.24)

are the vector spherical harmonics and⟨

ℓmℓ 1m∣

∣J M⟩

the Clebsch-Gordan coefficients. In

these expressions ℓ corresponds to the orbital angular momentum, J to the total angular mo-

mentum of the described system and M the projection of J on the quantisation axis. In

equation (3.23) the summation over ℓ is restricted by values of the Clebsch-Gordan coeffi-

cients to only three non-zero terms corresponding to ℓ = J , J ± 1. The expansion terms in

equation (3.23) can be simplified by introducing the functions XMJ (q x) = jJ (q x) Y M

J (x) and

XMJJ(q x) = jJ(q x) Y M

JJ(x). Evaluating the Clebsch-Gordan coefficients and using the prop-

erties of the Bessel functions, the expansion in equation (3.23) can be expressed in the form

[92]

em eiq·x = −√

2 π ×

∞∑

J≥0

BJ

√2

i

q∇X0

J(q x) : m = 0

∞∑

J≥1

BJ

[

mXmJJ(q x) +

1

q∇ × Xm

JJ(q x)

]

: m = ±1

(3.25)

where BJ = iJ√

2 J + 1. If the z-axis is not aligned with q the expansions on the right-hand-

side of equation (3.25) include a factor involving the Wigner D-functions.

To decompose the transition operator (3.17) into multipoles the operator is projected onto

the spherical basis (3.21). Since the charge operator is a scalar, its form is not affected by

the change of the coordinate system. As a result, its expansion into multipoles is simpler.

After projecting the vector components of the transition operator onto the spherical basis, the

electron and nuclear current three-vectors can be separated into longitudinal L and transverse

T components as

j = jL + jT and a = aL + aT . (3.26)

This labeling of the direction of these vectors is with reference to the direction of the transferred

47


momentum q. Then using the conditions

q · j(q) = ω ρ(q) and q · a = ω a0 (3.27)

which expresses the conservation of the nuclear and electron currents, the current transition

operator in equation (3.17) can be expressed in the form [93]

TJm(q) = T CJm(q) + T el

Jm(q) +m T magJm (q) (3.28)

where T CJm(q) is the Coulomb multipoles operator resulting from the longitudinal component

of the current whereas the electric multipoles operator T elJ (q) and the magnetic multipoles

operator T magJ (q) result from the transverse component of the current. These multipoles are

given by

T CJm(q) =

i

q

∫

[

∇XmJ (q x)

]

· j(x) dx (3.29)

T elJm(q) =

1

q

∫

[

∇ × XmJJ(q x)

]

· j(x) dx (3.30)

T magJm (q) =

∫

XmJJ(q x) · j(x) dx (3.31)

where the current operator is given by equation (3.18). These operators can be further simpli-

fied [94] using properties of the Bessel functions and spherical harmonics.

To obtain the transition amplitude the expectation value of the transition operator in equation

(3.28) is evaluated with the initial state∣

∣ JiMi

⟩

and the final state∣

∣ JfMf

⟩

of the system. The

transition amplitude is thus given by

TJm(q) =√

2 π ×

ω

qBJ

√2⟨

JfMf

∣

∣ T CJm(q)

∣

∣JiMi

⟩

: m = 0

− BJ

⟨

JfMf

∣

∣

[

T elJm(q) +m T mag

Jm (q)]

∣

∣ JiMi

⟩

: m = ±1

(3.32)

where use was made of the nuclear current conservation condition. The Wigner-Eckart theorem

48


[91] is used to express the elements of the transition amplitude in the form

⟨

JfMf

∣

∣ TJM(q)∣

∣ JiMi

⟩

= (−1)Jf−Mf⟨

JiMi J M∣

∣ Jf Mf

⟩ ⟨

Jf

∣

∣

∣

∣ TJ (q)∣

∣

∣

∣ Ji

⟩

. (3.33)

where TJM(q) is given by equation (3.28). This expression places a constraint on the transi-

tion multipoles allowed because of the requirement for the conservation of angular momentum

enforced by the Clebsch-Gordan coefficients. The coefficients vanish unless

∣

∣ Ji − Jf

∣

∣ ≤ J ≤ Ji + Jf . (3.34)

For physical significance, the matrix elements must be invariant under space-inversion and

time-reversal transformations [96]. Invariance under space-inversion requires the transitions to

conserve parity. Let πi (πf) be the parity of the initial (final) nuclear state. The conservation

of the space and time parities requires that

πJ =

πi · πf : space

πi/πf : time

(3.35)

where πJ is the parity of the multipoles, given by

πJ =

(

− 1)J

: charge/electric

(

− 1)J+1

: magnetic .

(3.36)

The conditions indicated by equations (3.34) and (3.35) identify multipoles that tend to be

favoured in transitions. The transitions involving multipoles that do not satisfy these conditions

are highly suppressed and, therefore, have very low probability of occurring. In the evaluation

of the transition matrix these selection rule lead to the non-interference of the Coulomb, electric

and magnetic transition multipoles [88].

49


3.4 The Differential Cross Section

A quantity related to the electron-nucleus scattering process that can be experimentally deter-

mined is the cross section. The differential cross section dσ for electrons scattering off nuclei is

obtained from the transition rate. It can be expressed in the form [88]

dσ =( 2 π )4

Fe

me

Ei

Mi

Ei

∑

i

∑

f

∣

∣

∣

⟨

Ψf

∣

∣ T (q)∣

∣Ψi

⟩

∣

∣

∣

2

× δ(ω + Ei − Ef ) δ( q + P i − P f ) dρf (3.37)

where Fe is the flux of the initial electrons and the mass-energy ratios result from normalisation

factors. The delta function ensures the conservation of energy and momentum in the process.

The symbol∑

i indicates averaging over the initial states of the electron and the target nucleus

while∑

f represents the usual sum over the final states of the scattering products. The form of

the factors determining the differential cross section depends on the frame of reference chosen

for the analysis and on the products of the scattering process. In the laboratory frame where

the target is stationary (P i = 0 and Ei = Mi) the flux is given by

Fe =ki

Ei(3.38)

which is the inverse of the electron speed. The expression for the density of final states is

determined by the products of the scattering process. For the nucleon knock-out processes the

density of states in given by

dρf =Mf

Ef

dP f

( 2 π )3

mx

Ex

dpx

( 2 π )3

me

Ef

dkf

( 2 π )3(3.39)

=Mf

Ef

dP f

( 2 π )3

mx px

( 2 π )3

me kf

( 2 π )3dEx dΩx dEf dΩe (3.40)

in the laboratory frame. In this frame the incident momentum of the electron is directed

along the z-axis and the direction of the momentum of the free nucleon is measured relative

to the initial momentum of the electron. For clarity the physical meaning of the geometrical

parameters and notation is illustrated in Figure 3.4. Using equations (3.38) and (3.40) the

50


σ

dσφ

θe

θxx

y

x

y

z

e− beam

( P i, Ei )

(px, Ex )

(ki, Ei )

(kf , Ef )

dΩe

dΩx

Figure 3.4: A schematic illustration of a coincident electron-nucleon scattering experiment.In the laboratory frame the initial momentum of the electron ki is directed along the z-axiswith the target cross section σ in the xy-plane.

differential cross section for electron-induced one-nucleon knock-out processes is given by

d6σ

dEx dΩx dEf dΩe= − 4m2

e e4

q4ν

kf

ki

Mf

Ef

mx px

( 2 π )3

∑

i

∑

f

∣

∣

∣

⟨

Ψf

∣

∣ T (q)∣

∣Ψi

⟩

∣

∣

∣

2

× δ(ω +Mi − Ex − Ef ) δ( q − px − P f ) dP f . (3.41)

To determine the expression for the differential cross section the transition probability ampli-

tude is evaluated as well as the summations over the initial and final states of the electron-

nucleus system.

A covariant formulation of the electron and nuclear current operators simplify the evaluation of

the elements for the transition amplitude. In the evaluations the angular momentum and parity

51


selection rules are considered, which reduces interferences among certain transition multipoles

[88]. Considering these selection rules, the differential cross section for unpolarised electrons

scattering off an unpolarised nucleus in the laboratory frame is given by [1, 92]

d6σ

dEf dΩe dEx dΩx=

(

dσ

dΩe

)

M

kf

ki

Mf

Ef

mx px

( 2 π )3V (ω, q, θe)

× δ(ω +Mi − Ex − Ef ) δ( q − px − P f ) dP f (3.42)

for the exclusive one nucleon knock-out processes. The Mott differential cross section

(

dσ

dΩe

)

M

=

(

e2

2 Ei

)2cos2(θe/2)

sin4(θe/2), (3.43)

is obtained when the electron interacts with a point charged target. In this parametrization of

the differential cross section

V (ω, q, θe) = KLRL +KT RT +KTT RTT +KTLRTL (3.44)

is the nuclear response function from the unpolarized electron interaction where L denotes

longitudinal and T transverse. The factors Kv depend on the kinematics of the process and are

given by [92]

KL =

(

q2µ

q2

)2

: KTL = − 1√2

q2µ

q2

[

tan2 θe

2− q2

µ

q2

]1/2

KT = tan2 θe

2− 1

2

q2µ

q2: KTT = −1

2

q2µ

q2

(3.45)

which depend only on the electron properties. The functions Rv(ω, q), given by

RL(ω, q) =∣

∣ ρ(q)∣

∣

2(3.46a)

RT (ω, q) =∣

∣ T+1(q)∣

∣

2+∣

∣ T−1(q)∣

∣

2(3.46b)

RTT (ω, q) = 2 Re[

T ∗+1(q) T−1(q)

]

(3.46c)

RTL(ω, q) = − 2 Re[

ρ∗(q)(

T+1(q) − T−1(q))]

. (3.46d)

52


depend only on the properties of the target nucleus. Information concerning the internal elec-

tromagnetic structure of the nucleus can be derived from these functions.

When the ejected nucleon is observed but its energy not measured the scattering process is

classified as semi-inclusive. For such processes the transition amplitude is independent of the

energy of the ejected nucleon. Then the dependence of the transition amplitude on the nucleon

energy is removed by integrating equation (3.42) over Ex. The differential cross section for a

semi-inclusive process has the form

d5σ

dEf dΩe dΩx=

(

dσ

dΩe

)

M

1

fr

kf

ki

Mf

Ef

mx px

( 2 π )3V (ω, q, θe) (3.47)

where

fr = 1 − Ex

Ef p2x

px · P f (3.48)

is the residual nucleus recoil factor resulting from integrating out the energy-conserving delta

function over the energy of the ejected nucleon. If only the scattered electron is observed

the scattering process is classified as inclusive. The dependence of the transition amplitude

on the geometrical parameters in equation (3.47) is again eliminated by integration over these

parameters. Since only RTT and RTL depend on the angle φ between the scattering and reaction

planes (Figure 3.3) [92] the resulting differential cross section has the form

d3σ

dEf dΩe

=

(

dσ

dΩe

)

M

1

fr

kf

ki

Mf

Ef

[

KL(ω, q, θe)RL(ω, q) +KT (ω, q, θe)RT (ω, q)]

(3.49)

where the recoil factor is now given by

fr = 1 + 2Ei

Ei

sin2 θe

2(3.50)

again obtained from integrating out the delta function over the momentum of the recoil nucleus

[34]. For elastic scattering fr −→ Ef/Ei.

53


3.5 Nuclear Current Operators

The nuclear electromagnetic current density j(q) and charge density ρ(q) can be combined into

a current four-vector density jµ = (i ρ, j) (µ = 0 , 1 , 2 , 3) with j0 = ρ. The current four-vector

satisfy the continuity equation

qµjµ = q · j − ω ρ = 0 (3.51)

which expresses the conservation of current in the nucleus. To reveal some of the properties of

the components of the current four-vector the continuity equation is expressed in terms of the

nuclear Hamiltonian H . Noting that the expectation value of the commutator[

H, ρ]

has the

form⟨

Ψf

∣

∣

[

H ρ− ρ H]∣

∣Ψi

⟩

=(

Ef − Ei

)⟨

Ψf

∣

∣ ρ∣

∣Ψi

⟩

(3.52)

where use was made of equation (3.11), the continuity equation can be written in the operator

form as

q · j = i[

H, ρ]

. (3.53)

The Hamiltonian of a many-body nuclear system has the general form

H =∑

n=1

H [n] (3.54)

where H [1] represents one-body operators, such as single-nucleon kinetic energy operators, and

H [m] (m > 1) the two-body, three-body through to n-body operators, such as kinetic energy

operators for relative motions and interaction potentials in the nucleus. To systematically

explain gauge invariance in terms of the nuclear Hamiltonian the total nuclear charge and

current operators must also be expanded in the form

ρ =∑

n=1

ρ[n] (3.55)

j =∑

n=1

j[n]

(3.56)

where, again, O[n] represents an n-body operator. Such a decomposition of the nuclear current

operators is also consistent with the meson-exchange theory of nucleon-nucleon interaction po-

54


tentials.

In terms of the n-body current operators the continuity equation decomposes into

q · j + i[

H, ρ]

= q · j[1] + i[

H [1], ρ[1]]

+

q · j[2] + i[

H [2], ρ[1]]

+ i[

H [1], ρ[2]]

+ i[

H [2], ρ[2]]

+

q · j[3] + i[

H [1], ρ[3]]

+ i[

H [2], ρ[3]]

+ i[

H [3], ρ[3]]

+ . . . (3.57)

where the maximum of n is determined by the number of nucleons in the nucleus considered. To

preserve the current conservation in the nucleus each set of n-body operators must separately

satisfy the current continuity equation. For a chosen set of n-body operators the current

conservation equation has the general form

q · j[n]=

n−1∑

m=1

[

H [n], ρ[m]]

+[

H [m], ρ[n]]

+[

H [n], ρ[n]]

(3.58)

where the sum is zero for the case n = 1. The one-body operators ρ[1] and j[1]

are the local

charge and current operators, respectively, of the nucleons. The two-body operators j[2]

arise

from the exchange of mesons by nucleons, hence they are called meson-exchange-current (MEC).

The two-body charge operators ρ[2] measure deformations to the local charges of the nucleons.

Similar physical significance hold for the higher order operators. For an A-body nuclear system

there are at least A nuclear charge and current operators.

3.5.1 One-Body Current Operators

Non-relativistic nucleon one-body charge and current operators are approximated from the

relativistic form of a free nucleon current operator. The most general on-shell relativistic

covariant single-nucleon current operator has the form [2]

jµ(pf ,pi) = u(pf)[

F1(q2µ) γµ − 1

4mF2(q

2µ)(

γµγν − γνγµ)

qν

]

u(pi) (3.59)

55


where γµ are the Dirac matrices and u(p) the Dirac spinors for the nucleon with initial mo-

mentum pi and final momentum pf . The functions F1(q2µ) and F2(q

2µ) are the Dirac and Pauli

nucleon form factors, respectively. The relevant Dirac spinor for the free nucleon has the form

u(p) =

(

m+ E

2m

)1/2

1

σ · pm+ E

=

√

1 + ε

2

1

σ · η1 + ε

(3.60)

where E =√

m2 + |p|2, ε = E/m and η = p/m. Applying this spinor and substituting in

the Dirac matrices in equation (3.59) the explicit expression for the relativistic nucleon current

operator is obtained [2]. Different versions of the non-relativistic nucleon one-body charge

and current operators are usually obtained as lower order terms in the series expansion of

the relativistic operator in powers of 1/m. The standard non-relativistic nucleon charge and

current operators are constructed in reference [97] by using q/m as the expansion parameter.

The required non-relativistic nucleon charge and current operators are then obtained as terms of

first-order in the series. Terms of second-order, and higher, in q/m are considered as relativistic

corrections to the non-relativistic one-body operators. In momentum space the operators can

be written in the form [97]

ρ0(q) =

(

1 − Q2

8m2

)

GE − i

4m2

[

2GM −GE

]

q · σ × p (3.61)

j0(q) =GE

2mp +

iGM

2mσ × q (3.62)

where GE and GM are the Sachs nucleon charge and magnetic form factors, respectively. In

terms of the Dirac (F1(Q2)) and Pauli (F2(Q

2)) form factors the Sachs form factors are given

by

GE(Q2) = F1(Q2) − Q2

4m2κF2(Q

2) (3.63)

and

GM(Q2) = F1(Q2) + κF2(Q

2) (3.64)

with κ as the anomalous magnetic moment of the nucleon. The form factors GE and GM give

an indication of the spatial distributions of the charge and magnetisation in the nucleon. This

56


form of the one-body operators is considered standard and is used in various investigations of

electromagnetic properties and structure of nuclei and nucleons [2, 26, 88].

The resulting series is convergent when the magnitude of the momentum transferred q is small

relative to the nucleon mass. However, it is now possible to generate, in experiment, energies

and transferred momenta with magnitudes comparable to and greater than the nucleon mass.

At such energies and momenta the meaning and accuracy of higher order terms in the operators

(3.61) and (3.62) is difficult to explain [98]. As an alternative, another form of the one-body

operators that result from involving the initial momentum of the nucleon pi/m in the expansion

parameter was discussed in [98]. The resulting operators in momentum space have the form

[98, 99]

ρ[1](q) =q

QGE +

i

4m2

2GM −GE√1 + η

σ · q × p (3.65)

j[1](q) =

√η

q

[

( 2GE + η GM) p − iGM

(

q × σ +ω

2mq · σ q × p

)]

(3.66)

where q = q/|q| and η = Q2/4m2. The series expansion defining jµ in powers of p/m is also

convergent when the magnitude of the initial momentum of the nucleon is smaller than the

nucleon mass. This form of the one-body operators may have the same shortcoming as in

equations (3.61) and (3.62) at high energies. However, these operators are shown to display a

behavior similar to that of the full relativistic version even at large energy and magnitudes of

transferred momenta. This version of the non-relativistic nucleon one-body charge and current

operators was applied in some investigations of the electromagnetic structure of nuclei [99, 100].

The description of the electromagnetic form factors of nucleons have been the subject of both

theoretical and experimental investigations for many years. Until recently most of the exper-

imental investigations were mainly based on unpolarised electron scattering off unpolarised

nucleon targets [101]. Analysis of the scattering data from such investigations suggested that

the nucleon charge and magnetisation have the same spatial distribution. However, new scat-

tering data from high precision polarisation transfer electron-proton scattering [14, 17] indicate

that the ratio of the proton charge form factor to the magnetic form factor linearly decrease

57


with increasing momentum transfers. The emergence of these new data called for improvements

on the existing models of nucleon electromagnetic form factors. One of the models improved

to fit the latest polarisation data is outlined in reference [102]. This model is based on dis-

persion relations and includes the 2 π , K K and ρ π continua contributions. In this thesis the

phenomenological model presented in reference [103] that bears some physical significance to

the in-medium nucleon structure is employed. The nucleon electromagnetic form factors in this

model are described by the form

GEN = ( 1 − bp )[

GuNE +GdN

E

]

+ bp

[

GuNE +GdN

E +GπE

]

(3.67)

GMN = GinM +Gout

M +GπM (3.68)

where

Gα =aα

0[

1 +Q2/aα1

]2 ; α ≡ qN, in, out , (3.69)

Gπ = aπ0

[

1 − 1

6

(

Q

aπ1

)2]

exp

[

−1

4

(

Q

aπ1

)2]

, (3.70)

with q ≡ quark, π ≡ pion and N ≡ nucleon. If N ≡ proton then N ≡ neutron. These

forms are based on the quark model of the nucleon and incorporate the pion cloud around the

nucleon. For the electric form factor the weight parameters a0 in equation (3.67) are given by

the relevant quark charges aup0 = 2 aun

0 = 4/3, adp0 = 2 adn

0 = −2/3 and aπ±

0 = ±1. All the other

Table 3.1: The parameters used to fit the electric form factors GEp and GEn for the protonand neutron, respectively.

Form aup1 adp

1 aun1 adn

1 aπ1 b

Factor (GeV/c)2 (GeV/c)2 (GeV/c)2 (GeV/c)2 GeV/c

GEp 1.008 2.54 2.54 1.008 0.203 0.11

GEn 1.008 2.54 6.2 5.3 0.203 0.086

58


Table 3.2: The parameters used to fit the magnetic form factors GMp and GMn for the protonand neutron, respectively.

Form aout0 aout

1 ain0 ain

1 aπ0 aπ

1

Factor (GeV/c)2 (GeV/c)2 GeV/c

GMp/µp 0.917 0.811 -0.0034 13.57 0.106 0.210

GMn/µn 1.363 1.173 -0.511 1.789 0.140 0.213

model parameters aαβ are given in Table 3.1 for the electric form factor and in Table 3.2 for the

magnetic form factor. This parametrisation of the nucleon electromagnetic form factors were

used in the study of nuclear charge and matter distribution in a variety of nuclei [105].

3.5.2 Effects of Two-Body Currents

For a realistic treatment of the electromagnetic interactions in a many-body nuclear system a

large number of nuclear charge and current operators must be considered. However, in practi-

cal applications only a limited number of operators is considered. A treatment involving a full

set of operators is inhibited by a number of factors. For example, the magnitudes of the ex-

pectation values of the Hamiltonian many-body operators appear to decrease rapidly with the

increase in the order of the operator, that is, the number of nucleons involved [66, 67]. Hence,

the contributions of higher many-body Hamiltonian operators are expected to be small and

are, in many cases, omitted in the construction of nuclear many-body Hamiltonian. In most

cases, only two-body and three-body interaction potentials are considered. Also, the complete

form or nature of the higher many-body Hamiltonian operators is not well understood, and

therefore, not well developed. The same uncertainties hold for the higher many-body current

operators.

Meson-exchange-currents are generally classified into two types, the model-dependent and model-

59


NN

π

NN

π

∆

( a ) ( b )

Figure 3.5: Processes considered in the two-body pion-exchange currents. (a) πNγ couplingand (b) π∆γ coupling.

NN

π π

NN

π ρ

( a ) ( b )

Figure 3.6: Processes considered in the model-dependent meson-meson coupling currents.(a) ππγ coupling and (b) πργ coupling.

60


independent exchange currents [106]. The model-independent currents are those that are related

to the nucleon-nucleon interactions through equation (3.58). Such currents are generated by

the components of the nuclear Hamiltonian for which

[

H [m], ρ[n]]

6= 0 : m > 1 . (3.71)

The components include the isospin-dependent and momentum-dependent interaction potential

[106]. The charge generated by the two-body and higher many-body deformation operators

ρ[m] do not contribute to the total charge of the nucleus [96]. This is the statement of Siegert’s

hypothesis which allows for the approximation

ρ[m] = 0 : m > 1 (3.72)

of the non-relativistic nuclear many-body charge operators. This is because the nuclear many-

body charge operators are model-dependent [106].

The AV4’ two-body potential can be written in the form

V (rij) =∑

α=c,σ

[

V α(rij) + V ατ (rij) τ i · τ j

]

Oαij (3.73)

where the operator are given by

Oαij ≡

1 , σi · σj

. (3.74)

The corresponding model independent two-body current operators can also be written in the

form

jij = jπij + j

ρij + jls

ij (3.75)

where π and ρ are the pseudo-scalar and vector mesons, respectively, exchanged. Figure 3.5 is a

diagram illustrating the electromagnetic couplings that generates model-independent exchange

currents. A criterion for generating gauge invariant forms of model-independent MEC operators

from a given potential is outlined in reference [106]. Using this criterion current operators for

61


different two-body, and three-body, potentials can be constructed so that the resulting operators

satisfy the charge conservation equation. Operators for the Argonne nucleon-nucleon potential

have also been derived [107, 108]. However, the effects of these type of exchange currents can be

implicitly taken into account using the Siegert hypothesis. Model-dependent meson-exchange-

currents like those involving the ∆ nucleon resonances, Figure 3.5(b), and those associated

with the electromagnetic coupling of different mesons, Figure 3.6 [106], are mainly transverse.

Such two-body current cannot be treated implicitly using Siegert theorem and can only be

treated explicitly. The Siegert theorem permits the representation transverse electric current

multipoles in terms of the charge multipoles at low energies [83]. Therefore, the application of

the theorem at high energies violates the conservation of transverse currents. To correct for

this violation an extended versions of the theorem that is applicable at high energy range was

proposed [84, 85]. This extended Siegert therem was shown to minimize the effects of current

conservation violation [86, 87].

62

Chapter 4

Transitions in Few-Nucleon Systems

The structure of few-nucleon systems and light nuclei is widely and continuously investigated,

both theoretically and experimentally. Reviews on a variety of aspects of nuclear structure,

including the electromagnetic structure, can be easily found in the literature [2, 26, 27]. Over

the years a variety of theoretical methods have been developed and refined in the study of elec-

tromagnetic processes in nuclei. Very accurate wave functions for bound and scattering states

in few-nucleon systems can now be constructed using realistic Hamiltonian for the systems. A

demonstration of the level of accuracy in describing ground state properties of the four-nucleon

system by seven different state-of-the-art methods is shown in reference [28]. The use of some

of these methods in the study of electromagnetic processes in three- and four-nucleon systems

is given in references [2, 26]. Although the construction of bound-state wave functions in all

these methods is almost standard, the construction of scattering-state wave functions still pose

a challenge in most of them. Scattering wave functions for two-body and three-body scattering

reactions can be treated accurately in the Faddeev formalism [2, 29]. In the hyperspherical

harmonics approach progress in being made towards the construction of accurate wave func-

tions for two-body scattering processes [30]. The Lorentz Integral Transform approach treats

bound states and scattering states in the same way [31, 32, 33]. In the Lorentz integral trans-

form approach the traditional solutions to the Schrodinger equation are not determined. The

approach generates integral transforms of the nuclear response functions from Schrodinger-like

equations. A number of microscopic simulation methods have also been used in the study of

light nuclei [34, 38, 40]. These methods are continuously developing. This thesis focuses on one

of the microscopic simulation method, the Antisymmetrized Molecular Dynamics.

In this chapter two of the main objectives of the thesis are addressed. The objectives are to

63


investigate

(i) the ability of the angular-momentum-projected and parity-projected AMD wave function

to describe ground-state properties, including electromagnetic form factors, of few-nucleon

systems and

(ii) the capability of the AMD-GA scattering wave function in predicting nuclear response

functions in the one-nucleon knock-out processes from few-nucleon systems.

A number of theoretical investigations relating to objective (i) have been conducted using

the AMD approach [65, 109]. However, most of the investigations employ phenomenological

NN potentials and do not consider some of the questions addressed in this thesis. The Glauber

multiple scattering approach is used in constructing wave functions for nuclear scattering states

in addressing objective (ii). In the processes studied in this thesis require ground state wave

functions as input in the determination of the transition amplitudes of the nuclei. As a result,

the chapter is commenced with a test of the ability of the AMD wave function to predict some of

the ground state properties of the 2H, 3H, 3He, 4He and 6Li nuclei, in Section 4.1. In Section 4.2

the AMD charge and magnetic form factors of the nuclei are compared with results of other

theoretical methods and some experimental data. In Section 4.3 the longitudinal and transverse

response functions of the electron-induced fragmentation processes in the four-nucleon system

are calculated.

4.1 Ground-State Properties

The construction of the AMD wave functions with parity and angular momentum projections

ΨJπ is explained in Chapter 2. In solving the cooling equations a random number generator

is used to set up the initial values of the variational parameters. The required wave function

is obtained when the energy-functional for the system is independent of the variations of the

parameters of the wave function. The parity projection of the wave functions is done before the

variation of the parameters and the angular momentum projection is done after the optimisation

of the parameters. The accuracy of a model wave function for a given system is often tested by

its ability to predict properties of the system. As a result, ground-state energies, root-mean-

64


square radii and magnetic moments of the selected nuclei are calculated using the AMD wave

functions with the Argonne V4’ (AV4) potential. This nucleon-nucleon potential is the reduced

version of the Argonne V18 potential. The AV4 potential has the form

V (rij) = Vc(rij) + Vσ(rij) σi · σj + Vτ (rij) τ i · τ j + Vστ (rij) σi · σj τ i · τ j (4.1)

where rij = ri − rj with ri, σi and τ i as the position, spin and isospin of the i-th nucleon,

respectively. The form factors Vc(rij), Vσ(rij), Vτ (rij) and Vστ (rij) are determined by fitting the

potential to limited experimental nucleon-nucleon scattering data [12, 110]. Only the VC1(pp)

component of the Coulomb part of the potential is used.

4.1.1 Energy, Radius and Magnetic Moment

The upper-bound to the ground-state energy of a nucleus is calculated as the variational energy

EJ±MK =

⟨

ΨJ±MK |H | ΨJ±

MK

⟩

⟨

ΨJ±MK

∣

∣ ΨJ±MK

⟩ (4.2)

=2 J + 1

8 π2 N J±MK

∫

DJ∗KK(Ω)

[

N (0) H(0) ±N (1) H(1)]

dΩ (4.3)

where H is the Hamiltonian of the nucleus and

N J±MK =

⟨

ΨJ±MK

∣

∣ ΨJ±MK

⟩

= N (0),JMK ±N (1),J

MK (4.4)

the normalisation of the total wave function. The expectation values of the Hamiltonian are

given by

H(n) =∑

ik

⟨

ψk | ti | ψi

⟩

B−1

ik +∑

ijkl,ν

Uνij Pν

ij BliBkj

[

B−1

il B−1

jk −B−1

ik B−1

jl

]

(4.5)

where⟨

ψk | ti | ψi

⟩

and Uνij Pν

ij are the expectation values of the kinetic and potential energies,

respectively. As explained in Chapter 2 the ground-state energies of the systems are adjusted

using the another free parameter of the AMD approach related to the center-of-mass and rel-

ative kinetic energies of the system, T0. In this thesis the value T0 = 10.0 MeV is adopted.

65


The results obtained using the parity projected wave function with α = 0.12 are presented

along with corresponding experimental data in Table 4.1. The value of α was chosen to satis-

factorily reproduce the experimental binding energy of the three-nucleon systems. As can be

observed from this table the theoretical prediction of the experimental binding energies of the

three-nucleon systems is satisfactory, as expected. However, the experimental binding energy

of the 2H nucleus is overestimated by about 60 %. The binding energy of the 4He nucleus is

underestimated by 10 % whereas that of the 6Li and 8Be nuclei are underestimated by 14 %.

The theoretical energy values differ significantly, in some cases, from experimental values. In

addition to the inadequate description of the nuclear Hamiltonian, there are two factors that

contribute to these differences. Firstly, the Gaussian width parameter is fixed for all the nucle-

ons and its determination does not depend on the nuclear size. For better results this parameter

needs to be independently optimized for a given nucleus. Secondly, the Gaussian shape for the

Table 4.1: The ground-state energies (E0), root-mean-square radii (√

〈 r2 〉) andmagnetic moments (µ) of selected few-nucleon systems. The values are obtainedwith the parity-projected wave function. The experimental data are taken fromreference [112].

E0 (MeV)√

〈 r2 〉 (fm) µ (µN)

AXπ AMD EXP AMD EXP AMD EXP

2H+ -3.53 -2.23 2.16 1.96 0.880 0.857

3H+ -8.12 -8.48 1.76 1.60 2.793 2.979

3He+ -7.78 -7.72 1.76 1.77 -1.913 -2.128

4He+ -25.50 -28.3 1.53 1.47 0.000

6Li+ -27.54 -31.99 2.70 2.4 0.880 0.822

8Be+ -49.38 -55.99 2.75 0.000

66


spatial wave function favours systems with high density. Low density systems, like the 2H

nucleus are better described by exponential wave functions with long ranges. The 8Be nucleus

is very unstable [111] and, therefore, very challenging to study experimentally. The major part

of this thesis is based on the three-nucleon and the four-nucleon systems. Therefore, the results

obtained with the parity and angular momentum projected wave function for these systems are

shown in Table 4.2. The rotation increases the theoretical energies of the three-nucleon sys-

tems by 10 % and decreases that of the four-nucleon system by 10 %, relative to the un-rotated

wave function. The theoretical energy for the 3H system overestimates the experimental energy

by 5 % and for the 3He system the experimental energy is overestimated by 12 %. Since the

nuclear systems are described with spherical wave functions, spatial rotations are not expected

to introduce significant modifications to the results presented in Table 4.1.

The same wave functions were used to calculate the root-mean-square radii of the nuclei using

the expression

⟨

r2⟩

MK=

1

A

〈ΨJ±MK | ∑A

i=1 ( ri − R )2 |ΨJ±MK 〉

〈ΨJ±MK |ΨJ±

MK 〉 (4.6)

=2 J + 1

8 π2AN J±MK

∫

DJ∗KK(Ω)

[

N (0) R(0) ±N (1) R(1)]

dΩ (4.7)

where A is the number of nucleons in the nucleus with

R(i) =∑

ik

⟨

ψk

∣

∣ r2i

∣

∣ ψi

⟩

B−1

ik . (4.8)

The results obtained with the parity projected wave function are presented in Table 4.1. As

can be observed from the table the AMD approach generates reasonable predictions of the ex-

perimental values for the root-mean-square radii of the nuclei. The deviation of the predicted

values from the experimental values for the nuclei range from 0 %, for the 3He nucleus, to 11 %,

for the 6Li nucleus. The theoretical radii of the 6Li and the 8Be nuclei are almost the same. The

experimental values of the radii are overestimated by the theoretical model for all the nuclei.

The results for the parity and angular momentum projected wave function are given in Table 4.2

for the three-nucleon and the four-nucleon systems. The predicted radius for the 3H system is

67


Table 4.2: The ground-state energies (E0), root-mean-square radii (√

〈 r2 〉) andmagnetic moments (µ) of the three-nucleon and four-nucleon systems. The values arefor the parity and angular momentum projected wave function. The experimentalvalues are taken from reference [112].

E0 (MeV)√

〈 r2 〉 (fm) µ (µN)

AX(

Jπ)

AMD EXP AMD EXP AMD EXP

3H(

12

+)

-8.95 -8.48 1.33 1.60 2.769 2.979

3He(

12

+)

-8.61 -7.72 1.33 1.77 -1.847 -2.128

4He(

0+)

-23.04 -28.30 1.16 1.47 0.000

about 16 % less and for the 3He system is about 22 % less than the experimental values. The

theoretical radius for the 4He system also underestimated the experimental value by about 21 %.

The magnetic moment µ of a nucleon in nuclear magnetons (µN) is given by [113]

µ = gℓ 〈 ℓ 〉 + gs 〈 s 〉 (4.9)

where 〈 ℓ 〉 (〈 s 〉) is the expectation value of the orbital (spin) angular momentum and gℓ (gs)

the orbital (spin) g-factor of the nucleon. The nucleon g-factors are constants, the values of

which are [113]

gℓ =

1 for proton

0 for neutron: gs =

5.585695 for proton

−3.826085 for neutron .(4.10)

The numerical magnetic moment of the nuclei µA is calculated by

68


µ±MK =

〈ΨJ±MK |∑A

i=1

[

gℓ ℓi + gs si

]

|ΨJ±MK 〉

〈ΨJ±MK |ΨJ±

MK 〉 (4.11)

=2 J + 1

8 π2N J±MK

∫

DJ∗KK(Ω)

[

N (0) µ(0) ±N (1) µ(1)]

dΩ (4.12)

where

µ(n) =∑

kl

[

−i gℓ

(

s∗k × sl

)

+ (−1)(1−λk)/2 gs

2

]

BklB−1

lk . (4.13)

The values of the magnitude µ±MK = |µ±

MK | for the parity projected wave function are also

presented in Table 4.1. Therefore, the indicated negative values are simply introduced. As

can be observed from the table the theoretical values. The calculated magnetic moments of

the three-nucleon systems are different from the experimental values by 6 % and 10 %. For

the three-nucleon systems the theoretical moments equal the magnetic moment of the unlike

nucleon in the system. This reason also explains why the calculated magnetic moments of the

2H and 6Li nuclei, which overestimate the respective experimental values by 3 % and 7 %, are

equal. The calculated magnetic moments of the 4He and 8Be nuclei are both equal to zero.

These results are consistent with the theoretical expectation when only the dominant S ground-

state wave function of the system is used in the calculations [114]. The results for the parity and

angular momentum projected wave function for the three-nucleon and four-nucleon systems are

given in Table 4.2. Projecting the wave function onto good angular momentum states reduce

the magnetic moments of the three-nucleon systems for the parity projected wave function by

about 2 %. In general, the AMD approach reproduces the experimental values for the magnetic

moment of the nuclei quite satisfactorily. The discrepancy can be rectified by including other

components of the ground-state wave function, like the mixed-symmetry S-state and the D-

state. Also, the implementation of the full version of the Argonne V18 potential including three-

nucleon forces, and the addition of relativistic corrections to the magnetic moment operator,

are expected to reduce the discrepancy between the theoretical results and experimental data.

69


4.1.2 Density Distributions

To discuss the density distribution in a system ofA particles define a set v = r1, r2, r3, . . . , rA of the position vectors of the particles in the system. Then introduce the subsets vi and vij

defined by

v =

ri , vi

(4.14)

=

ri , rj , vij

(4.15)

with the corresponding relations for the volume elements given by

d vi =

A∏

j 6=i

d rj and d vij =

A∏

k 6=j 6=i

d rk (4.16)

respectively. In terms of these sets the one-body and two-body density functions of the system

are given by

ρ(ri, r′i) =

∫

Ψ∗(ri,vi) Ψ(r′i,vi) dvi (4.17)

ρ(ri, rj , r′i, r

′j) =

∫

Ψ∗(ri, rj ,vij) Ψ(r′i, r

′j,vij) dvij , (4.18)

where Ψ(v) is the wave function describing the state of the system. The two-body density

function can also be expressed in terms of the relative coordinates of two particles as

ρ(rij , r′ij,Rij ,R

′ij) =

∫

Ψ∗(rij ,Rij,vij) Ψ(r′ij ,R

′ij,vij) dvij (4.19)

where rij = ri −rj and Rij = ( ri + rj )/2 are the relative and center-of-mass position vectors

of the pair of equal mass particles. In the case of particle of unequal masses these vectors are

constructed as outlined in Appendix A.

Intrinsic density distribution in a nucleus is determined by rotating the arbitrary space-fixed

coordinate axes of equation (4.17) onto the body-fixed principal axes of the nucleus. The

70


-6 -4 -2 0 2 4 6

X [fm]

-6

-4

-2

0

2

4

6

Y [f

m]

-6 -4 -2 0 2 4 6

X [fm]

-8

-6

-4

-2

0

2

4

Y [f

m]

0

0.2

0.4

0.6

0.8

1

1.2

2H 3H

Figure 4.1: The nuclear matter density for the 2H and 3H nuclei.

-6 -4 -2 0 2 4 6

X [fm]

-8

-6

-4

-2

0

2

4

Y [f

m]

-6 -4 -2 0 2 4 6

X [fm]

-8

-6

-4

-2

0

2

4

Y [f

m]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

3He 4He

Figure 4.2: The nuclear matter density for the 3He and 4He nuclei.

71


-8 -6 -4 -2 0 2 4 6 8

X [fm]

-6

-4

-2

0

2

4

6

8

10

Y [f

m]

-8 -6 -4 -2 0 2 4 6 8

X [fm]

-8

-6

-4

-2

0

2

4

6

8

Y [f

m]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

6Li 8Be

Figure 4.3: The nuclear matter density for the 6Li and 8Be nuclei.

rotation is generated by diagonalising the moment-of-inertia tensor

I =

Ixx Ixy Ixz

Iyx Iyy Iyz

Izx Izy Izz

(4.20)

of the nucleus where

Iµν =〈Ψ±

MK | ∑Ai=1 ( ri − R )µ ( ri − R )ν |Ψ±

MK 〉〈Ψ±

MK |Ψ±MK 〉 (4.21)

with R as the nuclear center-of-mass and µ, ν ≡ x, y, z . The diagonalisation of equation

(4.20) can be explained in terms of the unitary transformation

R I R−1 = I (4.22)

where R is the rotation matrix explained in Appendix C and I the diagonalized moment-of-

72


inertia tensor

I =

I1 0 0

0 I2 0

0 0 I3

. (4.23)

The elements I1, I2 and I3 are the principal moments of the nucleus. The intrinsic nuclear

matter density obtained with the parity and angular momentum projected wave function are

shown in Figure 4.1 for the 2H and 3H nuclei, in Figure 4.2 for the 3He and 4He nuclei and

in Figure 4.3 for the 6Li and 8Be nuclei. The ground-state densities for the 2H, 3H, 3He

and 4He systems are similar and appear spherical. This can be attributed to the use of only

spherically symmetric spatial wave functions (with zero orbital angular momentum) to describe

the systems. In addition, since these nuclei consist of four or less nucleons, the Pauli exclusion

principle restricts all of the nucleons to the lowest same state, the spherical 1 s-state. As can

be observed in Figure 4.3 the densities of the 6Li and 8Be nuclei display sizable degrees of

deformations. These deformations are consistent with two-cluster structures for the nuclei.

The deformations are explained by the Pauli exclusion principle. Since these nuclei consist of

more than four nucleons, the additional nucleons can only occupy one of the excited states in

the system. The first excited state is non-spherical 1 p-states. The clustering structure in the

6Li can be related to the (2H-4He) clustering and the clustering structure in the 8Be can be

identified with the (4He-4He) clustering. No clustering model assumption was made in obtaining

the results for these nuclei.

4.2 Elastic Electron-Nucleus Scattering

In elastic electron-nucleus scattering the electron transfers momentum q and energy ω to the

target nucleus. However, the nucleus does not undergo transitions to excited or continuum

states as a result [88]. Therefore, the initial and final states of the nucleus have the same

angular momentum, the ground state. This type of electron scattering is therefore used to

probe ground-state charge and magnetisation distributions in nuclei. The ground state of

the 3H and 3He systems have total angular momentum and parity Jπ = 12

+whereas the 4He

73


system has Jπ = 0+. Based on the angular momentum and parity selection rules the lowest

multipole transitions allowed in elastic electron scattering are the Coulomb monopole and

the magnetic dipole transitions in the three-nucleon systems and only the Coulomb monopole

transition in the four-nucleon system. In this section the charge and magnetic form factors of

3H, 3He and 4He nuclei are calculated in the plane wave impulse approximation (PWIA). The

electromagnetic form factors of three-nucleon and four-nucleon systems has been the subject of

many experimental and theoretical investigations for many years. Therefore, there are a number

of theoretical models and experimental data that the AMD form factors can be compared with.

4.2.1 Charge Form Factor

In elastic electron-nucleus scattering the charge distribution in the nucleus is inferred from the

induced electric transitions in the nucleus. The transitions are determined from the nuclear

charge operator. The transition amplitude is referred to as the charge form factor. The nuclear

charge form factor of a nucleus in a state |ΨJi±MK 〉 of parity ± and angular momentum Ji with

angular momentum projections MK is given by

Z Fch(q) =〈ΨJi±

MK | ρ(q) |ΨJi±MK 〉

〈ΨJi±MK |ΨJi±

MK 〉(4.24)

where Z is the charge on and ρ(q) the charge operator of the nucleus with q as the momentum

transferred to the nucleus by the electron. In the PWIA nuclear charge operator is formed by

the superposition of the individual nucleon charge operators. The nuclear charge operator is

given by

ρ(q) =A∑

k=1

[

q

QGN

Ek(Q2) − 2GN

Mk(Q2) −GN

Ek(Q2)

4M2N

√1 + τ

iσk · q × pk

]

exp(

i q · rk

)

(4.25)

where MN is the proton mass, τ = Q2/4M2N , Q2 = q2 − ω2, ω =

√

q2 +M2N − MN and

GNE (GN

M) the nucleon Sachs electric (magnetic) form factor. For the Sachs form factors the

phenomenological parametrisation derived in reference [103] is adopted. The transitions are

between states of definite angular momentum. The general multipole analysis of nuclear charge

74


form factors is given by [104]

Fch(q) =√

4 π

≤ 2J∑

L=0

〈JJL0|JJ〉F ρL(q) Y ∗

L0(q) (4.26)

where Y ∗LM(q) are the spherical harmonics, L the nuclear orbital angular momentum and

〈JJL0|JJ〉 Clebsch-Gordan coefficients. The summation is over even values of L only. For

three-nucleon systems Ji = 12.

The AMD nuclear charge form factor is calculated using the angular momentum and parity

projected wave function ΨJ±MK(S) as

Z Fch(q) =2 Ji + 1

8 π2N Ji±MK

∫

DJi∗MK(Ω)

[

N (0) F (0)ρ (q) ±N (1) F (1)

ρ (q)]

dΩ (4.27)

where the factors F (i)(q) are related to the functions F ρL(q) in equation (4.26). These factors

are determined from the nuclear charge operator as

F (i)ρ (q) =

∑

kj

[

CNRi (q, η′R) +

(

λk + λl

) (

s∗k × sl

)

zCSO

i (q, η′R)]

Bkj B−1

jk (4.28)

where η′R = ηR − ηcm with ηR and ηI as the real and imaginary components of η and ηcm

contributions of the center-of-mass. The non-relativistic (NR) and the spin-orbit (SO) contri-

butions are given by

CNRi (q, η′R) =

q

QGN

Ek(Q2) W0(q, η

′R) (4.29a)

CSOi (q, η′R) =

2GNMk(Q

2) −GNEk(Q

2)

4M2N

√1 + τ

W1(q, η′R) (4.29b)

where the function WJ(q, η′R) result from the Fourier transformation of the J-th multipole

components of the charge operator. The evaluation of the transformations is approximated

with the series

WJ(q ηR) = − exp

(

η21

2

) ∞∑

n=0

η2 n1 Iα(J, q, n) (4.30)

75


where ηi = s∗i + sj and Iα(J, q, n) the integral involving the spherical Bessel function, jJ(q r).

The integral is evaluated as in [94, 95] and has the form

Iα(J, q, n) =2αn+3/2

Γ(n+ 1) Γ(n+ 3/2)

∫ ∞

0

r2 n+2−JjJ(q r) e−2 α r2

dr (4.31)

= C(J, n)

(

q2

8α

)J/2

1F1

(

n+J + 3

2, J +

3

2,− q2

8 α

)

(4.32)

where 1F1(x, y, z) are the confluent hypergeometric functions and

C(J, n) =2n+1 Γ

(

n + J+32

)

Γ(n+ 1) Γ(

n+ 32

)

Γ(

J + 32

) . (4.33)

A formal proof of the convergence of the series in equation (4.30) is very difficult to locate in

the literature, if it exists. However, empirical observation indicates that the series converges

for any value of q and η. In the case of the non-relativistic component, the function

W0(qkj ηR) = j0

(

q η′R2√α

)

exp

(

η2I

2− q2

c

8α

)

(4.34)

is derived in Appendix C. The multipole decomposition of the spin-orbit term is explained

in reference [115]. The contribution of the center-of-mass are incorporated in the calculated

charge form factor. The wave function for the center-of-mass is given by

Gcm(S) = N exp

[

−Aα(

Rcm − Scm√Aα

)2

+1

2S2

cm

]

. (4.35)

Therefore, the contribution of the center-of-mass to the charge form factor has the form

Fcm(q) =

⟨

Gcm(S)∣

∣ eiq·Rcm∣

∣Gcm(S)⟩

⟨

Gcm(S)∣

∣Gcm(S)⟩ = exp

[

i q · SR√Aα

− q2

8Aα+ S∗ · S

]

(4.36)

where SR is the real component of Scm. The intrinsic charge form factor is obtained by dividing

the calculated charge form factor by Fcm(q).

The ground-state charge form factors of the 3H and 3He nuclei are calculated in the impulse

approximation. In this approximation the nucleons inside the target nucleus are assumed not

76


AMDFaddeev

q [fm−1]

∣ ∣

Fch

(q2)∣ ∣

76543210

100

10−1

10−2

10−3

10−4

10−5

Figure 4.4: The AMD charge form factor of the 3H nucleus compared with the impulseapproximation results of reference [117].

AMDFaddeev

q [fm−1]

∣ ∣

Fch

(q2)∣ ∣

6543210

100

10−1

10−2

10−3

10−4

10−5

Figure 4.5: The AMD charge form factor of the 3He nucleus compared with the IA resultsof reference [117]. In the displayed results of Ref. [117] the value of q1 is extrapolated.

77


to interact with one another during the nuclear interaction with the electron [116]. Then the

electron interacts with independent nucleons inside the nucleus. The results obtained for the

calculated charge form factors are displayed in Figure 4.4 and Figure 4.5. The presented charge

form factors are normalised such that Fch(0) = 1. In these figures the results of the theoretical

predictions of the AMD formalism are presented along with the IA theoretical results presented

in reference [117]. The calculations of [117] use the Faddeev approach with the SdT NN po-

tential [118] including a three-body force and employs standard parametrisation of nucleon

electromagnetic form factors. It is known that different theoretical methods generate different

predictions of nuclear electromagnetic form factors, much less with different input interaction

potentials, nucleon electromagnetic form factors and others. However, the general structure of

the form factors is similar. Therefore, the comparison of the AMD results and the Faddeev

results is only qualitative and is aimed at showing the general trend of the AMD results as

compared to results generated with conventional methods. For low momentum transfers, up to

the first diffraction minimum, predictions of the AMD are similar to those of reference [117].

For momentum transfers greater than the first diffraction minimum the AMD results are lower

than those of reference [117]. To further probe the general quality of the AMD model the

prediction of the position of the first diffraction minimum q1[fm−1] with the AV4’ potential is

shown together with the IA predictions of reference [119] that employed the Reid potential [120]

Table 4.3: The position of the first diffraction minimum q1[fm−1] of

the IA charge form factor for the three-nucleon systems obtainedwith different NN potential models. The AV4 results are from thiswork and all the other results are from reference [119].

Reid MT I-III MT I-IV AV4 EXP

3H 3.94 4.42 > 5.00 3.85 3.59

3He 3.87 4.06 4.90 3.85 3.46

78


and the Malfliet-Tjon potentials [121]. The results are shown in Table 4.3. As can be seen from

this table different NN potential models predict different values of q1 for the position of the

first diffraction minimum. However, all the models overestimate the experimental value of this

quantity. The AMD model generate values of q1 for the 3H and 3He nuclei that are consistent

with the predictions of other theoretical models. However, it is known that the overestimation

of the position of the diffraction minimum indicates the underestimation of the nuclear charge

radius.

In Figure 4.6, Figure 4.7 and Figure 4.8 the IA charge form factors for the 3H, 3He and 4He

nuclei, respectively, are compared with experimental data. The experimental data are taken

from reference [122, 123]. In the comparison phenomenological parametrisations that fit exper-

imental world data of the form factors for electron-nucleus scattering are also shown. In the

case of the three-nucleon systems the sum-of-Gaussian parametrisation [124, 125]

Fch(q2) = exp

[

−1

4( γ q )2

] 12∑

i=1

Qi

[

1 +2R2

i

γ2

]−1[

cos(

Ri q)

+2R2

i

γ2

sin(

Ri q)

Ri q

]

(4.37)

is used where γ, Ri and Qi are the fitting parameters. The best-fit values of the fitting param-

eters given in reference [124] are adopted. The parametrisation of reference [123],

Fch(q2) =

[

1 − ( a q )2 n ] exp[

−( b q )2]

(4.38)

where a = 0.316 fm, b = 0.681 fm and n = 6, is employed in the case of the four-nucleon

system. These parametrisations of the three-nucleon and four-nucleon systems describe the

form factors of these systems accurately for values of q2 < 25 fm−2. As can be observed in the

figures the AMD model give a good descriptions of the experimental charge form factors at

low momentum transfers for the nuclei. At intermediate momentum transfers the theoretical

model fails to describe the experimental results. At higher momentum transfers, greater than

the first diffraction minimum, the theoretical model underestimates the experimental results.

This behaviour is not unique to the AMD formalism. The charge form factors of the three-

nucleon and four-nucleon systems are analysed in references [107, 126, 127, 128, 129] using

79


AMDFit

q2 [fm−2]

∣ ∣

Fch

(q2)∣ ∣

2

20151050

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

Figure 4.6: The AMD charge form factor of the 3H nucleus compared with the theoreticalparametrisation that fits the experimental data of reference [124].

AMDFit

Exp

q2 [fm−2]

∣ ∣

Fch

(q2)∣ ∣

2

20151050

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

Figure 4.7: The AMD charge form factor of the 3He nucleus compared with the experimentaldata [123] and the theoretical fit for the experimental data of reference [124].

80


AMDFit

Exp

q2 [fm−2]

∣ ∣

Fch

(q2)∣ ∣

20151050

100

10−1

10−2

10−3

10−4

Figure 4.8: The AMD charge form factor of the 4He nucleus compared with the experimentaldata of reference [122].

conventional theoretical method that employ realistic nuclear Hamiltonian involving two-body

and three-body nuclear potentials. The results of the calculations of reference [126] are dis-

played in Figure 4.9, These results summarise the findings of many theoretical methods that

determine nuclear charge form factors rigorously. The IA of these methods display the same

description of the experimental data as the AMD. The disagreements between theoretical and

experimental results are corrected by the inclusion of two-body charge operators resulting from

meson-exchange-currents, isobar configurations [107, 128] and the effects of the three-body nu-

clear force [129].

The Coulomb distortions in electron-nucleus scattering are very small in light nuclei since the

nuclear charge Z e is small. For light nuclei the effects of such distortions can be taken into

accounted by employing the effective momentum approximation [130, 131, 132, 133, 134, 135,

81


Figure 4.9: The results of a comprehensive determination of the ground-state charge formfactors of the 3H and 4He nuclei taken from reference [126]. In the calculations meson-exchange-currents were considered.

136]. In this approximation the initial and final electron momenta are considered modified by

a term depending on the charge Ze on and the charge radius rc of the target nucleus. The

magnitude of the resulting initial and final effective electron momenta given by [132]

keffi = ki +

Zα fc

rcand keff

f = kf +Zα fc

rc(4.39)

where fc is a parameter depending on the shape of the charge distribution in the nucleus. In

the case of spherical charge distributions fc = 1.5. The corresponding effective momentum

transferred to the nucleus has the form [132]

qeff =

(

1 +Zα fc

rc kf

)

q +Zα fc

rc ki

(

1 − ki

kf

)

ki . (4.40)

82


In the case of elastic electron-nucleus kf = ki and the last term in equation (4.40) vanishes. The

effects of the Coulomb distortions are expected to simply decrease the depth of the diffraction

minimum [123]. Therefore, the Coulomb distortions are not investigated in this thesis.

4.2.2 Magnetic Form Factor

Magnetisation density distribution in nuclei is determined from magnetic transitions involving

transverse nuclear currents. The transition amplitude, the nuclear magnetic form factor, is

calculated as

µA Fmag(q) =〈ΨJ±

MfKf|µ(q) |ΨJ±

MiKi〉

√

N J±Mf Kf

N J±MiKi

(4.41)

where µ(q) is the magnetisation density operator and µA the nuclear magnetic dipole moment.

The PWIA transverse nuclear magnetisation density operator is given by

µ(q) =Q

2MN q

A∑

k=1

[

GNEk(Q

2) ℓk − iGNMk(Q

2) q × σ]

exp(

i q · rk

)

(4.42)

where ℓN is the nucleon orbital angular momentum [93]. The multipole expansion of nuclear

magnetic form factor has the form [104]

Fmag(q) =

√4 π

〈JJ10|JJ〉

≤ 2J∑

L=0

〈JJL0|JJ〉[

F µLL−1(q)Y0∗

LL−1(q) + F µLL+1(q)Y0∗

LL+1(q)]

(4.43)

where the summation is over odd values of L,

Y0∗LM(q) =

∑

m

〈Mm1 −m|L0〉 YMm(q) em . (4.44)

are the vector spherical harmonics and em spherical unit vectors. The general form of the

nuclear magnetic transition multipole operator can be derived as in reference [94] for a given

nuclear current operator.

For magnetic dipole transitions the initial and the final states of three-nucleon systems have

83


different spin projections. The AMD nuclear magnetic form factor is, therefore, calculated as

µA Fmag(q) =

[

4 π

N J±MfKf

N J±MiKi

]1/22 J + 1

8 π2〈JMi10|JMf〉

×∑

mµ m

〈JMi1mµ|Jm〉∫

DJ∗Mf m(Ω)

[

N (0) F (0)µ (q) ±N (1) F (1)

µ (q)]

dΩ (4.45)

where the coupling of the angular momenta is done as in reference [139]. The factors F (i)µ (q)

are given by

F (i)µ (q) =

2MN

q

∑

kl

[

J Lkl(q, η)

(

s∗k × sl

)

y+ (−1)(1−λk)/2 J S

kl(q, η)]

BklB−1

lk (4.46)

where

J Lkl(q, η) =

Q

2MN qGN

Ek(Q2)W0(q η

′R) (4.47a)

J Skl(q, η) =

Q

4MN

GNMk(Q

2)W0(q η′R) , (4.47b)

considering only the first term in the square brackets of equation (4.43). The contribution of

the orbital component J Lkl(q, η) is expected to be very small since only S-states are considered

for the three-nucleon systems. The intrinsic magnetic form factor of the systems is obtained

by dividing equation (4.45) with Fcm(q).

The results obtained for the calculated magnetic form factors of the three-nucleon systems are

displayed in Figure 4.10 and Figure 4.11. In these figures the AMD form factors are shown

together with the theoretical IA results of reference [117]. The calculated form factors are

normalised such that Fmag(0) = 1. As can be seen the AMD form factors are greater in magni-

tude than those of reference [117] at low momentum transfer, for both systems. At momentum

transfer greater than the diffraction minimum the AMD results are less than those of reference

[117]. As a result the AMD predicts a larger value for the diffraction minimum. In Figure 4.12

and Figure 4.13 the AMD form factors are compared with the parametrisation (4.37) of the

magnetic form factor that fits experimental data. The best-fit values of the fitting parameters

84


AMDFaddeev

q [fm−1]

∣ ∣

Fm

ag(q

2)∣ ∣

76543210

100

10−1

10−2

10−3

10−4

10−5

Figure 4.10: The AMD magnetic form factor of the 3H nucleus compared with the impulseapproximation results of reference [117].

AMDFaddeev

q [fm−1]

∣ ∣

Fm

ag(q

2)∣ ∣

6543210

100

10−1

10−2

10−3

10−4

10−5

Figure 4.11: The AMD magnetic form factor of the 3He nucleus compared with the IA resultsof reference [117]. In the displayed results of Ref. [117] the value of q1 is extrapolated.

85


are given in reference [124]. The AMD model reproduce the structure of the experimental data.

At low momentum transfers the AMD form factors generate a satisfactory prediction of the

experimental data. However, at intermediate and higher momentum transfers the AMD model

underestimate the experimental data as well as the position of the first diffraction minimum.

The AMD predictions at low momentum transfer are larger than the IA prediction of most

theoretical methods. Rigorous determination of the magnetic form factors of three-nucleon

systems has been carried out in the past [107, 117, 126, 127, 128, 129]. The IA results obtained

with conventional theoretical methods that employ realistic nuclear Hamiltonian generate IA

results that are similar to those of reference [117]. Again, the disagreements between theoreti-

cal results the experimental results are expected to be minimised by the inclusion of two-body

current operators, resulting from meson exchanges and isobar configurations [107].

AMDFit

q [fm−1]

∣ ∣

Fm

ag(q

2)∣ ∣

6543210

100

10−1

10−2

10−3

10−4

10−5

Figure 4.12: The AMD magnetic form factor of the 3H nucleus compared with the theoreticalparametrisation that fits the experimental data of reference [124].

86


AMDFit

q [fm−1]

∣ ∣

Fm

ag(q

2)∣ ∣

6543210

100

10−1

10−2

10−3

10−4

10−5

Figure 4.13: The AMD magnetic form factor of the 3He nucleus compared with the experi-mental data and the theoretical fit for the experimental data of reference [124].

4.3 Quasi-elastic Electron-Nucleon Scattering

The interaction of high energy electrons with a light nucleus can be described in terms of

the interaction of the electrons with quasi-free nucleons inside the nucleus. In the quasi-free

kinematics the electron is considered to interact with individual non-interacting nucleons which

may, subsequently, be ejected from the nucleus. When the reaction mechanism of the process

is not considered then the process can be treated as a direct nucleon knockout by the electron.

Such a process is referred to as a quasi-elastic electron-nucleon scattering. In quasi-elastic

electron-nucleon scattering experiments the ejected nucleon can be detected in coincidence with

the scattered electron. As a result different properties of the ejected nucleon can be directly

measured in the experiment. Therefore, this type electron-nucleus scattering can be used

to study momentum distribution in nuclei, nuclear interactions as well as nucleon properties

87


[140, 141]. Most of the theoretical and experimental (e, e′N) investigations are based on the

proton knockout. In this section 4He(e,e′ p)3H process is investigated. The differential cross

section for this process is determined by four nuclear response functions. All four of these

functions can be expressed in the form

Rv(q, ω) =∣

∣

⟨

Jπf

f Mf | O(q, ω) |Jπi

i Mi

⟩∣

∣

2(4.48)

=1

2N3H N4He

∣

∣

∣

⟨

ϕp(4) GΨJfπf

Mf ,3H

∣

∣

∣O(q, ω)

∣

∣

∣ΨJiπi

Mi,4He

⟩∣

∣

∣

2

(4.49)

where O(q, ω) is either the charge or current operator, ΨJiπi

Mi,4He

(

ΨJfπf

Mf ,3H

)

the ground-state wave

function of the 4He (3H) nucleus with the normalisation constant N4He (N3H), ϕp(4) the wave

function of the ejected proton and G the Glauber multiple-scattering operator.

The construction of the AMD wave functions is explained in Chapter. 2. It is convenient to

decompose the initial state of the target 4He nucleus into the form

|Jπi

i Mi 〉A =∑

mp

〈jpmp JoMo|JiMi 〉∣

∣jπp

p mp

⟩

⊗ |Jπo

o Mo 〉A−1 (4.50)

=∑

mp

〈jpmp JoMo|JiMi〉√

AN Jiπi

A

∣

∣

∣A[

ψjpπp

mp⊗ ΨJoπo

Mo,(A−1)p

]

⟩

(4.51)

for A = 4 where ψjpπpmp =

∣

∣jπpp mp

⟩

is the proton initial state, ΨJoπo

Mo,(A−1)pthe state of the remaining

(A − 1) nucleons and Mo = Mi −mp. The antisymetrisation operator A involve the different

configurations of the proton-(A−1)X system with angular momenta that couple to the angular

momentum configuration of the initial state. For the reaction 4He(e,e′ p)3H considered here the

angular momentum specifications are

|Jπi

i ;Mi 〉 =∣

∣ 0+; 0⟩

4He−→ 〈jpmp JfMf | 0 0〉

[

∣

∣

12

+; 1

2

⟩

3H⊗∣

∣

12

+;−1

2

⟩

p

]

(4.52)

for which the Clebsch-Gordan coefficients can be readily evaluated. The general response

88


function in equation (4.49) is, therefore, given by

Rv(q, ω) =G2

v(q, ω)

16N Jiπi4He N Jfπf

3H

∣

∣

∣

∣

∫

Djp∗mpmp

(Ω)DJf∗Kf Kf

(Ω)[

N (0)fi W

(0)v (q) + N (1)

fi W(1)v (q)

]

dΩ

∣

∣

∣

∣

2

(4.53)

where N (0)fi = 〈Ψf (S3H)|Ψi(S4He)〉, Gv(q, ω) a function depending on the electromagnetic form

factors of the proton and W(k)v (q) the AMD transition form factors. In equation (4.53) the

property [91]

DJMK(Ω) =

∑

m1k1

〈j1m1 j2m2|JM 〉〈j1k1 j2k2|JK 〉Dj1m1k1

(Ω)Dj2m2k2

(Ω) (4.54)

was used and Djp∗mpmp(Ω) acts only on the parameters of the ejected proton. The AMD form

factors are given by

W (n)v (q) = T (0)

n (q) +∑

ik

T (1)n (q)BkiB

−1

ik +∑

ijkl

T (2)n (q)BkiBlj

[

B−1

il B−1

jk − B−1

ik B−1

jl

]

(4.55)

where T (ν)n (q) are transition amplitudes evaluated as explained in Appendix C. In equation

(4.55), the first term represents the PWIA, the second and the third terms are due to the first

and second order FSI contributions, respectively. The transition amplitudes have the same

form for all the response functions. As shown in Appendix C the amplitudes can be cast in the

form

T (n)(q) =1

(2 π)3L(n)(q′,SR)K(n)(q′,SI) (4.56)

where q′ = q − pf and SR (SI) the real (imaginary) component of S. The final momentum

of the ejected proton is denoted by pf . The explicit forms of the functions L(n)(q′,S) and

K(n)(q′,S) are given in the appendix.

Energy and momentum conservation in the process are used in evaluating the integrals leading

to the final expressions of the response functions. As explained in Chapter. 3 the energy and

momentum conservation are given by

ω = ∆M + Tp + TA−1 and q = pf + P A−1 . (4.57)

89


where Tp (TA−1) is the non-relativistic energy of the proton (recoil nucleus) and ∆M = MA −MA−1 −MP the separation energy of a proton from the target nucleus. These expressions are

used to determine the angle θqp between q and pf as

2 q pf cos θqp = q2 +

(

1 +MA−1

m

)

p2f − 2MA−1

(

ω − ∆M)

. (4.58)

The magnitude of the proton momentum pf is restricted by the condition [142]

(

MA−1

Mp+ 1

)

pf ≥∣

∣Pω − q∣

∣ (4.59)

where

Pω =

2(

MA−1 +Mp

)MA−1

Mp

[

ω − ∆M − q2

2 (MA−1 +Mp)

]1/2

. (4.60)

In the next subsection the response functions for inclusive electron-nucleus scattering are calcu-

lated. It should be noted that the response functions are obtained by appropriate integrations

of the corresponding functions of the exclusive scattering process.

4.3.1 Inclusive Electron-Proton Scattering

The differential cross section for inclusive electron-nucleus scattering is determined by the

longitudinal and transverse nuclear response functions, RL(q, ω) and RT (q, ω), respectively.

These functions are given by

RL(q, ω) =1

2 Ji + 1

∑

f

∣

∣

∣

⟨

Jπf

f Mf | ρ(q, ω) |Jπi

i Mi

⟩

∣

∣

∣

2

δ (ω − ∆E ) (4.61)

RT (q, ω) =1

2 Ji + 1

∑

f

∣

∣

∣

⟨

Jπf

f Mf | j⊥(q, ω) |Jπi

i Mi

⟩

∣

∣

∣

2

δ (ω − ∆E ) (4.62)

where ∆E = Tp + TA−1 + ∆M . The nuclear charge and current operators, ρ(q, ω) and j(q, ω),

are respectively given by equations (3.65) and (3.66). The operator j⊥(q, ω) represents the

components of the current operator that are transverse to q. The coordinate system is oriented

such that q is directed along the z-axis, which is also chosen to be quantisation axis, so that

j⊥ = jx x + jy y. In the notation of reference [115] the nuclear charge and current operators

90


are written in the compact form

ρ(q, ω) = ρc(q, ω) + i ρso(q, ω)(

py σx − px σy

)

(4.63)

jx(q, ω) = Jc(q, ω) px + i Jm(q, ω)

(

q σy +ω

2Mppy σz

)

(4.64)

jy(q, ω) = Jc(q, ω) py − i Jm(q, ω)

(

q σx +ω

2Mppx σz

)

(4.65)

where pν (σν) are the Cartesian components of the nucleon linear momentum (spin) vectors.

The functions depending on the four-momentum transfer are given by

ρc(q, ω) =q

QGp

E(q, ω) , ρso(q, ω) =q

4M2p

2GpM(q, ω) −Gp

E(q, ω)√1 + τ

(4.66)

Jm(q, ω) =

√τ

qGp

M(q, ω) , Jc(q, ω) =

√τ

q

[

2GpE(q, ω) + τ Gp

M(q, ω)]

(4.67)

where Jc(q, ω) and Jm(q, ω) are obtained from the non-relativistic convection and magnetic

nuclear currents, respectively. The nuclear response functions are given by equation (4.53)

with

G2L(q, ω) = ρ2

c(q, ω) (4.68)

G2T (q, ω) = p2

⊥

[

J2c (q, ω) +

( ω

2Mp

)2

J2m(q, ω)

]

(4.69)

where p⊥ = px x + py y.

The longitudinal and transverse response function for the inclusive 4He(e,e′ p) process are cal-

culated for the kinematics of the experimental results presented in reference [143]. The PWIA

results including FSI for q = 400MeV/c are shown in Figure 4.14 for the longitudinal and

in Figure 4.15 for the transverse response. As can be seen in the figures the FSI effects are

quite significant, especially near the peaks. In Figure 4.16 the AMD results are compared with

experimental data from [143] for q = 300 MeV/c, q = 400 MeV/c and q = 500 MeV/c. The

numerical results of the longitudinal response function RL(q, ω) reproduce the general structure

of the experimental data in all the cases of the three values of q, with the value of the peak

91


DoubleSinglePWIA

ω [MeV]

RL

[GeV

−1]

200150100500

15.0

10.0

5.0

0.0

Figure 4.14: The AMD PWIA longitudinal response with the effects of Single and Double

scattering FSI.

DoubleSinglePWIA

ω [MeV]

RT

[GeV

−1]

200150100500

15.0

10.0

5.0

0.0

Figure 4.15: The AMD PWIA transverse response with the effects of Single and Double

scattering FSI.

92


0.0

5.0

10.0

15.0

0 50 100 150 200

RT [

Ge

V-1

]

q = 300 MeV/c

0.0

5.0

10.0

15.0

20.0

RL [

Ge

V-1

]

ExpPWIA

Full

50 100 150 200 250Em [MeV]

q = 400 MeV/c

ExpPWIA

Full

50 100 150 200 250 300

q = 500 MeV/c

ExpPWIA

Full

Figure 4.16: The longitudinal (RL) and transverse (RT ) AMD response function for the4He nucleus at q = 300MeV/c, q = 400MeV/c and q = 500MeV/c. The experimentaldata are from reference [143].

decreasing as q increases. For values of the energy transfer ω > 100 MeV the numerical results

reproduce the experimental data satisfactorily. However, for ω < 100 MeV the numerical results

overestimate the experimental data. The discrepancy between theory and experiment seem to

decrease as q increases. The contributions of the FSI to the theoretical response functions

decrease the disparity between the theoretical predictions and the experimental values. The

theoretical results for the transverse response functions RT (q, ω) overestimate the experimental

data at low energy transfers. Whereas the longitudinal response is well predicted for higher

values of q, the transverse response is generally well predicted for all the the values of q. The

same data was also analysed in reference [144] and the PWIA results of the calculations are

shown in Figure 4.17. These results display similar features as the AMD PWIA results except

93


Figure 4.17: The longitudinal (RL) and transverse (RT ) response function for the 4Henucleus at q = 300MeV/c, q = 400MeV/c and q = 500MeV/c. The figures are taken fromreference [143] and the curves are the results of the PWIA calculations of reference [144].

that the corresponding graph peaks are shifted toward higher energy values. Many theoreti-

cal nuclear models have been developed. Most of the models give a reasonable description of

empirical data of nuclear properties [107, 128, 145, 146]. The best description is achieved by

models that employ nuclear Hamiltonian with realistic two-body and three-body potentials,

and including FSI, MEC and IC in the formation of the nuclear currents and wave functions.

The contribution of these factors are found to be significant even at low momentum transfers.

It was also demonstrated [146, 147] that the differential cross section is sensitive to the input

nuclear two-body and three-body potential.

94

Chapter 5

Summary and Conclusions

In this thesis the AMD approach was used to study properties of few-nucleon systems. The

AMD wave function of a nucleus is constructed as a Slater determinant of single nucleon wave

functions. The wave functions are constructed with a set of complex variational parameters and

one free real parameter. The values of the variational parameters are determined by minimis-

ing the variational energy of the nucleus with respect to the variation of the parameters. The

value of the free parameter is chosen to reproduce the binding energies of certain light nuclei.

This form of the wave functions allows for the analytical evaluation of expectation values of

many operators related to the physical quantities of systems. To obtain the binding energies of

nuclear systems the possibility of cluster formation is taken into account, which introduces an

additional free parameter related to the energy of the relative motion of the possible clusters.

This form of the AMD approach is the simplest of the known versions of the AMD. This simple

version of the AMD incorporates Pauli correlations only and does not take into account other

properties, such as many-body correlation and more flexibility in the variational wave function.

Some process in nuclear systems are governed by angular momentum and parity selection rules.

Wave functions with definite parity and angular momentum should be used in investigating such

processes. In extracting physical properties of the systems the AMD wave function was first

projected onto wave functions that have definite angular momentum and parity. This thesis

has demonstrated how the variational parameters of the wave function can be determined with

realistic nucleon-nucleon potentials and how the wave function can be used to study electro-

magnetic processes in few-nucleon systems. Although the thesis focus on few-nucleon systems,

the methods are applicable to many-nucleon systems, as well.

To demonstrate the competency of the AMD model wave function in nuclear structure studies,

95


the angular momentum and parity projected AMD wave function was used to construct ground

state wave functions of selected few-nucleon systems. In the past the construction of AMD wave

functions was restricted to the use of phenomenological nucleon-nucleon potentials of Gaussian

radial form. In this thesis the nuclear Hamiltonian was described with the semi-realistic Ar-

gonne V4’ nucleon-nucleon potential. Ground-state energies, root-mean-square radii, magnetic

moments and clustering in the nuclei were calculated using the constructed wave functions and

results were compared with experimental data. The AMD wave functions reproduce the ex-

perimental values of these ground-state properties of light nuclei satisfactorily, given the crude

form of the model. The AMD wave function has limited flexibility which limits the accuracy

with which it can predict the nuclear properties. As was demonstrated in this thesis, it is

possible to construct the AMD wave function with realistic nuclear Hamiltonian. The use of a

more realistic nuclear Hamiltonian is important for a realistic description of nuclear systems.

However, because of the rigidity of the wave function, such a Hamiltonian may introduce very

little improvements in the accuracy of the theoretical results of the nuclear properties. A more

flexible variational wave function constructed with a more realistic nucleon-nucleon potentials,

including three-nucleon potentials may reduce the overall discrepancy between the experimen-

tal observation and theoretical AMD predictions of the nuclear properties.

To show other applications of the AMD formalism, electromagnetic properties of three-nucleon

and four-nucleon systems were investigated using electron-nucleus scattering. The properties

were determined in the non-relativistic formalism within the impulse approximation. Conven-

tional forms of the one-body nuclear charge and current operators were employed. Nucleon

electromagnetic form factors parametrised using recent experimental nucleon-nucleon scatter-

ing data were used. Except for the incomplete treatment of many-body current effects, this

approach is realistic. Firstly, the charge monopole and the magnetic dipole transitions in the

nuclei were determined from elastic electron scattering. These transitions are used to extract

information about ground-state charge and magnetisation distributions in the nuclei. The ob-

tained results were compared with predictions obtained using a conventional theoretical method

that employed a different nuclear Hamiltonian and different nucleon electromagnetic form fac-

tors. Such a comparison, though not realistic, helps judge the general quality of the AMD

96


model. The results were also compared with selected experimental data. The AMD approach

generates PWIA charge form factors for few-nucleon systems similar to those predicted by other

theoretical models. The calculated charge form factors for the nuclei are very close to the exper-

imental form factors for values of momentum transfers less than the first diffraction minimum.

For values of momentum transfers greater than the diffraction minimum the theoretical pre-

dictions underestimate the experimental data and overestimate the position of the diffraction

minimum. In the case of the magnetic form factors the theoretical results generally underesti-

mate the experimental data as well as the position diffraction minimum. The calculated AMD

ground-state charge and magnetisation distribution in the nuclei are consistent with the results

obtained with other conventional theoretical methods. The deviations of the theoretical results

from experimental data can be explained by the limitations of the electromagnetic operators

used. Therefore the AMD wave functions can describe nuclear states as accurately as most

conventional theoretical methods.

To extend the range of applicability of the AMD approach, the Glauber approximation was com-

bined with the angular-momentum and parity projected AMD to constructed realistic nucleon-

nucleus scattering wave functions. The constructed scattering wave function takes into account

the final-state-interactions in nuclear fragmentation processes in an approximate way. Such a

wave function does not satisfy gauge invariance conditions. The wave function is not orthogonal

to the ground-state wave function of the target nor is it an eigenfunction of the Hamiltonian of

the target. However, the wave function is a very good approximation of the exact solution of the

scattering problem. The electron-induced two-body break-up processes in four-nucleon systems

was investigated. Only the proton knock-out reactions was considered. The angular-momentum

and parity projected AMD wave functions were used to describe ground-states of the bound sys-

tems involved in the reactions. The nuclear response functions for unpolarised electron-nucleus

scattering were calculated under widely investigated kinematics. The response functions were

also used to determine the differential cross section for inclusive quasi-elastic electron-nucleon

scattering process. The results obtained were compared with experimental data as well as

results obtained using other theoretical methods. The AMD generated longitudinal and trans-

verse response functions in the PWIA are similar to those given by other theoretical methods.

97


In general, the AMD predictions give a satisfactory description of experimental data. However,

the agreement between theory and experiment deteriorates as momentum transfers increase.

The inclusion of the contributions of FSI improves the general agreement between theory and

experiment. The nuclear response functions for inclusive proton knock out 4He are satisfacto-

rily explained by the AMD model.

In the treatment of electromagnetic processes in few-nucleon systems investigated in this thesis,

a number of contributing factors were not considered in order to simplify the model. Such

factors are widely investigated in the literature [93, 147, 148, 149] and are found to contribute

very little to the non-relativistic electron-nucleus processes. The factors include :

• Many-Body Correlations : Explicit treatment of many-body correlations in the construc-

tion of nuclear wave functions or nuclear Hamiltonian.

• Many-Body Interactions : Including three-nucleon interactions in the description of nu-

clear Hamiltonian.

• Many-Body Currents : The interaction of the photon with the exchanged mesons by two

or more interacting nucleons, including the ejected nucleon.

• Coulomb Distortions : The distortions introduced to the cross section by the interactions

of the electron and nuclear electric charges.

• Indirect Processes : The detected nucleon is not the one that directly interacted with

the photon. That is, the photon interacted with another nucleon, the recoil nucleus, or a

nucleon resonance.

• Relativistic Effects : Employing relativistic kinematics and considering higher order terms

in the electron-nucleus interaction Hamiltonian. These involve the modification of nuclear

charge and current operators.

The individual contributions of the factors are relatively small. However, these factors are

necessary for constructing realistic nuclear Hamiltonian, realistic nuclear wave functions and

realistic nuclear current operators. All these factors can be readily incorporated in the described

98


AMD approach. The inclusion of these factors in the theoretical model is expected to bring

the theoretical predictions of nuclear properties in agreement with experimental observation.

These factors will be considered in future investigations.

99

Appendix A

Jacobi Coordinates

In this appendix some of the properties of Jacobi relative coordinates in configuration space

and momentum space are listed. To be more self-contained, the properties are given in a more

general form. Consider a system of N particles with masses m1, m2, m3, . . . , mN , individual

particle position vectors r1, r2, r3, . . . , rN and the corresponding individual particle momenta

p1, p2, p3, . . . , pN , in the laboratory coordinate system. The center of mass R and the total

momentum P of the system are defined as

R = RN =1

M

N∑

i=1

mi ri and P = P N =N∑

i=1

pi , (A.1)

respectively, where

M = MN =

N∑

i=1

mi (A.2)

is the total mass of the system. The reduced mass of the system µ, is defined by

1

µ=

1

µN=

N∑

i=1

1

mi. (A.3)

For a system composed of distinct clusters of particles these definitions still hold except for the

limits of the summations. For a given cluster, the summation is over the constituents of the

cluster and for the system the summation is over the constituent clusters.

To construct a set of primary relative position vectors in a system of a number of particles

one chooses a reference particle. Then the separation of the center-of-mass of the reference

particle and the center-of-mass of the closest particle defines the first relative position vector.

100

Jacobi Coordinates

ξ1

TTTTTTT

ξ2

\\\\\\\

aaaaaaaaaaaa

ξ3

@@

Figure A.1: The diagrammatic representation of the Jacobi vectors in the case oftwo-particle cluster formation in the system.

ξ1

###########

ξ2

bbbbbbbbbb

ξ3

DDDDDDDDDD

ξ4

##########

LLL

Figure A.2: The diagrammatic representation of the Jacobi vectors in the case ofno clusterisation in the system.

101

Jacobi Coordinates

The next relative position vector is defined as the separation of the center-of-mass of the first

two particles and the center-of-mass of the third particle. This procedure is followed until all

the particles in the system are accounted for. If there are distinct groups of particles (clusters)

in the system, then the relative position vectors may refer to the separation of the center-

of-mass of the clusters. In this configuration the vectors are referred to as secondary. The

number and complexity of the different arrangements of particles in a system increase with the

number of particles in the system. Consider a system that has a distinct number of clusters.

Then the relative position vector ξ between any two clusters, and the corresponding conjugate

momentum q, are given by [150]

ξi =

√

µi,j

m(Rj − Ri ) and qi =

√

µi,j

m

(

P j

Mj

− P i

Mi

)

, (A.4)

respectively, where m is an arbitrary normalization mass, here chosen as M/N and

µi,j =MiMj

Mi +Mj

(A.5)

the reduced mass of the two subsystems. Rτ and P τ is the center of mass and the corresponding

conjugate momentum of the τ -th cluster of total mass Mτ . An example of the position vectors

ξi for such a configuration in a system are illustrated in Figure A.1. In general the primary

set of relative position vectors ξτ and their respective conjugate momenta qτ can be generated

from the individual particle position vectors and momenta ri, pi through the transformations

[150]

ξi =∑

j

Aij αij rj and qi =∑

j

Bij αij pj (A.6)

where

αij =

√

µi,j

m. (A.7)

The numbers Aij and Bij are elements of the transformation mass matrices constructed as

outlined in reference [150]. In the case of a configuration where there are no clusters in the

system the transformation matrices for the system shown in Figure A.2 are explicitly given by

102

Jacobi Coordinates

A =

−1 1 0 · · · 0

−m1

M2

−m2

M2

1 · · · 0

−m1

M3−m2

M3−m3

M3· · · 0

......

.... . .

...

− m1

MN−1− m2

MN−1− m3

MN−1· · · 1

m1

MN

m2

MN

m3

MN

· · · mN

MN

(A.8)

for the position vectors and

B =

− 1

m1

1

m2

0 · · · 0

− 1

M2− 1

M2

1

m3· · · 0

− 1

M3− 1

M3− 1

M3· · · 0

......

.... . .

...

− 1

MN−1− 1

MN−1− 1

MN−1· · · 1

mN

1

MN

1

MN

1

MN

· · · 1

MN

(A.9)

for the momentum vectors. Using these matrices the expressions in equation (A.6) have the

explicit form

103

Jacobi Coordinates

Position Vectors Momentum Vectors

ξ1 = α12

[

r2 − r1

]

ξ2 = α23

[

r3 −m1 r1 +m2 r2

m1 +m2

]

...

ξj = αj,j+1

[

rj+1 −∑j

i=1mi ri

Mj

]

= αj,j+1

[

rj+1 − Rj

]

q1 = α12

[

p2

m2

− p1

m1

]

q2 = α23

[

p3

m3

− p1 + p2

m1 +m2

]

...

qj = αj,j+1

[

pj+1

mj+1−∑j

i=1 pi

Mj

]

= αj,j+1

[

pj+1

mj+1

− P j

Mj

]

(A.10)

where the mass-weight factors αij are given by equation (A.7). The expressions for the Jacobi

vectors in equation (A.6) represent just one of the many configurations or partitions of the

system. However, if the system is composed of identical particles then all the partitions of the

Jacobi vectors for the system depicted in Figure A.2 are identical.

Given the Jacobi vectors for a system, the corresponding individual particle vectors can be

constructed from Jacobi vectors using the inverse transformations

rj =∑

i

A−1ji xi and pj =

∑

i

B−1ji ρi (A.11)

where the definitions

xi =ξi

αji

and ρi =qi

αji

. (A.12)

are used with A−1ij and B−1

ij being the elements of the inverse of the matrices A and B, respec-

104

Jacobi Coordinates

tively. The inverse matrices are explicitly given by

A−1 =

−m2

M2

−m3

M3

−m4

M4

· · · −mN

MN

1

m1

M2−m3

M3−m4

M4· · · −mN

MN1

0M2

M3−m4

M4· · · −mN

MN1

......

.... . .

...

0 0 0 · · · MN−1

MN

1

(A.13)

and

B−1 =

1m1

M2

m1

M3· · · m1

MN−1

m1

MN

−1m2

M2

m2

M3· · · m2

MN−1

m2

MN

0 −1m3

M3

· · · m3

MN−1

m3

MN

......

.... . .

...

0 0 0 · · · −1mN

MN

(A.14)

respectively. The transformations given in equation (A.11) can be explicitly written as

105

Jacobi Coordinates

Position Vectors Momentum Vectors

r1 = R −∑

i=1

mi+1 xi

Mi+1

r2 = R −∑

i=2

mi+1 xi

Mi+1+m1 x1

M2

...

rj = R −∑

i=j

mi+1 xi

Mi+1

+Mj−1 xj−1

Mj

...

rN = R +MN−1 xN−1

MN

p1 =m1

MP +

∑

i=1

m1ρi

Mi

p2 =m2

MP +

∑

i=2

m2 ρi

Mi− ρ1

...

pj =mj

MP +

∑

i=j

mj ρi

Mi− ρj−1

...

pN =mN

MP − ρN−1

(A.15)

by using the matrices A−1 and B−1. In the case of identical particles, set mi = m, for all i, and

the reduced mass simplifies to

µj,j+1 =j

j + 1m (A.16)

and the reduced position vectors and their respective conjugate momenta are simplified.

106

Appendix B

Energy Expectation Values

In this appendix matrix elements related to the evaluation of expectation values of some oper-

ators using the AMD wave function are explained. Expressions of expectation values for most

of the operators considered can be found in the literature. However, because of the simplicity

of the AMD wave function, such expression can be easily derived. The AMD single-particle

wave function ψ(x) is expressed in the separable form

ψ(x) = φ(r)χ(σ) ξ(τ ) (B.1)

where φ(r), χ(σ) and ξ(τ ) are the spatial, spin and isospin components of the single–particle

wave functions, respectively. Define the elements of the overlap matrix of the wave function as

Bij = 〈ψi |ψj 〉 = 〈φi χi ξi | φj χj ξj 〉 = Rij Sij Tij (B.2)

where

Rij = 〈φi |φj 〉 , Sij = 〈χi |χj 〉 and Tij = 〈 ξi | ξj 〉 (B.3)

are the respective spatial, spin and isospin overlap matrix elements. To evaluate these single-

particle overlap matrices the explicit forms of the single-particle wave functions are required.

B.1 Overlap Matrix Elements

As indicated in Chapter 2 the single-particle spin and isospin wave functions have the form

χ(σ) =

| ↑ 〉 , | ↓ 〉

and ξ(τ ) =

| p 〉 , |n 〉

(B.4)

107


respectively where the vectors σ and τ are given by the Pauli matrices. Define the parameter

λ by

σz χi = λi χ (B.5)

which shows that λ = ±1. Expectation values of spin-isospin operators evaluated with these

functions can then be conveniently expressed in linear combinations of λk. The elements of the

spin overlap matrix have the form

Sij =1

2( 1 + λi λj ) = δλiλj

. (B.6)

A similar expression is obtained for Tij in the case of the isospin. It can be deduced that

Sij Tij = δαiαj(B.7)

where αi represents the spin-isospin combination of the i-th nucleon. If the expectation value

of the spin operator σ is denoted by

〈σ 〉ij = 〈χi |σ |χj 〉 , (B.8)

then the components of this expectation value have the form

〈 σx 〉ij =1

2( 1 − λi λj ) (B.9a)

〈 σy 〉ij =i

2(λj − λi ) (B.9b)

〈 σz 〉ij =1

2(λi + λj ) (B.9c)

in terms of the λ’s. For the product operator σi · σj one obtains the general expression

〈σ 〉ki · 〈σ 〉lj =1

4( 1 − λk λi ) ( 1 − λl λj ) +

1

2(λk λj + λl λi ) , (B.10)

for the arbitrary orientation of the spin, where use was made of equation (B.9). In the case of

dynamic spins the relations (B.6), (B.9) and (B.10) retain the same form with λ replaced by a

variational parameter. Similar relations hold for the isospin operators τ and τ · τ .

108


The spatial component of the single-particle wave function is chosen to be a Gaussian

φ(r) =

(

2α

π

)3/4

exp

[

−α

(

r − s√α

)2

+1

2s2

]

(B.11)

where s is a complex variational parameter and α a width parameter chosen as a free real

constant. This form of the wave function simplifies the evaluation of expectation values of

many operators. Note that the wave function for coupled two particles is constructed from

those of the single-particles by converting to relative coordinates discussed in appendix A. In

this coordinates the coupled two-particle state is given by

φ(r1)φ(r2) = φ(r)φ(R) (B.12)

with

φ(r) =( α

π

)3/4

exp

[

−α2

(

r − si − sj√α

)2

+1

4

(

s2i + s2

j

)

]

(B.13)

φ(R) =

(

4α

π

)3/4

exp

[

−2α

(

R − si + sj

2√α

)2

+1

4

(

s2i + s2

j

)

]

(B.14)

where R = (r1 + r2)/2 and r = r2 − r1 are the center-of-mass and relative coordinates of the

two particles. The elements of the overlap matrix of the spatial single-particle wave function

have the form

Rij = 〈φi |φj 〉 = exp

[

−1

2( s∗

i − sj )2 +1

2

(

s∗2i + s2

j

)

]

= exp ( s∗i · sj ) (B.15)

which is independent of the width parameter. The elements of the overlap matrix for combi-

nations of two-particle wave functions are given by

〈φk φl |φi φj 〉 = exp [ s∗k · si + s∗

l · sj ] = RkiRlj . (B.16)

Expressions for the case of three or more particles can be derived in the same way. The variation

109


of the elements Rij with respect to the parameters s are

∂Rij

∂s∗m

= sj Rij δmi = sj Rij (B.17)

which holds for each component of the vectors involved. For the elements of the inverse matrix

the variations have the form

∂ R−1ji

∂s∗m

= −∑

kl

R−1jk

∂ Rkl

∂s∗m

R−1li = −F s

ii R−1ji (B.18)

where the matrix F s with the elements

F sij =

∑

k

sk RikR−1kj , (B.19)

is introduced for convenience.

B.2 The Variational Energy

The ground state energy of a system described by the AMD wave function is approximated

from the energy functional

E(S,S∗) =〈Ψ(S) |H |Ψ(S) 〉〈Ψ(S) |Ψ(S) 〉 (B.20)

where H is the Hamiltonian of the system and S a set of the variational parameters si : i =

1 , . . . , A . The denominator in equation (B.20) is the norm of the total wave function, and

is given by

N = 〈Ψ |Ψ 〉 = det [Bij ] . (B.21)

To obtain the ground state energy the functional (B.20) is minimized with respect to the

variations of the set S. This is expressed as

∂ E

∂ si

=∂ E

∂ s∗i

= 0 : i = 1, 2, 3, . . . , A . (B.22)

110


Since the systems treated in this work are described by a Hamiltonian of the form

H =∑

i

[

−~2 p2

i

2Mi+∑

j>i

V (rij)

]

, (B.23)

one-body and two-body matrix elements need to be determined.

The single-particle kinetic energy operator is

ti = − ~2

2Mi∇2

i (B.24)

where Mi is the mass of the i-th particle. Therefore, the matrix elements of the single-particle

kinetic energies are computed as

Tij =〈φi | ti |φj 〉〈φi |φj 〉

= − ~2

2Mi

〈φi | ∇2i |φj 〉

〈φi |φj 〉(B.25)

=α ~

2

2Mi

[

3 − ( s∗i − sj )2] Bij B

−1ji (B.26)

where

( s∗i − sj )2 =

∑

σ

( s∗iσ − sjσ )2 (B.27)

with σ ≡ x, y, z. The total kinetic energy of the system is then given by

T =

A∑

ij=1

Tij =α ~

2

2M

[

3A−A∑

ij

( s∗i − sj )2Bij B

−1ji

]

(B.28)

for an equal-mass particle system, where use was made of the property∑A

ij Bij B−1ji = A. The

variation of the kinetic energy matrix elements with respect to the complex parameters s∗ is

given by

∂ T∂s∗mσ

= −α ~2

2M

A∑

ij

[

∂ ( s∗i − sj )2

∂s∗mσ

Bij B−1ji + ( s∗

i − sj )2

(

∂ Bij

∂s∗mσ

B−1ji +Bij

∂ B−1ji

∂s∗mσ

)]

=α ~

2

M

(

F smm,σ − s∗mσ

)

− α ~2

2M

A∑

ij

( s∗i − sj )2 ( sjσ δim − F s

mi,σ

)

Bij B−1jm . (B.29)

111


A similar expression is obtained for the variation with respect to sj.

The total potential energy of the system resulting from two-body interactions V2(rij) is given

by

V2 =〈Ψ | V2 |Ψ 〉〈Ψ |Ψ 〉 =

1

2

∑

ijkl

Ulk Plk RliRkj

[

B−1il B−1

jk − B−1ik B−1

jl

]

(B.30)

where Pij is the expectation value involving the spin-isospin operators and

Uij =( α

π

)3/2∫

exp

[

−α(

rij −η2

2√α

)2]

un(rij) drij (B.31)

=( α

π

)3/2

eη2

2/4

∫

exp[

−α r2ij +

√αη2 · rij

]

un(rij) drij (B.32)

with η2 = s∗l − s∗

k + si − sj . Consider the expansions

eη·r = 4 π∑

ℓm

(−i)ℓ jℓ(i η r) Y∗ℓm(η) Yℓm(r) (B.33)

= 4 π j0(i η r) Y∗00(η) Y00(r) =

√

π

2 η rI1/2(η r) for ℓ = 0 (B.34)

and

Iν(η r) =

∞∑

n=0

(

12η r)2n+ν

n! Γ(n+ ν + 1)(B.35)

where jℓ(z) (Iν(z)) is the spherical (modified) Bessel function and Yℓm(η) the spherical har-

monics. For potentials of general radial form the integrals in Uij can be approximated with the

series

Uij = exp

(

−η22

4

) ∞∑

m=0

(

η22

4

)m

Iα(m) (B.36)

where Iα(m) is the one-dimensional Talmi integral

Iα(m) =2αm+3/2

Γ(m+ 1)Γ(m+ 32)

∫ ∞

0

r2 m+2 u(r) e−αr2

dr (B.37)

112


involving the gamma function Γ(x). The integral in Iα(m) is evaluated numerically using a

Gaussian quadrature. If the range of the potential is not greater than 1/√

2α then the series in

equation (B.36) is expected to converge quite fast for all values of η2 [63]. It can be shown that

the values of these integrals for the different components of the Argonne V4’ potential decrease

rapidly with increasing values of m. The variation of the expectation value in equation (B.36)

with respect to the variational parameters has the analytical form

∂ Uij

∂ s∗lx=

(η2)x

2exp

(

−η22

4

)

∑

m

[

m

(

η22

4

)m−1

−(

η22

4

)m]

Iα(m) . (B.38)

Since the series in equations (B.36) and (B.38) converge quite rapidly for short-range potentials,

only the first few terms in the series tend to be significant. Thus, in practical applications the

series is truncated after N significant terms depending on the desired accuracy.

There are few forms of the radial function un(rij) for which the evaluation of the expectation

value Uij can be simplified. The form that occur often in nuclear potentials is the Gaussian

type

un(r) = vn exp(−an r2) . (B.39)

For this type of radial form the integral (B.32) can be evaluated exactly to obtain

Unij = vn ( 1 − ρn )3/2 exp

[

−ρnη2

1

4

]

(B.40)

∂Unij

∂s∗lx= −ρn

2

(

η2

)

xUn

ij (B.41)

where η1 = s∗l − s∗

k + sj − si and

ρn =an

an + α. (B.42)

As a result, it would be useful if the radial component of the potential can be expressed as

a sum of Gaussian functions because the expectation values can then be evaluated exactly.

For the Coulomb interaction of the Argonne V4’ only the VC1(pp) component is used. This

113


VC1

Vs

Vl

r [fm]

V(r

)[M

eV]

14121086420

8

6

4

2

0

-2

-4

-6

Figure B.1: The long-range Vl and the short-range Vs components of the VC1(pp) Coulombfor the Argonne V4’ interaction. The total potential is given by VC1 = Vl + Vs.

component can be expressed in the form

VC1(pp) = Vl(pp) + Vs(pp) (B.43)

where Vl represents the long-range part of the interaction and Vs the short-range part. The

spatial behaviour of these components is shown in Figure B.1. The expectation value of the

long-range interaction is evaluated using the integral transform

Vl(r) =1 − exp (−an r )

r=

2√π

∫ ∞

0

[

1 − e−a2n/4 q2 ]

exp[

−r2 q2]

dq (B.44)

114


to obtain

U lij =

2√π

∫ ∞

0

[

1 − ρ(q)]3/2 [

1 − e−a2n/4 q2 ]

exp

[

−ρ(q) η22

4

]

dq (B.45)

∂U lij

∂s∗lx= −

(

η2

)

x√π

∫ ∞

0

ρ(q)[

1 − ρ(q)]3/2 [

1 − e−a2n/4 q2 ]

exp

[

−ρ(q) η22

4

]

dq (B.46)

where

ρ(q) =q2

q2 + α. (B.47)

The integration over q is evaluated numerically. The range of the component Vs as observed

from Figure B.1 indicates that the expectation value of the short-range component can be eval-

uated using equation (B.36).

The expectation values of the relative position vector ri − rj , the mean-square radius r2ij and

the orbital angular momentum L = ~(

ri − rj

)

×(

pi − pj

)

/2 are given by

〈φk φl | ri − rj |φi φj 〉 =η2

2√αRliRkj (B.48)

〈φk φl | | ri − rj |2 |φi φj 〉 =3 + η2

2

4αRliRkj (B.49)

〈φk φl |L |φi φj 〉 = −i ~ ( s∗lk + sij ) × ( s∗

lk − sij ) /2RliRkj (B.50)

= i ~ ( s∗lk × sij ) RliRkj , (B.51)

where s∗lk = s∗

l − s∗k.

115

Appendix C

Transition Matrix Elements

In this appendix the evaluation of the expectation values of various operators with the angular

momentum projected AMD wave functions is explained. The expectation values of electromag-

netic transition operators involve the Fourier transformation of the AMD wave functions. In

Section C.1 the numerical angular momentum projection is explained. The matrix elements

for the potential energy and electromagnetic form factors are given in Section C.2. Section C.3

explains the matrix elements for the electron-induced nucleon ejection transition matrix.

C.1 Nuclear Rotations

To generate a wave function describing a state with definite total angular momentum J from

a wave function Ψ(S), that is not an eigenfunction of the total angular momentum operator

J , the Peielrs-Yoccoz angular momentum projection numerical techniques is employed. The

technique involve the use of the projection operator [73]

P JMK =

2 J + 1

8 π2

∫

dΩDJ∗MK(Ω) R(Ω) (C.1)

=2 J + 1

8 π2

∫ 2π

0

dα

∫ 2π

0

dγ

∫ π

0

dβ sin β DJ∗MK(α, β, γ) R(α, β, γ) (C.2)

where DJ∗MK(Ω) are the complex conjugates of the Wigner D-functions with M and K (ranging

form −J to J) being the projections of the angular momentum along the quantization axis.

The set of angles Ω = α, β,γ describing the rotation are called the Euler angles. The Wigner

D-functions are given by [91]

DJMK(α, β, γ) = e−i α M dJ

MK(β) e−i γ K (C.3)

116


where the functions

dJMK(β) =

[

(J +M)! (J +K)! (J −M)! (J −K)!]1/2

×∑

N

(−1)N(

cos 12β)2J−2N+M−K (

sin 12β)2N−M+K

N ! (J +M −N)! (J −K −N)! (K −M +N)!(C.4)

are referred to as the reduced Wigner d-functions. The summation run over the values of N for

which the factorials are defined. Various representations and properties of the Wigner functions

are given in reference [91]. The property

DJ∗MK(α, β, γ) = (−1)K−M DJ

−M −K(α, β, γ) (C.5)

is employed to make appropriate transformations of the Wigner functions.

The rotation is generated by the operator R(Ω) defined by

R(α, β, γ) = exp(

− iα Jz

)

exp(

− i β Jy

)

exp(

− i γ Jz

)

(C.6)

where Jτ = Lτ + Sτ (τ = y, z) are the components of the total angular momentum operator ob-

tained from the combination of the orbital angular momentum L and S spin angular momenta

operators. The rotation operator R(Ω) generates rotations in both the position space and spin

space. This operator can be decomposed into a product of a spatial rotation and spin rotation

operators. The rotations of position vectors are generated by the operator

Rr(α, β, γ) = exp(

− iα Lz

)

exp(

− i β Ly

)

exp(

− i γ Lz

)

(C.7)

and spin rotations are generated by the operator

Rσ(α, β, γ) = exp(

− iα Sz

)

exp(

− i β Sy

)

exp(

− i γ Sz

)

. (C.8)

Rotations in the two spaces can be carried out independently since the operators Rr(Ω) and

117


Rσ(Ω) commute. The rotation of the position space transforms the wave function to

Rr(Ω) Ψ(R,S) = Ψ(

R−1r (Ω)R,S

)

= Ψ (R, Rr(Ω)S) = Ψ(

R,S)

(C.9)

where

Rr(Ω)S ≡ Rr(Ω) s1 , Rr(Ω) s2 , Rr(Ω) s3 , . . . , Rr(Ω) sN . (C.10)

The rotated vectors

s = Rr(α, β, γ) s (C.11)

are generated by three successive rotations about the z-axis, y-axis and again the z-axis through

the Euler γ, β and α, respectively. The three independent rotations generate the total rotation

effected by the matrix [91]

Rr =

cos γ − sin γ 0

sin γ cos γ 0

0 0 1

cosβ 0 sin β

0 1 0

− sin β 0 cos β

cosα − sinα 0

sinα cosα 0

0 0 1

(C.12)

=

cosα cosβ cos γ − sinα sin γ − cosα cosβ sin γ − sinα cos γ cosα sin β

sinα cosβ cos γ + cosα sin γ − sinα cosβ sin γ + cosα cos γ sinα sin β

− sin β cos γ sin β sin γ cosβ

(C.13)

where the three 3×3 matrices in equation (C.12) each generate a single rotation about a given

axis through the indicated Euler angle. The transpose of equation (C.13) is used to generate

the inverse R−1r given by

R−1r =

cosα cosβ cos γ − sinα sin γ sinα cosβ cos γ + cosα sin γ − sin β cos γ

− cosα cosβ sin γ − sinα cos γ − sinα cosβ sin γ + cosα cos γ sin β sin γ

cosα sin β sinα sin β cos β

(C.14)

118


which generates rotation in the opposite direction to that generated by equation (C.13). From

equation (C.11) it can be deduced that the Jπ projection of the wave function affects only

the set S but not S∗. Therefore, the expectation values calculated with the projected wave

function have the same form as those calculated with the unprojected wave function except for

the replacement S∗ −→ S∗ and S −→ −S.

The complete rotation of the wave function is given by

R(Ω) Ψ(S) = Rσ(Ω) Ψ(

S)

(C.15)

where Rσ(Ω) is the spinor rotation matrix. Since the spin angular momentum depends on the

Pauli matrices σ, single rotations about a given axis in the spin-space are generated by 2×2

matrices. The rotation of the nucleon spin wave function is given by [91]

Rσ(Ω) |χj〉 =∑

χn

χnD1/2χnχj

(Ω) . (C.16)

The expectation value of the spin-rotation operator calculated with the spin states χ1 = | ↑ 〉and χ2 = | ↓ 〉 has the form

〈χi | Rσ(Ω) |χj〉 =∑

χn

〈χi | χn〉 D1/2χnχj

(Ω) = D1/2χiχj

(Ω) (C.17)

which is the rotation matrix for spin-1/2 particles. Therefore [91]

〈χi | Rσ(Ω) |χj〉 =

e−i 1

2γ 0

0 e+i 1

2γ

cos 12β − sin 1

2β

sin 12β cos 1

2β

e−i 1

2α 0

0 e+i 1

2α

(C.18)

=

e−i 1

2α cos 1

2β e−i 1

2γ − e−i 1

2α sin 1

2β e+i 1

2γ

e+i 1

2α sin 1

2β e−i 1

2γ e+i 1

2α cos 1

2β e+i 1

2γ

. (C.19)

119


The expectation value of a spin operator Oσ is evaluated as

〈χi | Oσ Rσ(Ω) |χj〉 =∑

χn

〈χi | Oσ |χn 〉〈χn | Rσ(Ω) |χj 〉 (C.20)

=∑

χn

〈χi | Oσ |χn 〉D1/2χnχj

(Ω) . (C.21)

Appropriate modification need to be made to the formulae used to calculate expectation values

with rotated wave functions. For example, the elements of the overlap matrix of the Jπ projected

wave function has the form

Bij = exp[

s∗i · sj

]

δτiτjD1/2

χiχj(Ω) (C.22)

when both the space and spin components of the wave function are rotated.

C.2 Nuclear Ground State

The matrix elements for the variational energy using the unprojected AMD wave function are

explained in Appendix B. Therefore, in this section only the modifications introduced by the

rotations will be addressed. The expectation values of the nuclear Hamiltonian H is given by

⟨

ΨJπMK(S) |H | ΨJπ

MK(S)⟩

=

∫

DJ∗MK(Ω)

⟨

Ψπ(S)∣

∣

[

T + V]∣

∣ Ψπ(S)⟩

dΩ (C.23)

where DJMK(Ω) are the Wigner D-function. The nuclear potential considered is this thesis has

the form

V (rij) =∑

ν

uν(rij)Oνij (C.24)

where uν(rij) are radial form factors and Oνij spin-isospin operators. Therefore (C.23) is evalu-

ated for the kinetic and potential energy operators as

⟨

ΨAMD(S)∣

∣T∣

∣ΨAMD(S)⟩

= N∑

ik

⟨

ψk | ti | ψi

⟩

B−1

ik (C.25)

⟨

ΨAMD(S)∣

∣V∣

∣ΨAMD(S)⟩

= N∑

ijkl,ν

Uνij Pν

ij

[

B−1

il B−1

jk −B−1

ik B−1

jl

]

(C.26)

120


where N =⟨

ΨAMD(S) |ΨAMD(S)⟩

and

Uνij =

⟨

φk φl |uν(rij) | φi φj

⟩

(C.27)

Pνij =

∑

χnχm

⟨

χkξk χlξl∣

∣Oνij

∣

∣ χnξi χmξj⟩

D1/2χnχi

(Ω)D1/2χmχj

(Ω) (C.28)

are used. The radial integral is evaluated as

Uνij = BkiBlj exp

(

−η22

4

) ∞∑

m=0

(

η22

4

)m

Iα(m) (C.29)

where η2 = s∗l − sj + s∗

k − si and Iα(m) is explained in Appendix B. The series in equation

(C.29) still converges with the rotated variational parameters.

C.3 Electromagnetic Form Factors

The nuclear electromagnetic operators are constructed as superposition of nucleons electromag-

netic operators. The charge form factor is evaluated as in equation (C.23) to obtain

F (q) =

∫

DJ∗MK(Ω)N

∑

ij

⟨

ψj

∣

∣Gi(q) exp(

i q · ri

)∣

∣ ψi

⟩

B−1

ji dΩ (C.30)

where Gi(q) are the elastic electron-nucleon electromagnetic form factors. The expectation

value in equation (C.30) is evaluated as

⟨

ψi

∣

∣ exp(

iq · r)∣

∣ψj

⟩

= N Bij

∫

exp

[

i q · r − 2α

(

r − η

2√α

)2]

dr (C.31)

= N Bij

∫

exp

[

2√αηR · r + i qc · r − 2α

(

r2 +η2

4α

)]

dr (C.32)

where N is the normalisation constant and qc = q + 2√αηI with ηR and ηI as the real and

imaginary components of η. To evaluate the integral in equation (C.32) the expansions [151]

eiu·r = 4 π∞∑

ℓ=0

∑

m

iℓjℓ(u r) Y∗ℓm(u) Yℓm(r) (C.33)

121


and

ev·r = 4 π∞∑

ℓ=0

∑

m

(−i)ℓjℓ(i v r) Y∗ℓm(r) Yℓm(v) (C.34)

where Yℓm(v) are the spherical harmonics and the transformations,

iℓjℓ(z) =

√

π

2 zJℓ+1/2(z) and (−i)ℓjℓ(i z) =

√

π

2 zIℓ+1/2(z) , (C.35)

are required where Jµ(z), Iµ(z) and jµ(z) are the Bessel, modified Bessel and spherical Bessel

functions, respectively. Using these expansions the equation (C.32) can be written in the form

⟨

ψi

∣

∣ exp(

iq · r)∣

∣ψj

⟩

=8 π3 N 2

√

2√α ηR qc

Bij e−η2/2∑

ℓm

Y ∗ℓm(qc) Yℓm(ηR)

×∫

e−2 α r2

Iℓ+1/2(2√α ηR r) Jℓ+1/2(qc r) r dr (C.36)

= Bij exp

[

η2I

2− q2

c

8α

]

∑

ℓm

√2 ℓ+ 1 jℓ

(

qc ηR

2√α

)

Yℓm(ηR) (C.37)

where qc = | qc |, ηR = |ηR | and use was made of properties of the Bessel functions [151]. The

quantisation axis is chosen to be along qc. Note that equation (C.31) can also be cast in the

form

⟨

ψi

∣

∣ exp(

iq · r)∣

∣ψj

⟩

= N Bij

∫

exp

[

−2α

(

r′ − i ηI

2√α

)2

+ i q · r′ +i q · ηR

2√α

]

dr′

= Bij exp

[

η2I

2− q2

c

8α

]

∑

ℓm

√2 ℓ+ 1 jℓ

(

q ηR

2√α

)

Yℓm(ηR) .(C.38)

where r′ = r − ηR/2√α. The calculation of magnetic dipole form factor follows the same

argument.

122


C.4 Nuclear Fragmentation Amplitude

Using the wave functions Ψi and Ψf for the initial and final states of the nucleus, the transition

amplitude for the electron-induced nucleon removal process is given by

T (q) = 〈Ψf | O(q) | Ψi 〉 =⟨

ΨJfπf

A−1 ϕAG(B)∣

∣

∣O(q)

∣

∣

∣ΨJiπi

A

⟩

(C.39)

where O(q) is an electromagnetic transition operator and B a set of components of the position

vectors of all the nucleons transverse to the momentum of the ejected nucleon. The Glauber

scattering operator can be expanded in combination with the electromagnetic transition oper-

ator into a series of interaction operators

G(B)O(q) = G(0)(B)O(1)(q) +G(1)(B)O(2)(q) +G(2)(B)O(3)(q) + · · · (C.40)

= T (0)(q) + T (1)(q) + T (2)(q) + T (3)(q) + · · · (C.41)

where the order of the transition operator O(n)(q) is determined mainly by the order of the

Glauber operator G(n)(B), and T (n)(q) = G(n)(B)O(n+1)(q) are the interaction operators. The

first three interaction operators are explicitly given by

T (0)(q) = GA(q) eiq·rA (C.42)

T (1)(q) =σ( 1 − i ǫ )

4 π βexp

(

−r2⊥

2 β

)

GA(q) eiq·rA (C.43)

T (2)(q) =

[

σ( 1 − i ǫ )

4 π β

]2

exp[

− 1

4 β

(

r2⊥ + 4 ρ2

⊥

)]

GA(q) eiq·rA (C.44)

where σ, ǫ and β are parameters obtained from elastic proton-nucleon scattering experimental

data. The vectors r and ρ are the Jacobi position vectors for the two and three interacting

nucleons with r⊥,ρ⊥ being elements of the set B. In equation (C.43) rA must be expressed

in terms of r whereas in equation (C.44) it must be expressed in terms of r and ρ. The n-th

123


order transition amplitude is given by

T (n)(q) =⟨

ΨJf πf

MfϕA

∣

∣

∣T (n)(q)

∣

∣

∣ΨJiπi

Mi

⟩

(C.45)

=〈 jpmp Jf Mf | JiMi〉√

ANA NA−1

⟨

ΨJfπf

MfϕA

∣

∣

∣T (n)(q)

∣

∣

∣A[

ψjpπpmp

⊗ ΨJoπo

Mo,(A−1)p

]⟩

(C.46)

where 〈 jpmp Jf Mf | JiMi〉 are Clebsch-Gordan coefficients, NX the normalisation constant for

the wave function of the X-particle system, (A− 1)p the combination of (A− 1) nucleons, with

total angular momentum Jo, that excludes the nucleon p. Using the properties of the angular

momentum projection operator the partial transition operators can be written in the form

T (n)p (q) =

∑

Kfkp

〈 jpmp Jf Mf | JiMi〉√ANA NA−1

∑

mp

∫

Djp∗mpkp

(Ω)DJf∗Mf Kf

(Ω)Nfi Tn(q) dΩ (C.47)

where

Tn(q) =⟨

Ψπf

AMD(S)ϕA(sA)∣

∣

∣T (n)(q)

∣

∣

∣ψπp(sA) Ψ

πf

AMD(S)⟩

(C.48)

and Nfi = 〈Ψ(S3H)|Ψ(S4He)〉. The form factor Tn(q) is formed by combinations of the Fourier

transforms of components of the wave function.

Only the first three of the AMD transition form factors in equation (C.47) are used. These

form factors are given by

T1(q) =⟨

ϕA

∣

∣T (0)(q)∣

∣ ψAν

⟩

(C.49)

T2(q) =∑

ki

⟨

ϕA ψk

∣

∣ θ(

zi − zA

)

T (1)(q)∣

∣ ψAν ψi

⟩

B−1ik (C.50)

T3(q) =∑

ijkl

⟨

ϕA ψkψl

∣

∣ θ(

zj − zA

)

θ(

zi − zA

)

T (2)(q)∣

∣ ψAν ψi ψj

⟩

×[

B−1il B−1

jk − B−1ik B−1

jl

]

(C.51)

where θ(

zi − zA

)

is the Heaviside step-function. The expectation values in equations (C.49),

(C.50) and (C.51) have the form

T (n)(q) =1

(2 π)3L(n)(q′,SR)K(n)(q′,SI) (C.52)

124


where q′ = q − pf and SR (SI) the real (imaginary) component of S. In equation (C.52)

L(0)(q′,SR) = exp

[

i q′

√α·(

sA

)

]

(C.53)

L(1)(q′,SR) = exp

[

i q′

3√α·(

Λ1 + ω‖ +4αβ ω⊥

4αβ + 3

)

− 1

2

ω2⊥

4αβ + 3

]

(C.54)

L(2)(q′,SR) = exp

[

i q′

5√α·(

Λ2 + κ‖ +4αβ κ⊥

4αβ + 5

)

− 1

4

κ2⊥

4αβ + 5− 1

4

(

η1 − η2

)2

⊥

4αβ + 1

]

(C.55)

where Λ1 = η1 + sA, Λ2 = η1 + η2 + sA and

K(0)(q′,SI) =

(

2 π

α

)3/4

exp

[

−( q′ + 2√α sA )

2

4α

]

(C.56)

K(1)(q′,SI) = C1 exp

[

−(

q′ + 2√αΛ1

)2

12α+

Λ21

3+

ω2⊥

6

]

× exp

−(

q′ +√αω

)2

‖

6α− 2 β

3

(

q′ +√αω

)2

⊥

4αβ + 3

Bki (C.57)

K(2)(q′,SI) = C2 exp

[

−(

q′ + 2√αΛ2

)2

20α+

Λ22

5+

κ2⊥

20− 1

4

(

η1 − η2

)2

⊥

4αβ + 1

]

× exp

−(

2 q′ +√ακ

)2

‖

20α− β

5

(

2 q′ +√ακ

)2

⊥

4αβ + 5

BkiBlj (C.58)

with ω = 2 sA − η1, κ = 4 sA − η1 − η2. The labels A‖ and A⊥ represent the component of

vector A that are, respectively, parallel and perpendicular to the momentum transfer q which

is directed along the z-axis. The coefficients C1 and C2 are determined by the nucleon-nucleon

scattering parameters as

C1 =√

2

(

2α

π

)1/4σ( 1 − i ǫ )

4αβ + 3and C2 =

1√2

(

2α

π

)5/4σ( 1 − i ǫ )

4αβ + 1

σ( 1 − i ǫ )

4αβ + 5.

(C.59)

The values of the parameters σ, ǫ and β are adopted from reference [43]. The factors involving

Λi are contributions from the center of mass of the interacting nucleons.

The dependence of the response functions of the directions of the ejected proton is removed by

125


integrating the factors L(n)(q′,SR) and K(n)(q′,SR) with respect to the angles of p. In the case

of L(n)(q′,SR) the partial wave expansion

ei(q−p)·u = 16 π2∑

ℓ1m1

∑

ℓ2m2

(i)ℓ1−ℓ2 jℓ1(q u) jℓ2(p u) Y∗ℓ1m1

(q) Yℓ2m2(p) Y ∗

ℓ1m1(u) Yℓ2m2

(u) (C.60)

is used. The relevant factors in the functions K(n)(q′,SR) are those that have the general form

Q(q,p) = Ci exp

[

−( q + A − p )2

V α

]

. (C.61)

where Ci and V are constants. Integrating out the angular directions of p leads to

Q(q, p) = Ci exp

[

−( q + A − p )2

V α

]

(C.62)

= 4 π Ci j0

(

i 2 |q + A| pV α

)

exp

[

−|q + A|2 + p2

V α

]

(C.63)

≈ Ci π V α|q + A| p exp

[

−(|q + A| − p )2

V α

]

(C.64)

where the final approximation is obtained because the argument of sinh(x) is x≫ 1. Using the

properties of the spherical harmonics, ℓ2 = 0 and since q is directed along the z-axis, ℓ1 = 0.

126

Bibliography

[1] T. W. Donnelly, A. S. Raskin : Ann. Phys. (N.Y.) 169, 247 (1986).

[2] J. Golak, R. Skibinski, H. Witala, W. Glockle, A. Nogga, H. Kamada :

Phys. Rep. C 415, 89 (2005).

[3] K. Amos, P. J. Dortmans, H. V. von Geramb, S. Karataglidis :

Adv. Nucl. Phys. 25, 275 (2000).

[4] S. Karataglidis, K. Amos, B. A. Brown and P. K. Deb : Phys. Rev. C 65, 044306 (2002).

[5] T. de Forest Jr. : Ann. Phys. (N.Y.) 45 365 (1967).

[6] T. W. Donnelly and I. Sick : Rev. Mod. Phys. 56, 461 (1984).

[7] W. N. Cottingham, M. Lacombe, B. Loiseau, J. M. Richard, R. Vinh Mau :

Phys. Rev. D 8, 800 (1973).

[8] M. M. Nagels, T. A. Rijken, J. J. de Swart : Phys. Rev. D 17, 768 (1978).

[9] R. Machleidt, K. Holinde, Ch Elster : Phys. Rep. 149, 1 (1987).

[10] I. E. Lagaris, V. R. Pandharipande : Nucl. Phys. A 359, 331 (1981).

[11] R. B. Wiringa, R. A. Smith, T. L. Ainsworth : Phys. Rev. C 29, 1207 (1984).

[12] B. S. Pudliner, V. R. Pandharipande, J. Carlson, Steven C. Pieper, and R. B. Wiringa :

Phys. Rev. C 56, 1720 (1997).

[13] L. S. Schroeder, S. A. Chessin, J. V. Geaga, J. Y. Grossiord, J. W. Harris, D. L. Hendrie,

R. Treuhaft, and K. Van Bibber : Phys. Rev. Lett. 43, 1787 (1979).

127

BIBLIOGRAPHY

[14] M. K. Jones et al. : Phys. Rev. Lett. 84, 1398 (2000).

[15] P. E. Ulmer et al. : Phys. Rev. Lett. 89, 062301 (2002).

[16] M. M. Sargsian et al. : J. Phys. G 29, Rl (2003).

[17] I. A. Qattan et al. : Phys. Rev. Lett. 94, 142301-1 (2005).

[18] K. S. Egiyan et al. : Phys. Rev. Lett. 96, 082501 (2006).

[19] K. S. Egiyan et al. : Phys. Rev. Lett. 98, 262502 (2007).

[20] R. Subedi et al. : Science 320, 1476 (2008).

[21] L. L. Frankfurt and M. I. Strikman : Phys. Rept. 76, 215 (1981).

[22] L. L. Frankfurt, M. I. Strikman, D. B. Day and M. Sargsian :

Phys. Rev. C 48, 2451 (1993).

[23] E. Piasetzky, M. Sargsian, L. Frankfurt, M. Strikman and J. W. Watson :

Phys. Rev. Lett. 97, 162504 (2006).

[24] R. Schiavilla, R. B. Wiringa, S. C. Pieper and J. Carlson :

Phys. Rev. Lett. 98, 132501 (2007).

[25] M. Alvioli, C. Ciofi degli Atti and H. Morita : Phys. Rev. Lett. 100, 162503 (2008).

[26] J. Carlson, R. Schiavilla : Rev. Mod. Phys, 70, 743 (1998).

[27] O. Benhar, D. Day, I. Sick : Rev. Mod. Phys. 80, 189 (2008).

[28] H. Kamada, A. Nogga, W. Glockle, E. Hiyama, M. Kamimura, K. Varga, Y. Suzuki, M.

Viviani, A. Kievsky, S. Rosati, J. Carlson, S. C. Pieper, R. B. Wiringa, P. Navratil, B. R.

Barrett, N. Barnea, W. Leidemann, G. Orlandini : Phys. Rev.C 64, 044001 (2001).

[29] S. P. Merkuriev, C. Gignoux, A. Laverne : Ann. Phys. 99, 30 (1976).

[30] A. Kievsky, S. Rosati, M. Viviani, L. E. Marcucci, L. Girlanda :

J. Phys. G: Nucl. Part. Phys. : 35, 063101 (2008).

128

BIBLIOGRAPHY

[31] V. D. Efros, W. Leidemann, G. Orlandini, N. Barnea :

J. Phys. G: Nucl. Part. Phys. 34, R459 (2007).

[32] V. D. Efros, W. Leidemann, G. Orlandini : Phys. Lett. B 338, 130 (1994).

[33] A. Lapiana, W. Leidemann : Nucl. Phys. A 677, 423 (2000).

[34] S. Boffi, M. Bouten, C. Ciofi degli Atti, J. Sawicki : Nucl. Phys. A 120, 135 (1968).

[35] J. Aichelin, H. Stocker : Phys. Lett. B 176, 14 (1986).

[36] J. Aichelin : Phys. Rep. 202, 233 (1991).

[37] R. Sabu, K. H. Bhatt and D. P Ahalpara : J. Phys. G: Nucl. Part. Phys. 16, 733 (1990).

[38] T. Neff, H. Feldmeier and R. Roth Nucl. Phys. A 752, 321c (2005).

[39] Y. Kanada-En’yo, M. Kimura, and H. Horiuchi : C. R. Physique 4, 497 (2003).

[40] A. Ono and H. Horiuchi : Prog. Part. Nucl. Phys. 53, 501 (2004).

[41] R. J. Glauber : Phys. Rev. 100, 242 (1955).

[42] Mitsuaki OBU : Prog. Theor. Phys. 48, 1934 (1972).

[43] H. Morita, C. Ciofi degli Atti, D. Treleani : Phys. Rev. C 60, 034603-1 (1999).

[44] L. L. Frankfurt, M. M. Sargsian, M. I. Strikman : Phys. Rev. C 56, 1124 (1997).

[45] M. M. Sargsian : Int. J. Mod. Phys. E 10, 405 (2001).

[46] T. W. Chen : Phys. Rev. C 30, 585 (1984).

[47] S. Drozdz, J. Okolowcz and M. Ploszajczak : Phys. Lett. B 109, 145 (1982).

[48] E. Caurier, B. Grammaticos and T. Sami : Phys. Lett. B 109, 150 (1982).

[49] H. Feldmeier : Nucl. Phys.A 515, 147 (1990).

[50] H. Horiuchi : Nucl. Phys. A 522, 257c (1991).

[51] A. Ono, H. Horiuchi, T. Maruyama and A. Ohnishi : Prog. Theor. Phys. 87, 1185 (1992).

129

BIBLIOGRAPHY

[52] E. I. Tanaka, A. Ono, H. Horiuchi, T. Maruyama, A. Engel : Phys. Rev. C 52, 316 (1995).

[53] A. Engel, E. I. Tanaka, T. Maruyama, A. Ono, H. Horiuchi :

Phys. Rev. C 52, 3231 (1995).

[54] Y. Kanada-En’yo and H. Horiuchi : Phys. Rev. C 55, 628 (1995).

[55] A. Dote, H. Horiuchi : Prog. Theor. Phys. 103, 261 (2000).

[56] A. Dote, Y. Kanada-En’yo, H. Horiuchi, Y. Akaishi, K. Ikeda :

Prog. Theor. Phys. 115, 1069 (2006).

[57] M. Kimura : Phys. Rev. C 69, 044319 (2004).

[58] A. Dote, H. Horiuchi, and Y. Kanada-En’yo : Phys. Rev. C 56, 1844 (1997).

[59] T. Togashi, K. Kato : Prog. Theor. Phys. 117, 189 (2007).

[60] T. Togashi, T. Murakami, K. Kato : Prog. Theor. Phys. 121, 299 (2009).

[61] T. Watanabe and S. Oryu : Prog. Theor. Phys. 116, 429 (2006).

[62] T. Watanabe, M. Oosawa, K. Saito, S. Oryu :

J. Phys. G : Nucl. Part. Phys. 36, 015001 (2009).

[63] D. Brink : Proceedings of the International School of Physics “Enroco Fermi,”

Course 36, Edited by C. Bloch, (Academic, New York, 1965) p247.

[64] A. Ono and H. Horiuchi : Phys. Rev. C 53, 2958 (1996).

[65] T. Watanabe and S. Oryu : Prog. Theor. Phys. 116, 429 (2006).

[66] E. Epelbaum : Phys. Lett. B 639, 456 (2006).

[67] A. Deltuva, A. C. Fonseca, P. U. Sauerb : Mod. Phys. Lett. A 24, 785 (2009).

[68] M. Lacombe, B. Loiseau, J. M. Richard, R. Vinh Mau, J. Cote, P. Pires, R. de Tourreil :

Phys. Rev. C 21, 861 (1979).

130

BIBLIOGRAPHY

[69] V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, J. J. de Swart :

Phys. Rev. C 49, 2950 (1994).

[70] R. Machleidt : Phys. Rev. C 63, 024001 (2001).

[71] I. E. Lagaris, V. R. Pandharipande : Nucl. Phys. A 359, 349 (1981).

[72] R. B. Wiringa, V. G. J. Stoks, R. Schiavilla : Phys. Rev. C 51, 38 (1995).

[73] R. E. Peierls and J. Yoccoz : Proc. Phys. Soc. A 70 , 381 (1957).

[74] G. Bohannon, L. Zamick, E. Moya de Guerra : Nucl. Phys. A 334, 278 (1980).

[75] C. K. Lin, L. Zamick : Phys. Rev. C 23, 2338 (1981).

[76] P. Kramer, M. Saraceno : Geometry of the Time-Dependent Variational Principle in Quan-

tum Mechanics, Lecture Notes in Physics 140, (Springer-Verlag, Berlin, Germany) (1981).

[77] John A. Wheeler : Phys. Rev. 52, 1107 (1937).

[78] I. Reichstein, D. R. Thompson and Y. C. Tang : Phys. Rev. C 3, 2139 (1971).

[79] O. D. Sharma and B. B. Srivastava : J. Phys. : Nucl. Phys. 6, 209 (1980).

[80] S. J. Wallace : Adv. Nucl. Phys. 12, 135 (1981).

[81] R. H. Bassel and C. Wilkin : Phys. Rev. 174, 1179 (1968).

[82] V. S. Barashenkov and V. D. Toneev : Sov. Phys. Uspekhi 13, 182 (1970); Usp. Fiz. Nauk

100 425 (1970).

[83] A. J. F. Siegert : Phys. Rev. 52, 787 (1937).

[84] F. Partovi : Ann. Phys. (N.Y.) 27 114 (1964).

[85] J. L. Friar, S. Fallieros : Phys. Rev. C 29, 1645 (1984).

[86] R. Brizzolara and M. M. Giannini : Few-Body Systems 5, 1 (1988).

[87] S. Karataglidis, P. Halse, and K. Amos : Phys. Rev. C 51, 2494 (1995).

131

BIBLIOGRAPHY

[88] T. de Forest Jr. and J. D. Walecka : Adv. Phys. 15 1 (1966).

[89] R. R. Roy and B. P. Nigam : Nuclear Physics, Theory and Experiment

(Wiley, New York) (1967).

[90] J. M. Eisenberg and W. Greiner : Nuclear Theory 2, Excitation Mechanisms of the Nucleus,

Third Revised Edition. (Elsevier, North-Holland) (1988).

[91] D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii : Quantum Theory of Angular

Momentum, (World Scientific, Singapore) (1988).

[92] A. S. Raskin, T. W. Donnelly : Ann. Phys. (N.Y.) 191, 78 (1989).

[93] C. Ciofi Degli Atti : Prog. Part. Nucl. Phys. 3, 163 (1980).

[94] R. S. Willey : Nucl. Phys. 40, 529 (1963).

[95] G. N. Watson : Theory of Bessel Functions, 2nd Edition

(Cambrodge University Press, London) (1966).

[96] M. M. Giannini, G. Ricco : Nuovo Cim. 8, 1 (1985).

[97] K. M. McVoy and L. Van Hove : Phys. Rev. 125, 1034 (1962).

[98] S. Jeschonnek, T. W. Donnelly : Phys. Rev. C 57, 2438 (1998).

[99] J. E. Amaro, J. A. Caballero, T. W. Donnelly, A. M. Lallena, E. Moya de Guerra,

J. M. Udias : Nucl. Phys. A 602, 263 (1996).

[100] R. Schiavilla, O. Benhar, A. Kievsky, L. E. Marcucci, M. Viviani :

Phys. Rev. C 72, 064003-1 (2005).

[101] R. Hofstadter : Electron Scattering and Nucleon and Nuclear Structure, A Collection of

Reprints with an Introduction, (W. A. Benjamin, Inc., New York) (1963).

[102] M. A. Belushkin, H. W. Hammer and U. G. Meissner : Phys. Rev. C 75, 035202 (2007).

[103] J. Friedrich and Th. Walcher : Eur. Phys. J. A 17, 607 (2003).

132

BIBLIOGRAPHY

[104] H. Uberall : Electron Scattering from Complex Nuclei: Part A, Academic - New York

(1971).

[105] A. N. Antonov, D. N. Kadrev, M. K. Gaidarov, E. Moya de Guerra, P. Sarringuren, J.

M. Udias, V. K. Lukyanov, E. V. Zemlyanaya, and G. Z. Krumova :

Phys. Rev. C 72, 044307 (2005).

[106] D. O. Riska : Phys. Rep. 181, 207 (1989).

[107] R. Schiavilla, V. R. Pandaripande, D. O. Riska : Phys. Rev. C 40, 2294 (1989).

[108] M. Viviani, R. Schiavilla, A. Kievsky : Phys. Rev. C 54, 534 (1996).

[109] T. Togashi, T. Murakami and K. Kato : J. Phys.: Conf. Series 111 012026 (2008).

[110] R. B. Wiringa and S. C. Pieper : Phys. Rev. Lett. 89, 182501-1 (2002).

[111] G. Audi, O. Bersillon, J. Blachot and A.H. Wapstra : Nucl. Phys. A 729, 3 (2003).

[112] Y. Suzuki, W. Horiuchi, M. Orabi, K. Arai : Few-Body Systems. 42, 33 (2008).

[113] Samuel S. M. Wong : Introductory Nuclear Physics, 2nd Edition

(John Wiley and Sons, Inc. New York) (1998).

[114] A. C. Phillips : Rep. Prog. Phys. 40, 905 (1977).

[115] J. E. Amaro, J. A. Caballero, T. W. Donnelly, E. Moya de Guerra :

Nucl. Phys. A 611, 163 (1996).

[116] G. F. Chew, G. C. Wick : Phys. Rev. 85, 636 (1952).

[117] E. Hadjimichael, B. Goulard, R. Bornais, Phys. Rev. C 27, 831 (1983).

[118] R. De Tourreil, B. Rouben, D. W. L. Sprung : Nucl. Phys. A 242, 445 (1975)

[119] W. M. Kloet, J. A. Tjon : Phys. Lett. B 49, 419 (1974).

[120] R. V. Reid : Annals of Physics. 50, 411 (1960).

133

BIBLIOGRAPHY

[121] R. A. Malfliet and J. A. Tjon : Nucl. Phys. A 127, 161 (1969);

Annals of Physics. 61, 425 (1970).

[122] R. F. Frosch, J. S. McCarthy, R. E. Rand, and M. R. Yearian : Phys. Rev. 160, 874

(1967).

[123] J. S. McCarthy, I. Sick, R. R. Whitney : Phys. Rev. C 15, 1396 (1977).

[124] A. Amroun, V. Breton, J. -M. Cavedon, B. Frois, D. Goutte, F. P. Juster, Ph. Leconte,

J. Martino, Y. Mizuno, X. -H. Phan, S. K. Platchkov, I. Sick, S. Williamson :

Nucl. Phys. A 579, 596 (1994).

[125] I. Sick : Nucl. Phys. A 218, 509 (1977).

[126] M. J. Musolf, R. Schiavilla, T. W. Donnelly : Phys. Rev. C50, 2173 (1994).

[127] J. Carlson, V. R. Pandharipande, R. B. Wiringa : Nucl. Phys. A 401, 59 (1983).

[128] Ch. Hajduk, P. U. Sauer and W. Strueve : Nucl. Phys. A 405, 581 (1983).

[129] T. Katayama, Y. Akaishi, H. Tanaka : Prog. Theor. Phys. 67, 236 (1982).

[130] L. I. Schiff : Phys. Rev. 103, 443 (1956).

[131] M. Traini, S. Turck-Chieze, and A. Zghiche : Phys. Rev. C 38, 2799 (1988).

[132] Yanhe Jin, H. P. Blok, and L. Lapikas : Phys. Rev. C 48, R964 (1993).

[133] K. S. Kim, L. E.Wright, Yanhe Jin, and D.W. Kosik : Phys. Rev. C 54, 2515 (1996).

[134] K. S. Kim and L. E. Wright : Phys. Rev. C 56, 302 (1997).



[137] R. A. Brandenburg, Y. E. Kim, A. Tubis : Phys. Rev. Lett. 32, 1325 (1974).

[138] R. Schiavilla : Nucl. Phys. A 499, 301 (1989).

[139] G. Ripka : Adv. Nucl. Phys. 1, 183 (1968).

134

BIBLIOGRAPHY

[140] T. de Forest Jr : Ann. Phys. 45, 365 (1967).

[141] J. J. Kelly : Adv. Nucl. Phys. 23, 75 (1996).

[142] T. Muto and T. Sebe : Prog. Theor. Phys. 18, 621 (1957).

[143] K. F. von Reden, et al : Phys. Rev. C 41, 1084 (1990).

[144] C. Ciofi degli Atti : Nucl. Phys. A 463, 127c (1987).

[145] V. D. Efros : Phys. Atom. Nucl. 71, 1239 (2008).

[146] S. Bacca : Euro. Phys. J. 71, 1239 (2008).

[147] L. Howell : PhD Thesis, University of South Africa, unpublished (1997).

[148] J. F. J. van den Brand : PhD Thesis, Universiteit van Amsterdam, unpublished (1988).

[149] S. Frullani and J. Mougey : Adv. Nucl. Phys. 14, 1 (1984).

[150] D. W. Jepsen and J. O. Hirschfelder : Proc. Natl. Acad. Sci. U.S.A. 45, 249 (1959).

[151] J. Zhang, A. C. Merchant, W. D. M. Rae : Nucl. Phys. A 613, 14 (1997).

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