Electron and proton scattering oﬀ nuclei with neutron and ... the frameworks of electron and...

UNIVERSITA DEGLI STUDI DI PAVIA

Dottorato di Ricerca in Fisica - XXVIII Ciclo

Electron and proton scattering off nuclei with neutron

and proton excess

Matteo Vorabbi

Tesi per il conseguimento del titolo

Contents

1 Introduction 1

2 Relativistic Mean-Field Models 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 The Density-Dependent Model . . . . . . . . . . . . . . . . . . . 9

2.2.1 The Lagrangian Density . . . . . . . . . . . . . . . . . . 9

2.2.2 Stationary Solutions of the Equations of Motion . . . . . 12

2.2.3 Rotationally Invariant Systems . . . . . . . . . . . . . . 14

2.2.4 Covariant Density Functional Theory . . . . . . . . . . . 15

2.2.5 The Density-Dependence of the Couplings . . . . . . . . 17

2.3 Pairing Correlations . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Non-Relativistic Theory . . . . . . . . . . . . . . . . . . 17

2.3.2 Covariant Density Functional Theory with Pairing . . . . 20

2.4 Results for Neutron and Proton Densities . . . . . . . . . . . . . 22

3 Electron Scattering 29

3.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 The Nuclear Response . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Elastic and Quasi-Elastic Scattering on Exotic Nuclei . . . . . . 33

3.4 Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.1 Theoretical Framework . . . . . . . . . . . . . . . . . . . 36

3.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5 Parity-Violating Elastic Scattering . . . . . . . . . . . . . . . . 44

3.5.1 Theoretical Framework . . . . . . . . . . . . . . . . . . . 45

3.5.2 Results for Oxygen and Calcium Isotopic Chains . . . . . 46

3.5.3 Neutron Density of 208Pb . . . . . . . . . . . . . . . . . . 49

3.6 Inclusive Quasi-Elastic Scattering . . . . . . . . . . . . . . . . . 54

3.6.1 Relativistic Green’s Function Model . . . . . . . . . . . . 55

3.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 65

iii

CONTENTS

4 Theoretical Foundations for Proton Scattering 694.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 The First-Order Optical Potential . . . . . . . . . . . . . . . . . 73

4.2.1 The Spectator Expansion . . . . . . . . . . . . . . . . . 734.2.2 The Impulse Approximation . . . . . . . . . . . . . . . . 754.2.3 The KMT Multiple Scattering Theory . . . . . . . . . . 774.2.4 Fixed Beam Energy Approximation . . . . . . . . . . . . 814.2.5 The Optimum Factorization Approximation . . . . . . . 814.2.6 Kinematical Variables . . . . . . . . . . . . . . . . . . . 83

4.3 The Nucleon-Nucleon Potential . . . . . . . . . . . . . . . . . . 854.3.1 The CD-Bonn Potential . . . . . . . . . . . . . . . . . . 854.3.2 The Chiral Potential . . . . . . . . . . . . . . . . . . . . 87

4.4 The Nucleon-Nucleon Transition Matrix . . . . . . . . . . . . . 944.5 Theoretical Results for Nucleon-Nucleon Amplitudes . . . . . . 97

5 Relativistic Kinematics and the Scattering Observables 1035.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.1.1 Relativistic Kinematics . . . . . . . . . . . . . . . . . . . 1045.1.2 The Møller Factor . . . . . . . . . . . . . . . . . . . . . 105

5.2 The Transition Amplitude in Partial Wave Representation . . . 1065.3 The Scattering Observables . . . . . . . . . . . . . . . . . . . . 1075.4 Treatment of the Coulomb Potential . . . . . . . . . . . . . . . 1115.5 Results for Scattering Observables . . . . . . . . . . . . . . . . . 1155.6 Conclusions and Future Perspectives . . . . . . . . . . . . . . . 119

6 Summary and Conclusions 139

A The Singlet-Triplet Scattering Amplitudes 143

B Numerical Solution of the Lippmann-Schwinger Equation 145B.1 The Matrix-Inversion Method . . . . . . . . . . . . . . . . . . . 145

B.1.1 The Nucleon-Nucleon Scattering . . . . . . . . . . . . . . 148B.2 The relativistic Lippmann-Schwinger equation . . . . . . . . . . 150

C Angular Momentum Projections 153

D Abbreviations 157

iv

Chapter 1Introduction

The study of atomic nuclei has a long history and represents one of the mostexciting and difficult many-body problems, described in terms of nucleons, i.e.,neutrons and protons. Because of the impossibility to derive a nucleon-nucleon(NN) interaction from fundamental principles, i.e. from Quantum Chromo-dynamics (QCD), neutrons and protons are taken as the relevant degrees offreedom for present nuclear theories. The properties of nuclei are thus char-acterized by the number of nucleons and by the ratio between neutron andproton numbers.

In nature we have found only 263 stable nuclei surrounded by radioactiveones. Some of these unstable nuclei can be found on Earth, some are manufac-tured and several thousand nuclei are the yet unexplored exotic species. A largenumber of these radioactive nuclei are characterised by beta decay, while, forheavier nuclei, processes like emission of alpha particles or spontaneous fissiondominate due to the important role played by the electromagnetic interaction.In the nuclear chart, moving away from stable nuclei by adding either protonsor neutrons, we reach the boundaries for nuclear particle stability, called driplines. The nuclei beyond the drip lines are unbound to nucleon emission, thatis, for these systems the strong interaction is unable to bind all the A nucleonsin one nucleus.

Nuclear life far from stability is different from that around the stability line.The promised access to completely new combinations of proton and neutronnumbers offers prospects for new structural phenomena.

Since neutrons do not carry an electric charge and do not repel each other,many neutrons can be added to nuclei starting from the valley of stability. Asa result, the nuclear landscape separating the neutron drip line and the valleyof stability is large and difficult to probe experimentally. In this region of thenuclear chart we can find new unexpected phenomena which are important notonly for nuclear physics but also for astrophysics and cosmology. The structureof the nuclei is expected to change significantly as the limit of nuclear stabilityis approached in neutron excess. Due to the systematic variation in the spatialdistribution of nucleonic densities and the increased importance of the pairing

1

1. Introduction

field, the average nucleonic potential is modified when approaching the neutrondrip line. With the increase of the neutron number, the single-particle neutronpotential becomes very shallow and for drip-line nuclei this results in a newshell structure.

On the proton-rich side of the valley of stability, physics is different thanin nuclei with a large neutron excess. The Coulomb barrier tends to localizethe proton density in the nuclear interior and thus nuclei beyond the protondrip line are quasi-bound with respect to proton emission. In spite of this, theeffects associated with the weak binding are also present in proton drip-linenuclei. Moreover, N = Z proton-rich nuclei are very important for the studyof nuclear pairing. Due to the large spatial overlaps between neutron andproton single-particle wave functions, these systems are expected to exhibitunique manifestations of proton-neutron pairing. Nuclei beyond the protondrip line are ground-state proton emitters. Proton radioactivity is an excellentexample of the elementary three-dimensional quantum mechanical tunnelingand its investigation will open up a wealth of exciting new physics.

Compared with most physical systems, nuclei are difficult to study andthe reason lies in the strength of the nuclear interaction, which results in avery tightly bound system. Accordingly, we cannot proceed as easily as inatomic physics, where the wavefunctions of individual electrons are studied bybombarding the atom with photons or electrons or X-rays and then measuringthe energies, momenta and angular momenta of the ejected electrons. In thecase of nuclei, an equivalent investigation would require photons or particles ofhigh enough energy to overcome the nuclear binding, which is tipically 8 MeVper nucleon.

There are basically two ways to investigate nuclear properties, namely ra-dioactive decay and nuclear reactions. The former is an important source ofinformation: it often provides the first information on any newly producednucleus and it is the basis for precision tests of the Standard Model and otherthings. Thus it is extremely useful but unfortunately we have no control overradioactive decay. Decay rates can be altered under some circumstances but,in general, this is not a profitable way to proceed.

The nuclear response to an external probe is a powerful tool for investi-gating the structure of hadron systems such as atomic nuclei and their con-stituents. Electron- and nucleon-scattering reactions are the most importantones historically employed for such a purpose.

Electron-scattering reactions have provided the most complete and de-tailed information on nuclear and nucleon structure. Electrons predominantlyinteract with nuclei through the electromagnetic interaction, which is wellknown from Quantum Electrodynamics (QED) and is weak compared withthe strength of the interaction between hadrons. The scattering process istherefore adequately treated assuming the validity of the Born approximation,i.e., the one-photon exchange mechanism between the electron and the target.The virtual photon, like the real one, has a mean free path much larger than the

2

target dimensions, thus exploring the whole target volume. This contrasts withhadron probes, which are generally absorbed at the target surface. Moreover,the ability to vary independently the energy and momentum of the exchangedvirtual photon transferred to the nucleus makes it possible to map the nuclearresponse as a function of its excitation energy with a spatial resolution thatcan be adjusted to the scale of processes that need to be studied.

On the other hand, the scattering of neutrons and protons from nuclei hasan even longer history and represents a very important tool to investigate theinteraction of a nucleon with a many-nucleon system. In the case of elasticscattering the reaction may be cast in the form of an optical potential whichis used as input for the Schrodinger or Lippmann-Schwinger (LS) equationdepending on the coordinate or momentum space where we work. The solu-tion of these equations permits then to compute the elastic differential crosssection. The comparison of the calculated elastic scattering observables withavailable data is the first crucial test of an optical potential. After its validityhas been checked in comparison with elastic nucleon-nucleus (NA) scatteringdata, the optical potential can then be used for calculations on a wide varietyof inelastic processes and nuclear reactions, where it is a crucial and criticalinput. In particular, in electron scattering the optical potential is the basicingredient to describe final-state interactions (FSI) in quasi-elastic (QE) exclu-sive and inclusive electron-nucleus scattering. This makes nucleon scatteringvery important for many purposes and, in particular, a microscopic theoret-ical approach to this computational tool allows us to get more insight aboutnuclear structure.

Several decades of experimental and theoretical work on these reactionshave provided a wealth of information on the properties of stable nuclei. Theuse of electron and nucleon probes can be extended to exotic nuclei. Thedetailed study of the properties of nuclei far from the stability line and theevolution of nuclear properties with respect to the asymmetry between thenumber of neutrons and protons is one of the major topics of interest in modernnuclear physics. In the next years the advent of radioactive ion beam facilitieswill provide a large amount of data on unstable nuclei. These facilities willoffer unprecedented opportunities to study the structure of exotic unstablenuclei through electron and nucleon scattering. In particular, for electronscattering, kinematically complete experiments will become feasible for thefirst time, allowing a clean separation of different reaction channels as wellas a reduction of the unavoidable radiative background seen in conventionalexperiments. Therefore, even applications using stable isotope beams will beof interest.

In this work we present and discuss theoretical results obtained withinthe frameworks of electron and proton scattering off nuclei with neutron andproton excess. In particular, our aim is to study the evolution of nuclearproperties with the increase of neutron and proton number. For this reason, weperformed our calculations along isotopic and isotonic chains. The dissertation

3

1. Introduction

is organized as follows.

In Chapter 2 we discuss the theoretical framework of relativistic mean-field(RMF) models used to describe the ground-state properties of nuclei and weshow the results obtained for neutron and proton densities of the nuclei con-sidered in the next chapters. The study of the evolution of nuclear propertiesrequires a good knowledge of nuclear-matter distributions for neutrons andprotons separately. The ground-state densities reflect the basic properties ofeffective nuclear forces providing fundamental nuclear structure informationand represent the basic ingredient to compute electron- and proton-scatteringobservables.

In Chapter 3 we present and discuss the theoretical results obtained for elec-tron scattering. In particular, we consider elastic and QE reactions that allowus to investigate bulk and single-particle properties of nuclei. For these calcu-lations we consider both isotopic and isotonic chains. The chapter is dividedin three sections devoted to describe the theory and the results for elastic scat-tering, parity-violating elastic scattering, and inclusive QE scattering. Elasticelectron scattering makes it possible to measure with excellent precision onlycharge densities and therefore proton distributions. It is much more difficultto measure neutron distributions. Our present knowledge of neutron densitiescomes primarily from hadron-scattering experiments, the analysis of whichrequires always model-dependent assumptions about strong nuclear forces atlow energies. A model-independent probe of neutron densities is provided byparity-violating elastic electron scattering, where direct information on theneutron density can be obtained from the measurement of the parity-violatingasymmetry Apv parameter, which is defined as the difference between the crosssections for the scattering of right- and left-handed longitudinally polarizedelectrons. This quantity is related to the radius of the neutron distributionand provides a robust model-independent measurement of it. In QE scatteringthe response of the nucleus to the electron probe is dominated by the single-particle dynamics and by the process of one-nucleon knockout, where the probeinteracts with only one nucleon which is then ejected from the nucleus witha direct knockout mechanism. The emitted nucleon can be detected in coin-cidence with the scattered electron and suitable kinematics conditions can beenvisaged where we can assume that the residual nucleus is left in a discreteeigenstate. This is the case of the exclusive scattering, where the final nuclearstate is completely determined. In the inclusive scattering only the scatteredelectron is detected, the final nuclear state is not determined and the cross sec-tion includes all the available final nuclear state, but the main contribution inthe region of the QE peak still comes from the interaction on single nucleons.The inclusive scattering corresponds to an integral over all available nuclearstates and it is directly related to the dynamics of the initial nuclear groundstate.

In Chapter 4 we introduce the general NA scattering problem stated inmomentum space. In this space the description of the interaction between

4

the incoming nucleon and the target nucleus is described by the transitionmatrix, obtained solving the general many-body LS equation. This problemcan be cast in a simple one-body equation for the general transition matrixand a complicated many-body one for the optical potential that enters thefirst equation. The optical potential operator can be written as a sum of Amany-body terms, where the first one is a two-body term given by the sumover all nucleons of the interaction between the projectile nucleon and the i-th target nucleon inside the nucleus. The first term of this expansion definesthe first-order optical potential and leads to a manageable equation that isfurther simplified using the optimum factorization approximation, in which,the form of the optical potential is factorized into the product of the NNtransition matrix and the neutron and proton densities. The rest of the chapteris devoted to the computation of the NN transition matrix that represents,with the nucleon densities, the second basic ingredient for the constructionof a microscopic theoretical optical potential. Particular attention is givento the different NN potentials necessary to the construction of the transitionmatrix. In fact, a novelty of our calculations is represented by the use of Chiralpotentials as input in the NN LS equation to obtain the transition matrix. Atthe end of the chapter we present and discuss the results obtained with differentNN potentials for the Wolfenstein amplitudes, which are proportional to thecentral and spin-orbit part of the NN transition matrix.

In Chapter 5 we describe the calculational framework needed to employ theoptical potential developed in Chapter 4 in calculations of the elastic scatteringobservables: the differential cross section, the analyzing power, and the spinrotation. The calculation is performed in the NA center-of-mass frame andthe optical potential and the transition matrix are decomposed in the partialwave repesentation. A section is also dedicated to the algorithm employed toinclude in the model the Coulomb interaction between the projectile nucleonand the target nucleus. Finally, we present and discuss the theoretical resultsof the scattering observables obtained with different NN potentials. First,calculations are performed for stable nuclei and compared with available ex-perimental data, then they are extended to nuclei with neutron excess. To beconsistent with the work of Chapter 3 we take under consideration the sameoxygen and calcium isotopic chains.

Finally, in Chapter 6 we draw our conclusions and give some perspectivesfor possible improvements and future developments in the construction of amicroscopic optical potential.

5

1. Introduction

6

Chapter 2Relativistic Mean-Field Models

2.1 Introduction

The atomic nucleus presents one of the most challenging many-body problems[1]. Experimental and theoretical studies of nuclei far from stability are atthe forefront of nuclear science. Recent experiments have been probing nucleiunder extreme conditions of high spin, large deformations, high excitationenergies and neutron-to-proton ratios at their limit of stability. A wealth ofexperimental data on such nuclei is expected in the near future from newradioactive-beam facilities.

In neutron-rich nuclei the weak binding energy of the outermost neutronsand the effects of the coupling between bound states and particle continuumproduce different exotic phenomena allowing the growth of regions in nucleiwith very diffuse neutron density in which we can find the formation of aneutron skin for heavy nuclei or halo structures for light ones. With the in-crease of the number of neutrons the effective nuclear potential is modified andthis produces a suppression of shell effects with the resulting disappearance ofmagic numbers. Moreover, very neutron-rich nuclei are important for nuclearastrophysics because they offer the opportunity to study pairing phenomena insystems with strong density variations that determine the astrophysical con-ditions for the formation of neutron-rich stable isotopes.

On the proton-rich side of the valley of stability physics is different than innuclei with a large neutron excess. Extremely proton-rich nuclei are importantboth for nuclear structure studies and in astrophysical applications. Because ofthe Coulomb barrier, which tends to localize the proton density in the nuclearinterior, nuclei beyond the proton drip line are quasi-bound with respect toproton emission. These systems are characterized by exotic ground-state decaymodes, such as direct emission of charged particles and β-decays with large Q-values. The phenomenon of proton emission from the ground-state has beenextensively investigated in medium-heavy and heavy spherical and deformednuclei. The properties of many proton-rich nuclei play an important role in

7

2. Relativistic Mean-Field Models

the process of nucleosynthesis by rapid-proton capture.

Nuclei with an equal number of protons and neutrons are very importantin studies of exotic nuclear structure. They permit to study the details ofthe effective off-shell neutron-proton interaction because of the high symmetrybetween the degrees of freedom of protons and neutrons that occupy the sameshell-model orbitals. Light N = Z nuclei are β-stable, while heavier nuclei arefound close to the proton drip line and they display a variety of structure phe-nomena: shape coexistence, super-deformation, alignment of proton-neutronpairs, and proton radioactivity from highly excited states.

During the last decades nuclear structure theory has evolved from themacroscopic and microscopic descriptions of structure phenomena in stablenuclei, towards more exotic ones far from the valley of β-stability. The devel-opment of mean-field theories has reported significant progress, in particular,in describing the structure phenomena of medium-heavy and heavy nuclearsystems for which an accurate description is obtained by mean-field calcula-tions based on nuclear energy density functionals.

These models are based on concepts of non-renormalizable effective rela-tivistic field theories and density functional theory and provide an interestingtheoretical framework for studies of nuclear structure phenomena at and farfrom the valley of β-stability. A well known example of an effective theoryof nuclear structure is Quantum Hadrodynamics (QHD): a field theoreticalframework of Lorentz-covariant, meson-nucleon or point-coupling models ofnuclear dynamics. A variety of nuclear phenomena have been described withQHD based models such as nuclear matter or properties of finite spherical anddeformed nuclei.

The effective Lagrangians of QHD is compound by known long-range inter-actions constrained by symmetries and a set of generic short-range interactions.All QHD based models are consistent with the symmetries of QCD: Lorentzinvariance, parity invariance, electromagnetic gauge invariance, isospin andchiral symmetry. However, pions are not explicitly included in the RMF calcu-lation and the effects of correlated two-pion exchange are implicitly taken intoaccount through the phenomenological scalar, isoscalar σ mean-field. RMFmodels of nuclear structure are phenomenological, with parameters adjustedto reproduce the nuclear matter equation of state and a set of global propertiesof spherical closed-shell nuclei.

Lorentz scalar and four-vector nucleon self-energies are at the basis of allQHD models. Although it is not possible to experimentally verify the presenceof the large scalar and vector potentials, from nuclear matter saturation andspin-orbit splittings in finite nuclei the magnitude of these potentials can beestimated around several hundred MeV in the nuclear interior. In applicationto finite nuclei, the scalar and the vector nucleon self-energies are treated phe-nomenologically and in some models they arise from the exchange of σ and ωmesons. The strength parameters are determined by nuclear matter saturationand nuclear structure data. However, in order to reproduce the empirical bulk

8

2.2. The Density-Dependent Model

properties of finite nuclei it is necessary to consider density-dependent couplingconstants adjusted to Dirac-Brueckner self-energies in nuclear matter.

In applications to open-shell nuclei pairing correlations are very importantand they have to be included in the RMF framework. In the BCS1 model,the pairing force is taken into account phenomenologically from experimentalodd-even mass differences. For exotic nuclei far from the valley of stabilitythis is, however, only a poor approximation. For nuclei close to the drip linesa unified and self-consistent treatment of mean-field and pairing correlationsis required. This led to the development of the relativistic Hartree-Bogoliubov(RHB) model that represents the relativistic extension of the conventionalHartree-Fock-Bogoliubov (HFB) [2] framework. Such a model has been suc-cessfully employed in analyses of structure phenomena in exotic nuclei far fromthe valley of stability and it is crucial for the description of ground-state prop-erties in weakly bound systems.

In Section 2.2 we present the characteristics of the density-dependent meson-exchange (DDME) RMF model: we start from the Lagrangian density and wederive the Dirac equation for the nucleon and the Klein-Gordon equations forall mesonic fields. In order to obtain a description of ground-state propertiesof nuclei, the equations of motion are then derived for the static case and as-suming spherical symmetry for all nuclei. In Section 2.3 we present the RHBmodel based on a covariant density functional approach that gives a unifieddescription of mean-field and pairing correlations. In this model the pairingforce is treated non-relativistically, but at this stage this approximation is fullyjustified. Finally, in Section 2.4 we present our theoretical results for neutronand proton densities of oxygen and calcium isotopic chains and N = 14, 20,and 28 isotonic ones, considered in Refs. [3,4] for a study of nuclear propertieswith electron scattering calculations and presented in Chapter 3.

2.2 The Density-Dependent Model

2.2.1 The Lagrangian Density

In the standard representation of QHD the nucleus is described as a system ofDirac nucleons coupled to the exchange mesons and the electromagnetic fieldthrough an effective Lagrangian. The isoscalar scalar σ-meson, the isoscalarvector ω-meson, and the isovector vector ρ-meson build the minimal set ofmeson fields that together with the electromagnetic field is necessary for aquantitative description of bulk and single-particle nuclear properties [5–8].The model is defined by the Lagrangian density

L = LN + Lm + Lint . (2.1)

1The name derives from the Bardeen-Cooper-Schrieffer theory of superconductivity ap-

plied in nuclear physics to describe the pairing interaction between nucleons in the atomic

nucleus.

9


The first term LN denotes the Lagrangian of the free nucleon

LN = ψ(iγµ∂µ −m)ψ , (2.2)

where m is the bare nucleon mass and ψ denotes the Dirac spinor. The termLm is the Lagrangian of the free meson fields and the electromagnetic field

Lm =1

2∂µσ∂

µσ − 1

2m2

σσ2 − 1

4ΩµνΩµν +

1

2m2

ωωµωµ

− 1

4~Rµν

~Rµν +1

2m2

ρ~ρµ~ρµ − 1

4FµνF

µν ,(2.3)

with the corresponding masses mσ, mω, mρ, and Ωµν , ~Rµν , Fµν are field tensors

Ωµν = ∂µων − ∂νωµ ,

~Rµν = ∂µ~ρν − ∂ν~ρµ ,

Fµν = ∂µAν − ∂νAµ ,

(2.4)

where arrows denote isovectors. Boldface symbols will be used for vectors inordinary space. In the previous equations, σ, ωµ, and ~ρµ are the scalar, vectorisoscalar, and vector isovector meson fields, while Aµ is the electromagneticfield. In Lint is contained the minimal set of interaction terms

Lint = −gσψσψ − gωψγµωµψ − gρψ~τγ

µ~ρµψ − eψ1 − τ3

2γµAµψ , (2.5)

where τ3 is the third isospin matrix and gσ, gω, gρ, and e are the couplingconstants.

Already in the earliest applications of the RMF framework it was realized,however, that this simple model, with interaction terms only linear in themeson fields, does not provide a quantitative description of complex nuclearsystems. An effective density dependence was introduced [9] by replacing thequadratic σ-potential 1

2m2

σσ2 with a quartic potential

U(σ) =1

2m2

σσ2 +

g23σ3 +

g34σ4 . (2.6)

This potential includes the non-linear σ self-interactions with two additionalparameters g2 and g3. This particular form of the non-linear potential hasbecome standard in applications of RMF models, although additional non-linear interaction terms, both in the isoscalar and isovector channels, have alsobeen considered [10–13]. The problem with including additional interactionterms, however, is that the empirical data set of bulk and single-particle prop-erties of finite nuclei can only determine six or seven parameters in the generalexpansion of the effective Lagrangian in powers of the fields and their deriva-tives [14]. Moreover, implementation of the covariant density functional (seesection 2.2.4) with non-linear meson couplings has no direct physical meaning.Therefore, it seems more natural to follow an idea of Brockmann and Toki [15]

10


and use density-dependent couplings: gσ, gω, and gρ are assumed to be vertexfunctions of Lorentz-scalar bilinear forms of the nucleon operators. In mostapplications of the density-dependent hadron field theory the meson-nucleoncouplings are functions of the vector density

ρv =√

jµjµ with jµ = ψγµψ . (2.7)

In a relativistic framework the couplings could also depend on the scalar densityψψ. Nevertheless, expanding in ψ†ψ is the natural choice, for several reasons.The vector density is connected to the conserved baryon number, unlike thescalar density for which no conservation law exists. The scalar density is a dy-namical quantity, to be determined self-consistently by the equations of motion,and expandable in powers of the Fermi momentum. For the meson-exchangemodels it has been shown that the dependence on vector density alone pro-vides a more direct relation between the self-energies of the density-dependenthadron field theory and the Dirac-Brueckner microscopic self-energies [16]. Inthe following the vector density dependence of the meson-nucleon couplingswill be assumed. The single-nucleon Dirac equation is derived by variation ofLagrangian (2.1) with respect to ψ

[

γµ(

i∂µ − Σµ

)

−(

m + Σ)]

ψ = 0 , (2.8)

with the nucleon self-energies defined by the following relations:

Σ = gσσ ,

Σµ = gωωµ + gρ~τ · ~ρµ + e(1 − τ3)

2Aµ + ΣR

µ .(2.9)

The density dependence of the vertex functions gσ, gω, and gρ produces therearrangement contribution ΣR

µ to the vector self-energy

ΣRµ =

jµρv

(

∂gω∂ρv

ψγνψων +∂gρ∂ρv

ψγν~τψ · ~ρν +∂gσ∂ρv

ψψσ

)

, (2.10)

where jµ = ρvuµ and uµ is the four-velocity. The inclusion of the rearrangementself-energies is essential for the energy-momentum T µν conservation:

T µν = −gµνL +∑

i

∂L∂(∂µφi)

∂νφi , φi = ψ, ψ, σ, ωµ, ρµ, Aµ . (2.11)

If the rearrangement terms are omitted, the relation ∂µTµν = 0 is no longer

valid [17].The variation of Lagrangian (2.1) with respect to meson fields and the

photon field results with the set of Klein-Gordon equations for mesons and thePoisson equation for the electromagnetic potential

(

+m2σ

)

σ = −gσ 〈ψψ〉 ,(

+m2ω

)

ωµ = gω 〈ψγµψ〉 ,(

+m2ρ

)

~ρµ = gρ 〈ψγµ~τψ〉 ,

Aµ = e 〈ψγµ1 − τ32

ψ〉 .

(2.12)

11


The lowest order of the quantum field theory is the mean-field approximation:the meson-field operators are replaced by their expectation values in the nuclearground state. The A nucleons, described by a Slater determinant |Φ〉 of single-particle spinors ψi, (i = 1, 2, . . . , A), move independently in the classical mesonfields. The densities can be expressed as simple sums of bilinear products ofbaryon amplitudes. In applications to nuclear matter and finite nuclei, thedensities are calculated in the no-sea approximation: the Dirac sea of stateswith negative energies does not contribute to the densities and currents. Fora nucleus with A nucleons

〈ψΓmψ〉 =

A∑

i=1

ψi(r, t) Γm ψi(r, t) , (2.13)

where the summation is performed only over occupied orbits in the Fermi seaof positive energy states and Γm represents the model vertices. The set ofcoupled equations (2.8) and (2.12) define the RMF model.

RMF models can also be formulated without explicitly including mesonicdegrees of freedom. Meson-exchange interactions can be replaced by localfour-point interactions between nucleons. It has been shown that the relativis-tic point-coupling models [18–21] are completely equivalent to the standardmeson-exchange approach. In order to describe properties of finite nuclei ata quantitative level, the point-coupling models include also some higher orderinteraction terms. For instance, six-nucleon vertices (ψψ)

3, and eight-nucleon

vertices (ψψ)4

and [(ψγµψ)(ψγµψ)]2.

2.2.2 Stationary Solutions of the Equations of Motion

In this dissertation we are interested in the description of ground state proper-ties of nuclei and in order to do it we look for static solutions of the equationsof motion 2.8 and 2.12. In this case the nucleon spinors are eigenvectors of thestationary Dirac equation which gives the single-particle energies ǫi:

[

α(−i∇− V (r)) + βm∗(r) + V (r) + ΣR(r)]

ψi(r) = ǫiψi(r) . (2.14)

The effective mass m∗ is determined by the scalar field

m∗(r) = m+ gσσ(r) . (2.15)

The potentials V (r) and V (r) denote the time-like and space-like componentsof the vector self-energy Σµ, respectively. The term ΣR is the rearrangementself-energy of Eq. (2.10). Due to the charge conservation only the 3rd com-ponent of the isovector ρ-meson contributes. The equations of motion can befurther simplified if we restrict ourselves to systems with time-reversal invari-ance, for example even-even nuclei in the ground state. In this case there areno net currents and the spatial components of the vector self-energy vanish.The Dirac equation (2.14) reduces to

[

− iα∇ + βm∗(r) + V (r) + ΣR(r)]

ψi(r) = ǫiψi(r) . (2.16)

12


The Klein-Gordon equations and the Poisson equation reduce to(

− +m2σ

)

σ(r) = −gσρs(r) ,(

− +m2ω

)

ω0(r) = gωρv(r) ,(

− +m2ρ

)

ρ03(r) = gρρtv(r) ,

−A0(r) = eρc(r) .

(2.17)

The Eqs. (2.17) can be solved analitically using the Green’s function method:

σ(r) = −∫

gσ(ρv(r))Dσ(r, r′)ρs(r′) d3r′ ,

ω0(r) =

∫

gω(ρv(r))Dω(r, r′)ρv(r′) d3r′ ,

ρ03(r) =

∫

gρ(ρv(r))Dρ(r, r′)ρtv(r

′) d3r′ ,

A0(r) = e

∫

Dc(r, r′)ρc(r

′) d3r′ .

(2.18)

The propagator for the massive meson field reads

Dφ(r, r′) =e−mφ|r−r′|

4π|r − r′| , (2.19)

and for the photon

Dc(r, r′) =

1

4π|r − r′| . (2.20)

The densities which enter Eqs. (2.17) read

ρs(r) =

A∑

i=1

ψ†i (r) β ψi(r) ,

ρv(r) =

A∑

i=1

ψ†i (r)ψi(r) ,

ρtv(r) =A∑

i=1

ψ†i (r) τ3 ψi(r) ,

ρc(r) =

A∑

i=1

ψ†i (r)

1 − τ32

ψi(r) .

(2.21)

The set of coupled equations (2.16) and (2.17) has to be solved by iteration.Starting with an initial guess for the scalar and vector potentials we solve theDirac equation (2.16) for the spinors ψi. The results are used to calculate thedensities of Eqs. (2.21) which form the sources of Eqs. (2.17). From this set ofequations we calculate the meson fields and construct a new set of scalar andvector potentials. We repeat this cycle until the convergence is achieved.

13


The total energy of the system is calculated by integrating the T 00 compo-nent of the energy-momentum tensor (2.11) over the entire nucleus:

ERMF =

A∑

i=1

∫

d3r ψ†i

(

− iα∇ + βm)

ψi

+1

2

∫

d3r(

gσρsσ + gωρvω0 + gρρtvρ

03

)

.

(2.22)

When the solution of the self-consistent equations has been obtained, the totalbinding energy is corrected with the microscopic estimate for the energy ofcenter-of-mass motion

Ecm = −〈P 2cm〉

2Am, (2.23)

where Pcm is the total momentum of a nucleus with A nucleons [22].

2.2.3 Rotationally Invariant Systems

A nucleus with a proton or neutron closed major shell always exhibits sphericalsymmetry. The corresponding density and fields depend only on the radialcoordinate r. The Dirac spinor is characterized by the single-particle angularmomentum quantum numbers j and m, the parity π, and the isospin projectionmt = ±1/2 for neutron and proton, respectively:

ψ(r, s, t) =

(

f(r) Φl,j,m(θ, φ, s)ig(r) Φl,j,m(θ, φ, s)

)

χt,mt, (2.24)

where χt,mtis the isospin wave function. The orbital angular momentum l for

the large, and l for the small component of the Dirac spinor are determinedby the angular momentum j and the parity π:

l = j +1

2, l = j − 1

2for π = (−1)j+

1

2 ,

l = j − 1

2, l = j +

1

2for π = (−1)j−

1

2 .(2.25)

The spin-angular function Φl,j,m is a two-dimensional spinor with the angularmomentum quantum numbers ljm,

Φl,j,m(θ, φ, s) =∑

ml,ms

(

lml12ms

∣

∣j m)

Y ml

l (θ, φ)χ 1

2,ms

, (2.26)

where Y ml

l (θ, φ) are the spherical harmonics and χ 1

2,ms

the spin wave functions.The Dirac equation reduces to a set of coupled ordinary differential equationsfor the radial functions f(r) and g(r)

[m∗(r) + V (r)]f(r) +

(

∂r −κ− 1

r

)

g(r) = ǫf(r) ,

−(

∂r +κ+ 1

r

)

f(r) − [m∗(r) − V (r)]g(r) = ǫg(r) ,

(2.27)

14


where κ = ±(j + 1/2) for j = l ∓ 1/2. In spherical coordinates the densitiesare expressed in terms of the radial functions

ρs(r) =

A∑

i=1

(2ji + 1)(

|fi(r)|2 − |gi(r)|2)

,

ρv(r) =A∑

i=1

(2ji + 1)(

|fi(r)|2 + |gi(r)|2)

,

ρtv(r) =A∑

i=1

ti(2ji + 1)(

|fi(r)|2 + |gi(r)|2)

,

ρc(r) =

A∑

i=1

(1 − ti)(2ji + 1)(

|fi(r)|2 + |gi(r)|2)

,

(2.28)

where ti = 1 for neutrons, and ti = −1 for protons. The densities define thesource terms for the spherical Klein-Gordon equations

(

− ∂2

∂r2− 2

r

∂

∂r+m2

φ

)

φ(r) = sφ(r) , (2.29)

where mφ are the meson masses for φ = σ, ω, ρ and zero for the photon. Thesource term reads

sφ(r) =

−gσ(ρv(r))ρs(r) for the σ-field,

gω(ρv(r))ρv(r) for the ω-field,

gρ(ρv(r))ρtv(r) for the ρ-field,

eρc(r) for the Coulomb field.

(2.30)

2.2.4 Covariant Density Functional Theory

The modern interpretation of QHD is based on concepts of the Effective FieldTheory (EFT) and the Density Functional Theory (DFT). The DFT enablesa description of the nuclear many-body problem in terms of a universal energydensity functional and the mean field approach to nuclear structure repre-sents an approximate implementation of the Kohn-Sham DFT [23–25], whichis successfully used in the treatment of different quantum many-body prob-lems. Energy functionals of the ground-state density are the basis of the DFTapproach and for RMF models they are functionals of the ground-state scalardensity and baryon currents. In the description of nuclear ground-state prop-erties we employ the self-consistent mean-field implementation of QHD: theRHB model. By employing global effective interactions, adjusted to repro-duce empirical properties of symmetric and asymmetric nuclear matter, andbulk properties of simple, spherical and stable nuclei, the current generation ofself-consistent mean-field methods has achieved a high level of accuracy in thedescription of the properties of ground and excited states in arbitrarily heavy

15


nuclei, exotic nuclei far from β-stability, and in nuclear systems at the nucleondrip lines.

The equations of motion of the RMF model can be derived starting from adensity functional. The energy-momentum tensor determines the total energyof the nuclear system and for the static case we have:

ERMF[ψ, ψ, σ, ωµ, ~ρµ, Aµ] =

A∑

i=1

∫

d3r ψ†i

(

α · p + βm)

ψi

+1

2

∫

d3r[

(∇σ)2 +m2σσ

2]

− 1

2

∫

d3r[

(∇ω)2 +m2ωω

2]

− 1

2

∫

d3r[

(∇ρ)2 +m2ρρ

2]

− 1

2

∫

d3r (∇A)2

+

∫

d3r[

gσρσσ + gωjµωµ + gρ~jµ~ρ

µ + ejcµAµ]

.

(2.31)

By using the definition of the relativistic single-nucleon density matrix

ρ(r, r′, t) =

A∑

i=1

|ψi(r, t)〉〈ψi(r′, t)| , (2.32)

the total energy can also be rewritten as a functional of the density matrix ρand of the meson fields

ERMF[ρ, φm] = Tr[(

α·p+βm)

ρ]

±1

2

∫

d3r[

(∇φm)2 +m2φφ

2m

]

+Tr[(

Γmφm

)

ρ]

.

(2.33)The trace operation involves a sum over the Dirac indices and an integral incoordinate space. The index m is used as generic notation for all mesons andthe photon. The equations of motion are obtained from the classical variationalprinciple

δ

∫ t2

t1

dt 〈Φ|i∂t|Φ〉 −ERMF[ρ, φm] = 0 . (2.34)

The equation of motion for the density matrix reads

i∂tρ =[

h(ρ, φm), ρ]

(2.35)

and the single-particle Hamiltonian h is obtained from the functional derivativeof the energy with respect to the single-particle density matrix ρ

h =δE

δρ. (2.36)

16

2.3. Pairing Correlations

2.2.5 The Density-Dependence of the Couplings

The density dependence of the meson-nucleon couplings can be parameterizedin a phenomenological way by the functionals of Refs. [26, 27]. The couplingof the σ-meson and ω-meson to the nucleon field reads

gi(ρ) = gi(ρsat)fi(x) for i = σ, ω , (2.37)

where

fi(x) = ai1 + bi(x + di)

2

1 + ci(x+ di)2 (2.38)

is a function of x = ρ/ρsat, and ρsat denotes the baryon density at saturationin symmetric nuclear matter. The eight real parameters in Eq. (2.38) are notindependent. The five constraints fi(1) = 1, f ′′

σ (1) = f ′′ω(1), and f ′′

i (0) =0, reduce the number of independent parameters to three. Three additionalparameters in the isoscalar channel are: gσ(ρsat), gω(ρsat), and mσ, the mass ofthe phenomenological sigma-meson. For the ρ-meson coupling the functionalform of the density dependence is suggested by Dirac-Brueckner calculationsof asymmetric nuclear matter [28]

gρ(ρ) = gρ(ρsat) exp[−aρ(x− 1)] . (2.39)

The isovector channel is parameterized by gρ(ρsat) and aρ. For the massesof the ω and ρ mesons the free values are used: mω = 783 MeV and mρ =763 MeV. The eight independent parameters, seven coupling parameters andthe mass of the σ-meson, are adjusted to reproduce properties of symmetricand asymmetric nuclear matter, binding energies, charge radii and neutronradii of 12 spherical nuclei.

2.3 Pairing Correlations

2.3.1 Non-Relativistic Theory

Pairing correlations are essential for a correct description of structure phe-nomena in spherical open-shell nuclei and in deformed nuclei and they haveto be included in the model. For nuclei close to the β-stability line, pairinghas been included in the RMF model in the form of the simple BCS approx-imation [29]. In this approximation for open-shell nuclei pairing is treatedphenomenologically using a schematic ansatz in which the constant gap ap-proximation is made and the empirical value of the gap parameter ∆ is givenby the five-point formula [30]

∆(5)(N) = −1

8

[

M(N + 2) − 4M(N + 1)

+ 6M(N) − 4M(N − 1) +M(N − 2)]

,(2.40)

17


where M(N) is the mass for a nucleus with N neutrons. We used this approx-imation in Ref. [3] where we considered oxygen and calcium isotopic chainsand consequently only the neutron gap was needed.

However, for nuclei far from stability the BCS model presents only a poorapproximation. In particular, in drip-line nuclei the Fermi level is found closeto the particle continuum and the BCS model is unable to correctly describethe scattering of nucleonic pairs from bound states to the positive energy con-tinuum allowing the partial occupation of the high levels in the continuum.

The HFB theory [2] provides a unified description of ph- and pp-correlationsat a mean-field level by using two average potentials: the self-consistent Hartree-Fock field Γ which encloses all the long-range ph-correlations, and a pairingfield ∆ which sums up the pp-correlations. The ground state of a nucleus isdescribed by a generalized Slater determinant |Φ〉 which represents the vacuumwith respect to independent quasiparticles. The quasiparticle operators are de-fined by the unitary Bogoliubov transformation of the single-nucleon creationand annihilation operators:

α+k =

∑

l

Ulk c+l + Vlk cl , (2.41)

where Ulk, Vlk are the HFB wave functions. The index l denotes an arbitrarybasis, for instance the harmonic oscillator states. In the coordinate spacerepresentation l = (r, σ, τ), where σ and τ are the spin and isospin indices,respectively. The HFB wave functions determine the Hermitian single-particledensity matrix

ρll′ = 〈Φ|c+l′ cl|Φ〉 =(

V ∗ V T)

ll′(2.42)

and the antisymmetric pairing tensor

κll′ = 〈Φ|cl′cl|Φ〉 =(

V ∗ UT)

ll′. (2.43)

According to Valatin [31] these two densities can be combined into the gener-alized density matrix

R =

(

ρ κ−κ∗ 1 − ρ∗

)

. (2.44)

In the case of a nuclear Hamiltonian of the form

H =∑

l

ǫlc+l cl +

1

4

∑

ll′mm′

vlm′l′mc+l c

+m′cmcl′ (2.45)

with a general NN interaction v, the expectation value 〈Φ|H|Φ〉 can be ex-pressed as a function of the Hermitian density matrix ρ and the antisymmetricpairing tensor κ. From the variation of this energy functional with respect toρ and κ, the HFB equations are derived

(

h− λ ∆

−∆∗ −h∗ + λ

)(

Uk

Vk

)

= Ek

(

Uk

Vk

)

. (2.46)

18


The single-nucleon Hamiltonian reads

h = ǫ + Γ . (2.47)

The two self-consistent potentials Γ and ∆ are obtained as

Γll′ =∑

mm′

vlm′l′m ρmm′ (2.48)

and∆ll′ =

∑

m<m′

vll′mm′ κmm′ . (2.49)

The chemical potential λ is determined by the particle number subsidiarycondition in such a way that the expectation value of the particle numberoperator in the ground state equals the number of nucleons. The columnvectors denote the quasiparticle wave functions, and Ek are the quasiparticleenergies.

If the nuclear many-body problem is formulated in the framework of theDFT, one does not start from a two-body interaction as given in Eq. (2.45), butrather from an energy functional E[R] = E[ρ, κ] that depends on the densitiesρ and κ. The generalized Hamiltonian H is obtained as a functional derivativeof the energy with respect to the generalized density

H =δE

δR =

(

h ∆

−∆∗ −h∗)

. (2.50)

The single-particle Hamiltonian h results from the variation of the energyfunctional with respect to the Hermitian density matrix ρ

h =δE

δρ(2.51)

and the pairing field is obtained from the variation of the energy functionalwith respect to the pairing tensor

∆ =δE

δκ. (2.52)

The relativistic extension of the HFB theory was introduced in Ref. [32].Starting from the effective Lagrangian (2.1) it is possible to develop a fullyrelativistic theory of the pairing force and it can be shown that the RHBequations can be rewritten in the same form as Eqs. (2.46). However, it mustbe emphasized that pairing in nuclei is a fully non-relativistic phenomenon inwhich pairing effects are restricted to an energy window of a few MeV aroundthe Fermi level. Moreover, the energy scale of pairing effects is well separatedfrom the scale of binding energies, which are in the range of several hundred,and for heavy nuclei even more than thousand MeV. Up to now there is noexperimental evidence for any relativistic effect in the nuclear pairing field ∆.From these considerations it is therefore justified to use a hybrid model witha non-relativistic pairing interaction in a phenomenological approach based ona DFT, such as the RHB framework used in this dissertation.

19


2.3.2 Covariant Density Functional Theory with Pairing

The RHB model can be derived within the framework of covariant DFT. Whenpairing correlations are included, the energy functional depends not only onthe density matrix ρ and the meson fields φm, but in addition also on thepairing tensor:

ERHB[ρ, κ, φm] = ERMF[ρ, φm] + Epair[κ] , (2.53)

where ERMF[ρ, φm] is the RMF functional defined in Eq. (2.33), and the pairingenergy Epair[κ] is given by

Epair[κ] =1

4Tr

[

κ∗ V pp κ]

. (2.54)

The term V pp denotes a general two-body pairing interaction. The equationof motion for the generalized density matrix reads

i∂tR = [H(R),R] (2.55)

and the generalized Hamiltonian H is obtained as a functional derivative ofthe energy with respect to the generalized density

HRHB =δERHB

δR =

(

hD −m− λ ∆

−∆∗ −h∗D + m+ λ

)

. (2.56)

The self-consistent mean field hD is the Dirac Hamiltonian. In the static casewith time-reversal symmetry

hD = −iα ·∇ + V0(r) + β[m+ S(r)] (2.57)

and the pairing field is an integral operator with the kernel (2.49)

∆ab(r, r′) =

1

2

∑

c,d

V ppabcd(r, r

′) κcd(r, r′) , (2.58)

where a, b, c, d denote quantum numbers that specify the Dirac indices of thespinors, and V pp

abcd(r, r′) are the matrix elements of a general two-body pairing

interaction.The stationary limit of Eq. (2.55) describes the ground state of an open-

shell nucleus [33]. It is determined by the solutions of the Hartree-Bogoliubovequations

(

hD −m− λ ∆

−∆∗ −h∗D +m + λ

)(

Uk(r)Vk(r)

)

= Ek

(

Uk(r)Vk(r)

)

. (2.59)

The chemical potential λ is determined by the particle number subsidiary con-dition in order that the expectation value of the particle number operator in

20


the ground state equals the number of nucleons. The column vectors denotethe quasiparticle wave functions, and Ek are the quasiparticle energies. Thedimension of the RHB matrix equation is two times the dimension of the cor-responding Dirac equation. For each eigenvector (Uk, Vk) with positive quasi-particle energy Ek > 0, there exists an eigenvector (V ∗

k , U∗k ) with quasiparticle

energy −Ek. Since the baryon quasiparticle operators satisfy fermion commu-tation relations, the levels Ek and −Ek cannot be occupied simultaneously. Forthe solution that corresponds to a ground state of a nucleus with even particlenumber, one usually chooses the eigenvectors with positive eigenvalues Ek.

The RHB equations are solved self-consistently, with potentials determinedin the mean-field approximation from solutions of static Klein-Gordon equa-tions (2.17) for the σ-meson, the ω-meson, the ~ρ-meson and the photon field,respectively. The source terms in Eqs. (2.17) are sums of bilinear products ofbaryon amplitudes

ρs(r) =∑

k>0

V †k (r)γ0Vk(r) ,

ρv(r) =∑

k>0

V †k (r)Vk(r) ,

ρtv(r) =∑

k>0

V †k (r)τ3Vk(r) ,

ρc(r) =∑

k>0

V †k (r)

1 − τ32

Vk(r) ,

(2.60)

where the sum over positive-energy states corresponds to the no-sea approxi-mation. The self-consistent solution of the Dirac-Hartree-Bogoliubov integro-differential equations and Klein-Gordon equations for the meson fields deter-mines the ground state of a nucleus.

The eigensolutions of Eq. (2.59) form a set of orthogonal (normalized) singlequasiparticle states. The corresponding eigenvalues are the single quasiparticleenergies Ek. The self-consistent iteration procedure is performed in the basis ofquasiparticle states. In order to obtain a better understanding of the structureof the Hartree-Bogoliubov wave function |Φ〉, the self-consistent quasiparticleeigenspectrum is then transformed into the canonical basis of single-nucleonstates. By definition the canonical basis |ϕµ(r)〉 diagonalizes the single-nucleon density matrix ρ in Eq. (2.42)

ρ |ϕµ〉 = v2µ |ϕµ〉 . (2.61)

The transformation to the canonical basis determines the energies and pairingmatrix elements

ǫµ = 〈ϕµ|hD −m|ϕµ〉 and ∆µ = 〈ϕµ|∆|ϕµ〉 (2.62)

21


and the occupation probabilities of single-nucleon states

v2µ =1

2

1 − (ǫµ − λ)√

(ǫµ − λ)2 + ∆2µ

. (2.63)

The self-consistent solution |Φ〉 for the ground state of a nucleus has the struc-ture of a BCS state with these occupation probabilities [2].

In Ref. [33] it was suggested that the pairing part of the well known andvery successful Gogny force [34] should be employed in the pp-channel:

V pp(1, 2) =∑

i=1,2

e−[(r1−r2)/µi]2 (Wi +BiPσ −HiP

τ −MiPσP τ ) (2.64)

with the set D1S [34, 35] for the parameters µi, Wi, Bi, Hi, and Mi (i = 1, 2).This force has been very carefully adjusted to the pairing properties of finitenuclei all over the periodic table. Moreover, because of the finite range, theGogny force has the basic advantage to automatically guarantee a proper cut-off in momentum space.

2.4 Results for Neutron and Proton Densities

As already stated, the RHB model permits to achieve a unified and self-consistent treatment of mean-field and pairing correlations. The details ofthe calculated ground-state properties will depend on the coupling constantsand masses of the effective Lagrangian, on the pairing interaction, and on thecoupling between bound and continuum states. In order to obtain a betterdescription of open-shell nuclei, in our calculations we have predominantlyadopted the density-dependent RHB model using, in particular, two differentsets of parameters: the DDME1 [26] and DDME2 [36]. In all calculationswe have used the DIRHB package [37], that consists of three Fortran com-puter codes for the calculation of ground-state properties of even-even nucleifor different choices of spatial symmetry: DIRHBS, DIRHBZ, and DIRHBTcodes are used to calculate nuclei with spherical symmetry, axially symmet-ric quadrupole deformation, and triaxial quadrupole shapes, respectively. Asa first step, in this dissertation we have assumed spherical symmetry for allnuclei and thus all the theoretical results are obtained with the DIRHBS code.

Neutron and proton distributions are one of the basic ingredients for com-puting the scattering observables. In Refs. [3,4] we gave theoretical predictionsfor elastic and QE electron scattering cross sections on oxygen and calcium iso-topic chains and on N = 14, 20, and 28 isotonic ones. In this section we presentand discuss the theoretical predictions for nucleon distributions for all thesenuclei.

As a first step we present the numerical results obtained from calculationson isotopic chains. The calculations are performed in a RMF framework and

22

2.4. Results for Neutron and Proton Densities

0 2 4 6r [fm]

0

0.02

0.04

0.06

0.08

0.1

ρ n [fm

-3]

14O

16O

18O

20O

22O

24O

26O

28O

0 2 4 6r [fm]

0

0.02

0.04

0.06

0.08

0.1

ρ p [fm

-3]

(a) (b)

Figure 2.1: Neutron, panel (a), and proton, panel (b), distributions for thevarious oxygen isotopes we have considered. The densities have been computedwith DDME2 parametrization [36].

pairing is treated in the BCS approximation, estimating the constant gap pa-rameter with the five-point formula of Eq. (2.40). In Figs. 2.1 and 2.2 we plotthe neutron and proton density distributions ρn and ρp as a function of theradial coordinate r for 14−28O and 36−56Ca isotopic chains, respectively. Thesedensity distributions are the sum of the squared moduli of the single-particleneutron or proton wave functions. All the nuclei we have investigated resultedto be bound. From the experimental point of view it seems rather well estab-lished that the neutron drip line for the oxygen isotopes starts with 26O [38]and, therefore, 28O should not be bound.

For both isotopic chains the differences between the proton and neutrondensities in Figs. 2.1 and 2.2 generally increase with the neutron number.When the number of neutrons increases there is a gradual increase of theneutron radius. The differences of the neutron density profiles in the nuclearinterior display pronounced shell effects. The effect of adding neutrons is topopulate and extend the neutron densities and, to a minor extent, also theproton densities. In the case of protons, however, there is a decrease of thedensity in the nuclear interior to preserve the normalization to the constantnumber of protons.

In Figs. 2.3, 2.4, and 2.5 we present the numerical results for the N =28, 20, 14 isotonic chains, respectively. In this case all the calculations are

23


0 2 4 6 8 10r [fm]

0

0.02

0.04

0.06

0.08

0.1

ρ n [fm

-3]

36Ca

38Ca

40Ca

42Ca

44Ca

46Ca

48Ca

50Ca

52Ca

54Ca

56Ca

0 2 4 6 8r [fm]

0

0.02

0.04

0.06

0.08

0.1

ρ p [fm

-3]

(a) (b)

Figure 2.2: The same as in Fig. 2.1, but for calcium isotopes.

performed in the RHB model treating mean-field and pairing correlations in aunified way. This allows us to obtain a better description of the ground-stateproperties of nuclei close to the drip lines.

In panel (a) of Fig. 2.3 we plot the proton density distributions ρp as func-tions of the radial coordinate r. All the nuclei that we consider result to bebound but, experimentally, there are proton-deficient nuclei (from 40Mg to46Ar), stable nuclei (from 48Ca to 54Fe) and one proton-rich nucleus (56Ni).The most significant effect of adding protons is to populate and extend theproton densities. In panel (b) of Fig. 2.3 we plot the neutron densities for theN = 28 isotonic chain. The evolution of the neutron density distribution alongan isotonic chain is less significant than that of the proton density distribution;generally, there is a decrease of the density in the nuclear interior and an ex-tension towards large r to preserve the normalization to the constant numberof neutrons.

The differences of the proton density profiles in the nuclear interior displaypronounced shell effects; the most relevant proton single particle levels in ouranalysis of the N = 28 chain are the 2s1/2 and the 1f7/2. In the case of theheavier nuclei, starting from 48Ca up to 56Ni, we obtain similar results for ρpin the central region whereas the differences at large r can be ascribed to thefilling of the 1f7/2 shell which starts from 50Ti.

The proton density of the proton-deficient 46Ar nucleus also gets a nonnegligible contribution from the 2s1/2 shell, which has an occupation number

24


0 2 4 6 8r [fm]

0

0.02

0.04

0.06

0.08

0.1ρ p [

fm-3

]

40Mg

42Si

44S

46Ar

48Ca

50Ti

52Cr

54Fe

56Ni

0 2 4 6 8r [fm]

0

0.02

0.04

0.06

0.08

0.1

ρ n [fm

-3]

(a) (b)

Figure 2.3: Proton, panel (a), and neutron, panel (b), distributions alongthe N = 28 isotonic chain. The densities have been computed with DDME2parametrization [36].

of 0.330, and, as a consequence, in the central region it is approximately 20%larger than those of lighter isotones. In our model the 42Si nucleus, with 14protons, behaves like a magical nucleus and its proton density does not get anycontribution from neither the 2s1/2 nor the 1d3/2 shells which, on the contrary,contribute to 40Mg and to 44S densities which are, therefore, larger in thecentral region. However, the experimental evidence of a 2+

1 state at 770 ± 19keV [39], much smaller than for other nuclei in this chain, has been interpretedas a signal of the disappearance of the N = 28 shell closure around 42Si andhas suggested that proton-core excitations and the tensor interactions cannotbe neglected [40, 41].

In panel (a) of Fig. 2.4 we present our results for the proton density dis-tributions as functions of r along the N = 20 isotonic chain. In panel (b) ofFig. 2.4 we present the evolution of the neutron density distribution along theN = 20 isotonic chain. For the heavier isotones we observe a decrease of thedensity in the nuclear interior and a corresponding enhancement away fromthe center to preserve the normalization to the constant number of neutrons.

In our model all these nuclei result to be bound but, along this chain,there are proton-deficient nuclei (from 28O to 34Si), stable nuclei (36S, 38Ar,and 40Ca) and proton-rich nuclei (42Ti, 44Cr, and 46Fe). The densities of theproton-rich isotones are significantly extended toward larger r with respect to

25


0 2 4 6 8r [fm]

0

0.02

0.04

0.06

0.08

0.1ρ p [

fm-3

]

28O

30Ne

32Mg

34Si

36S

38Ar

40Ca

42Ti

44Cr

46Fe

0 2 4 6r [fm]

0

0.02

0.04

0.06

0.08

0.1

ρ n [fm

-3]

(a) (b)

Figure 2.4: The same as in Fig. 2.3, but for N = 20 isotonic chain.

those of the proton-deficient ones. Also in this chain pronounced shell effectsare visible in the nuclear interior. In particular, we find large differences in theradial profiles of the proton density of 34Si and 36S; in our model the protonoccupation number of the 2s1/2 shell of 36S is not negligible and, owing tothe observation that the squared wave function of the 2s1/2 state has a mainpeak at the center, the proton central density is enlarged. We obtain thatalso the proton occupation number of the 1d3/2 state for 36S is significant; thecorresponding peak of the squared wave function is away from the center and,therefore, the filling of the 1d3/2 state by protons increases the density awayfrom the center. In the case of the heavier nuclei, starting from 38Ar up to 46Fe,we obtain similar results for ρp in the central region, whereas the differencesat large r can be ascribed to the addition of protons that populate and extendthe densities.

In panel (a) of Fig. 2.5 we present our results for the proton density distri-butions as functions of r along the N = 14 isotonic chain. The densities in thenuclear interior of proton-deficient isotones up to the stable nucleus 28Si arevery similar. For all these nuclei only the two protons in the 1s1/2 state con-tribute to the density at the center. Starting from 30S also the protons in the2s1/2 state contribute and there is an evident transition in the density profile.As usual, the densities of the proton-rich isotones are significantly extendedtowards larger r with respect to those of the proton-deficient ones. In panel(b) of Fig. 2.5 we present the evolution of the neutron density distribution

26


0 2 4 6 8r [fm]

0

0.02

0.04

0.06

0.08

0.1ρ p [

fm-3

]22

O24

Ne26

Mg28

Si30

S32

Ar34

Ca

0 2 4 6r [fm]

0

0.02

0.04

0.06

0.08

0.1

ρ n [fm

-3]

(a) (b)

Figure 2.5: The same as in Fig. 2.3, but for N = 14 isotonic chain.

along the N = 14 isotonic chain. The profiles of proton-deficient and proton-rich isotones are very different, in particular in the nuclear interior, but thenormalization to the constant number of neutrons is always preserved.

27


28

Chapter 3Electron Scattering

3.1 Historical Introduction

Nuclear physics with electromagnetic probes [42] has its roots deep in the earlyyears of modern physics. Soon after the discovery of the Dirac equation [43]for the relativistic electron, Mott [44] derived the cross-section formula forthe scattering of Dirac particles by point-like nuclei. The effects of deviationsfrom the point Coulomb interaction in the scattering of relativistic electronswere studied first by Guth (1934) and the possibility of exploring the chargedistribution in nuclei was independently pointed out by Rose [45] much later.It was not until the early 1950s that the first electron-nucleus scattering ex-periment was performed by Lyman et al. [46] with the 22 MeV betatron ofthe University of Illinois at Urbana. This experiment opened up a new eraof nuclear structure investigations by eleastic and inelastic electron scattering.The earliest communication on electron scattering at energies above 100 MeVand deviation from the Mott formula were given by Hofstadter et al. [47] work-ing in the High Energy Physics Laboratory at Stanford with electrons of 116MeV energy. The finite proton size was first appreciated by observing devia-tions from the Rosenbluth [48] cross section for elastic electron scattering bya point-like Dirac particle [49]. An extensive quantitative investigation of thesize and structure of nucleons and nuclei then began and was crowned by theNobel Prize given to Hofstadter in 1961.

Several decades of experimentation have proved that electron scattering isprobably the best tool for investigating the structure of hadron systems suchas atomic nuclei and their constituents. The electromagnetic interaction isknown from QED and is weak compared with the strength of the interactionbetween hadrons. Thus electron scattering is adequately treated assuming thevalidity of the Born approximation, i.e., the one-photon exchange mechanismbetween electron and target. By scattering an electron with initial energy Eand momentum k to a final energy E ′ and momentum k′ at a deflection angleθ, the target receives an energy transfer ω = E−E ′ and a momentum transfer

29

3. Electron Scattering

q = k − k′. In the one-photon exchange approximation, ω and q are theenergy and momentum of the virtual photon absorbed by the target. We canalso form an invariant quantity, Q2 = |q|2 − ω2, which is often referred to asthe negative mass squared of the virtual photon. For high-energy electrons,Q2 = 4EE ′ sin2 θ

2, so that in the limit of small scattering angles Q2 approaches

zero and we have the fixed relationship between energy and momentum of areal photon, ω = |q| (~ = c = 1). Therefore we should expect the resultsof electron scattering to approach photon absorption as the scattering angleapproaches zero. The ability to vary independently the momentum and theenergy transferred to the target makes it possible to map the target response asa function of its excitation energy, with a spatial resolution that can be adjustedto the scale of processes that need to be studied. This scale is related to themomentum transfer by the wavelength of the exchanged photon, λ ∼ 1/|q|. Inthe early days of the Hofstadter experiments the maximum momentum transferwas sufficiently high (|q| ∼ 1 − 1.5 fm−1) to allow the measurement of basicquantities such as the radius and surface thickness of the charge distributionof nucleons and nuclei, but was still too low to reveal the details of the internalstructure.

Elastic and inelastic electron scattering data were taken in parallel with thedevelopment of the appropriate theoretical background. Several decades of ex-perimental and theoretical work on electron scattering have provided a wealthof information on static and dynamic properties of stable nuclei. In particu-lar, nuclear charge density distributions and charge radii have been determinedfrom the analysis of elastic electron scattering data [50,51]. The general pictureemerged of a nucleus formed by nucleons moving in a self-consistent mean-fieldpotential. The shell model found a spectacular confirmation by the first (e, e′p)experiment on complex nuclei performed by Amaldi et al. [52] with the internalbeam of the Frascati electron synchrotron. By such experiments access was ob-tained to the bound nucleon wave function in momentum space distinguishingamong different nuclear shells [53]. With the technological progress, descrip-tions of few-body systems as well as of complex nuclei took great advantageof the accurate determination of elastic and transition form factors in electronscattering [54–56]. Extensive investigation of electron-nucleon scattering in thenuclear medium started with inclusive and semi-inclusive experiments underQE conditions [57, 58]. The simple shell-model picture was shown to be in-sufficient to explain the data on nucleon knockout and violation of the energysum rule [59, 60] clearly indicated the role of correlations even in a quasi-freeprocess such as the (e, e′p) reaction [61]. Separation of the longitudinal andtransverse responses in the inclusive (e, e′) cross section confirmed that thepicture of the nucleus responding as a collection of non-interacting nucleonshad to be improved.

30

3.2. The Nuclear Response

3.2 The Nuclear Response

The key element for understanding the structure and dynamics of hadronicmatter is its response to an external probe as a function of energy and mo-mentum transfer (ω, q). Depending on the excitation energy and the rangeof explored momenta, different degrees of freedom come into play. Single-nucleon excitations and collective motion are appropriate descriptions of nucleiat low (ω, q). In the framework of intermediate-energy nuclear physics, how-ever, other degrees of freedom become important. They are ultimately relatedto the composite structure of nucleons. The definition and relevance of suchdegrees of freedom in the nuclear many-body theory is guided by accumulat-ing knowledge of the behavior of free nucleons. However, when embedded inthe nuclear environment nucleons may behave differently because of mediumeffects. Thus intermediate-energy nuclear physics is a many-body problemintersecting nuclear and particle physics with great intellectual challenge.

The electromagnetic probe is the best reliable tool to investigate the nuclearresponse. It is weak enough to be treated in first-order perturbation theory. Inelectron scattering this corresponds to the one-photon exchange approximationso that in the cross section the electron current, completely determined byQED, is clearly separated from the hadron current describing the transition ofthe target system induced by the electron. It is therefore possible in principleto access the relevant information about the hadron structure contained in thetarget response to the electromagnetic probe. In addition, the virtual photon,like the real one, has a mean free path much larger than the target dimensions,thus exploring the whole target volume. This contrasts with hadron probeswhich are absorbed at the target surface.

With respect to real photon absorption, electron scattering allows inde-pendent variation of energy and momentum transfer as well as of the longi-tudinal and transverse polarizations of the exchanged virtual photon. Thislarge flexibility of the electron probe is reflected in the cross section, wherea correspondingly large number of structure functions appear, related to thedifferent ways the target can absorb the virtual photon. Thus, separation ofthe longitudinal and transverse components of the transition matrix elementsis in principle possible with detailed information on the target dynamics.

However, the full exploration of all these potentialities is limited by thesmallness of the cross sections involved (proportional to α2, where α = e2/4π ≃1/137 is the fine structure constant). Thus coincident experiments, neces-sary to detect specific final states, require high duty factors and high electroncurrents. Measurements of polarization observables, absolutely required for acomplete determination of the transition amplitudes, imply sophisticated tech-niques in preparing polarized targets and/or detecting polarized recoil parti-cles.

For a proton target, elastic scattering occurs at ω = Q2/2M , M being thenucleon mass. The magnitude of the response is a function of Q2 asymptoti-cally decreasing as Q−4. The Q2 dependence is governed by the finite extension

31


of the proton charge and spin distribution. At an excitation energy of about300 MeV the ∆-resonance peak corresponds to the first nucleon excitation.Other peaks at higher energy are produced by other baryon resonances. Inter-nal dynamics of free nucleons at such excitation energies can only be modelledalong lines dictated by the basic requirements of QCD. Owing to the rich spec-trum predicted by hadron models we expect that many overlapping resonancescontribute to each observed peak. To disentangle their contribution and to un-derstand the underlying dynamics it is necessary to look at the decay modesby coincident detection of the decay products.

Deep inelastic scattering occurs at very high energy and momentum trans-fers. In the limit of both Q2 ≫ M2 and an invariant hadronic mass W ≫ M(W 2 = M2 −Q2 + 2Mω), the proton response is said to scale, i.e. becomes afunction of the single Bjorken variable x = Q2/2Mω. This behavior is due tothe quark-gluon content of the proton and in this regime perturbative QCD isan adequate description of hadrons.

Remarkable differences are observed between the nuclear and the nucleonresponse. Elastic electron scattering by a nucleus with A nucleons occurs atω = Q2/2MA. Bound nucleons may produce low-energy excited states as a re-sult of single-particle transitions and/or collective motion. Spectroscopic anal-ysis of such excitation mechanisms is the traditional field of nuclear physics.The systematic investigation done in the past by means of inelastic electronscattering has given the necessary support to the foundation of many-bodytheories applied to the nuclear system. The giant resonance, which is thefirst striking feature in a plot versus energy of the nuclear response in photonabsorption, also occurs in electron scattering. Here, the possibility of inde-pendently varying Q2 and ω, in conjunction with the coincident detection ofnucleons emitted in the decay process, allows a detailed analysis of the varioustypes of collective motions that are responsible for contributions of differentmultipolarities to such an enhanced resonance. Again many-body theorieswith nucleon degrees of freedom and the basic NN interaction are successfulin accounting for the data.

A large broad peak occurs in the nuclear response at about ω = Q2/2M .Its position corresponds to the elastic peak in electron scattering by a free nu-cleon. It is quite natural to assume that a quasi-free process is responsible forsuch a peak with a nucleon emitted quasi-elastically. The striking differencein the peak width is primarily due to the Fermi motion which causes Dopplerbroadening. Also a small shift in the peak position is observed and explainedas due to the nucleon binding. The quasi-free peak is not present in photonabsorption as a consequence of energy-momentum conservation. Coincidenceexperiments in the quasi-free region confirm such a picture and represent avaluable source of information on single-nucleon degrees of freedom inside nu-clei.

Coincidence (e, e′p) experiments in the QE region represent a very cleantool to explore the proton-hole states. A large amount of data for the ex-

32

3.3. Elastic and Quasi-Elastic Scattering on Exotic Nuclei

clusive (e, e′p) reaction have confirmed the assumption of a direct knockoutmechanism and has provided accurate information on the single-particle struc-ture of stable closed-shell nuclei [42, 58, 62–67]. The separation energy andthe momentum distribution of the removed proton, which allows to determinethe associated quantum numbers, have been obtained. From the comparisonbetween experimental and theoretical cross sections it has been possible to ex-tract the spectroscopic factors, which give a measurement of the occupation ofthe different shells and, as a consequence, of the effects of nuclear correlations,which go beyond a mean field description of nuclear structure.

In the inclusive (e, e′) process only the scattered electron is detected andthe final nuclear state is undetermined, but the main contribution in the re-gion of the QE peak still comes from the interaction on single nucleons. Incomparison with the exclusive (e, e′p) process, the inclusive (e, e′) scatteringcorresponds to an integral over all available nuclear states and consequentlyprovides less specific information, but it is more directly related to the dy-namics of the initial nuclear ground state. The width of the QE peak cangive a direct measurement of the average momentum of nucleons in nuclei, theshape depends on the distribution in energy and momentum of the initiallybound nucleons. Precise measurements can give direct access to integratedproperties of the nuclear spectral function which describes this distribution. Aconsiderably body of QE data for light-to-heavy nuclei in different kinematicsituations has been collected [42, 68, 69]. Not only differential cross sections,but also the contribution of the separate longitudinal and transverse responsefunctions have been considered. From the theoretical point of view, many ef-forts have been devoted to the description of the available data and importantprogress has been achieved in terms of experimental results and theoreticalunderstanding [42, 58, 68].

3.3 Elastic and Quasi-Elastic Scattering on Ex-

otic Nuclei

The use of electron scattering can be extended to the study of exotic nucleifar from stability line. The evolution of nuclear properties with the increasingasymmetry between the number of protons and neutrons is one of the mostinteresting topics of nuclear physics. Of particular interest is the behaviorof the single-particle properties, with a consequent modification of the shellmodel magic numbers. The exclusive (e, e′p) reaction would be the best suitedtool for this study [70,71]. However, the measurement of (e, e′p) cross sectionsrequires a double coincidence detection which is very difficult. Experimentsfor elastic scattering, and possibly inclusive QE scattering, appear easier toperform and are therefore to be considered as a first step.

In the next years radioactive ion beam (RIB) facilities [72–74] will producea large amount of data on unstable nuclei. In particular, electron-RIB colliders

33


using storage rings are under construction at RIKEN (Japan) [75–77] and GSI(Germany) [78]. Proposals have been presented for the ELISe experiment atFAIR in Germany [79–81] and the SCRIPT project in Japan [82, 83].

From the theoretical point of view, several studies of electron scatteringon unstable nuclei have already been published [70, 71, 84–99]. In Ref. [3] wehave presented and discussed numerical predictions for elastic and inclusive QEelectron scattering on oxygen and calcium isotopic chains, with the aim of in-vestigating the evolution of some nuclear properties with increasing asymmetrybetween the number of neutrons and protons. The elastic electron scatteringgives information on the global properties of a nucleus and, in particular, onthe charge density distributions and on the properties of proton wave functions.The inclusive QE scattering is the integral of the spectral density function overall the available final states. As such, it is affected by the dynamical propertiesand preferably exploits the nuclear single-particle aspects.

It is much more difficult to measure neutron density distributions. Directaccess to the neutron distribution can be obtained from the parity-violatingasymmetry parameter Apv, which is defined as the difference between the crosssections for the scattering of right- and left-handed longitudinally polarizedelectrons [100–102]. This quantity is related to the radius of the neutron dis-tribution Rn, because Z0-boson exchange, which mediates the weak neutralinteraction, couples mainly to neutrons and gives a model-independent mea-surement of Rn.

The first measurement of Apv was performed by the PREX experiment[103,104] on 208Pb with a poor resolution. More stringent results are expectedfrom an improved experiment (PREX-II [105]) which has been recently ap-proved. A recent result for the neutron skin on 208Pb has been extracted fromcoherent pion photoproduction cross sections measured at MAMI Mainz elec-tron microtron [106]. These data, combined with the results of the CREXexperiment on 48Ca [107], which has also been conditionally approved, willprovide an important test on the validity of microscopic models concerningthe dependence of Rn on the mass number A and the properties of the neutronskin.

In Ref. [3] we have compared our calculations for the asymmetry parameterApv with the result of the first PREX experiment on 208Pb and have obtained agood agreement with the empirical value. Moreover, we have provided numer-ical predictions for the future experiment CREX on 48Ca and we have studiedthe behavior of Apv along oxygen and calcium isotonic chains. In Ref. [4] weextended our study to isotonic chains, in which we presented and discussednumerical predictions for the cross sections of elastic and inclusive QE elec-tron scattering and for the parity-violating asymmetry parameter Apv on theN = 14, 20, and 28 isotonic chains. Electron scattering on an isotopic chaingives information on the dependence of the charge density distribution andof the proton wave functions on the neutron number. In an isotonic chainwe can investigate the behavior of the charge distribution and of the proton

34

3.3. Elastic and Quasi-Elastic Scattering on Exotic Nuclei

single-particle states when a new proton is added to the nucleus. Nuclei withboth neutron and proton excess are considered in our study. Our choice of theisotonic chains is motivated by the fact that data for light nuclei, such as thosewith N = 14, are likely to be obtained in future electron scattering facilitiessuch as SCRIT and ELISe; the N = 20 and 28 isotonic chains correspond to nu-clei with an intermediate mass and a magic number of neutrons. Medium-sizenuclei are very interesting to study the details of nuclear forces. In particu-lar, semi-magic N = 20 and 28 isotones are composed by a moderately largenumber of nucleons with orbits clearly separated from the neighboring ones, sothat smooth but continuous modifications in nuclear shapes can be observedwhen protons are added or removed. Therefore, the peculiar features of nuclearforces between nucleons moving in orbits with specific quantum numbers aremuch easier to disentangle in medium than in heavy nuclei, where the mod-ification of shell structures occurs only when many nucleons are involved. Arecent review of the evolution of the N = 28 shell closure far from stabilitycan be found in Ref. [40].

The basic ingredients of the calculations for both elastic and QE scatter-ing are the ground-state wave functions of proton and neutron single-particlestates. Models based on the RMF approximation have been successfully ap-plied in analyses of nuclear structure from light to superheavy nuclei. RMFmodels are phenomenological because the parameters of their effective La-grangian are adjusted to reproduce the nuclear matter equation of state anda set of global properties of spherical closed-shell nuclei [5, 6, 8, 108]. In re-cent years effective hadron field theories with additional non-linear terms anddensity-dependent coupling constants adjusted to Dirac-Brueckner self-energiesin nuclear matter have been able to obtain a satisfactory nuclear matter equa-tion of state and to reproduce the empirical bulk properties of finite nuclei [1].To obtain a proper description of open-shell nuclei a unified and self-consistenttreatment of mean-field and pairing correlations is necessary. As stated inChapter 2, the RHB model provides a unified treatment of the nuclear mean-field and pairing correlations, which is crucial for an accurate description ofthe properties of the ground and excited states in weakly bound nuclei, andhas been successfully employed in the analyses of exotic nuclei far from thevalley of stability. For most nuclei here considered, however, pairing effectsare negligible or small and the RHB model gives wave functions practicallyequivalent to the ones that can be obtained with the relativistic Dirac-Hartreemodel used in Ref. [3].

The cross sections for elastic electron scattering are obtained solving thepartial wave Dirac equation and include Coulomb distortion effects. For the in-clusive QE electron scattering, calculations are performed with the relativisticGreen’s function (RGF) model, which has already been widely and successfullyapplied to the analysis of QE electron and neutrino-nucleus scattering data ondifferent nuclei [109–116].

In Section 3.4 we present the theoretical results for the elastic scatter-

35


ing cross sections calculated for oxygen and calcium isotopic chains and forN = 14, 20, and 28 isotonic ones. In order to give a better description of theexperimental data, the calculations are performed in the distorted-wave Bornapproximation (DWBA) in order to properly include Coulomb distortions inthe model. In Section 3.5 we present the theoretical results for the elasticscattering parity-violating asymmetry. First of all we show the predictions forboth isotopic and isotonic chains considered in Section 3.4 and then we showthe theoretical results for 208Pb compared with the experimental measurementof PREX [103, 104] and Mainz [106]. In Section 3.6 we present the theoret-ical results for the inclusive QE scattering for the same chains considered inthe previous sections. They are computed with the RGF model, developedand successfully employed for describing experimental data in the QE region.Finally, in Section 3.7 we summarize our results and present our conclusions.

3.4 Elastic Scattering

Measuring the angular distribution of an elastically scattered probe is the firstsource of information on a system and, in particular, on its size, charge andmatter distribution, magnetization density and the role of its constituents. Inthe case of electron scattering this information is contained in the differentialcross section in terms of the so called electromagnetic form factors.

3.4.1 Theoretical Framework

In the one-photon exchange approximation and neglecting the effect of thenuclear Coulomb field on incoming and outgoing electrons, i.e., in the plane-wave Born approximation (PWBA), the differential cross section for the elasticscattering of an electron with kinetic energy E and momentum transfer q offa spherical spin-zero nucleus with Ze charge is given by

dσ

dΩ= σM |Fp(q)|2 , (3.1)

where Ω is the scattered electron solid angle, σM is the Mott cross section[42, 58]

σM(θ) =

(

Ze2

2E

)2cos2(θ/2)

sin4(θ/2), (3.2)

and

Fp(q) =

∫

d3r j0(qr)ρch(r) (3.3)

is the charge form factor for a spherical nuclear charge density ρch(r) and j0is the zeroth-order spherical Bessel function. The nuclear charge density isdefined as

ρch(r) =

∫

dr′GE(|r − r′|) ρp(r′) , (3.4)

36

3.4. Elastic Scattering

30 40 50 60 70-910

-810

-710

-610

-510

-410

-310

-210

-110 [

mb

/sr]

Ω/dσd

[deg]θ

10 ×O 16

Ca 40

O data 16

Ca data 40

Figure 3.1: Differential cross section for elastic electron scattering on 16O atan electron energy ε = 374.5 MeV and 40Ca at ε = 496.8 MeV as a function ofthe scattering angle θ. Experimental data from [119] (16O) and [120] (40Ca).

where ρp is the point proton density obtained from a RMF model and theelectric form factor of the proton is

GE(r) =Λ3

8πe−Λr , Λ = 4.27 fm−1 . (3.5)

The PWBA is, however, not adequate for medium and heavy nuclei where thedistortion produced on the electron wave functions by the nuclear Coulomb po-tential V (r) from ρch(r) can have significant effects. The DWBA cross sectionsare obtained from the numerical solutions of the partial-wave Dirac equation(see the seminal works [117, 118]).

3.4.2 Results

The cross sections for elastic electron scattering have been calculated in theDWBA and with the self-consistent relativistic ground state charge densitiescomputed with the RMF models described in Chapter 2. In the PWBA thecross section is proportional to the Fourier transform of the proton chargedensity (see Eq. (3.1)) and reflects its behavior also when Coulomb distortionis included in the calculations. In different studies of the charge form factorsalong isotopic chains [89, 93, 95, 123, 124] it has been found that, when thenumber of neutrons increases, the squared modulus of the charge form factorand the position of its minima show, respectively, an upward trend and asignificant inward shifting in the momentum transfer.

37


30 40 50 60 70 80 90 100 110 120 130θ [deg]

1×10-7

1×10-5

1×10-3

1×10-1

1×101

1×103

dσ/d

Ω [

mb/

sr]

40Ca

42Ca x 10

1

44Ca x 10

2

48Ca x 10

3

(a)

Figure 3.2: Differential cross sections for elastic electron scattering on calciumisotopes at an electron energy ε = 250 MeV as functions of the scattering angleθ. Experimental data from [121].

An example of the comparison [3] between theoretical and experimentaldifferential cross sections is displayed in Fig. 3.1 for elastic electron scatteringon 16O at an electron energy ε = 374.5 MeV and on 40Ca at ε = 496.8 MeV.The general trend of the experimental data is reasonably reproduced by thecalculations. Both experimental cross sections considered in the figure are welldescribed at low scattering angles. For 40Ca there is a fair agreement betweentheory and data also at larger angles, while for 16O data beyond the minimumare somewhat underestimated by the theoretical results. Other examples of thecomparison between theoretical and experimental differential cross sections areshown in Fig. 3.2 for four calcium isotopes (40,42,44,48Ca) at an electron energyε = 250 MeV and in Fig. 3.3 for three chromium isotopes (50,52,54Cr) at ε = 200MeV and for 48Ti at ε = 250 MeV. Also in these cases the DWBA calculationsare able to reproduce the general trend of the data and give a good descriptionof the experimental cross sections considered, except for small discrepancies atlarge scattering angles.

The calculated differential cross sections for elastic electron scattering onvarious oxygen isotopes (14−28O) at ε = 374.5 MeV and on calcium isotopes(36−56Ca) at ε = 496.8.5 MeV [3] are shown in Figs. 3.4 and 3.5, respectively.With increasing neutron number the positions of the diffraction minima shifttowards smaller scattering angles, i.e., towards smaller values of the momentumtransfer. The shift of the minima towards smaller q is in general accompanied

38


30 40 50 60 70 80 90 100θ [deg]

1×10-7

1×10-5

1×10-3

1×10-1

1×101

1×103

dσ/d

Ω [

mb/

sr]

50Cr

52Cr x 10

1

54Cr x 10

2

48Ti

(b)

Figure 3.3: Differential cross sections for elastic electron scattering on 50,52,54Crat ε = 200 MeV and 48Ti at ε = 250 MeV as functions of θ. Experimentaldata from [121] (48Ti) and [122] (50,52,54Cr).

20 40 60 80 100 120

-1110

-1010

-910

-810

-710

-610

-510

-410

-310

-210

-1101

[m

b/s

r]Ω

/dσd

[deg]θ

O 14

O 16

O18

O 20

O 22

O 24

O 26

O 28

Figure 3.4: Differential cross section for elastic electron scattering on 14−28Oat ε = 374.5 MeV as a function of θ.

39


20 30 40 50 60 70 80 90-1110

-1010

-910

-810

-710

-610

-510

-410

-310

-210

-110

[m

b/s

r]Ω

/dσd

[deg]θ

Ca 36

Ca 38

Ca40

Ca 42

Ca 44

Ca48

Ca 52

Ca 56

Figure 3.5: Differential cross section for elastic electron scattering on 36−56Caat ε = 496.8 MeV as a function of θ.

by a simultaneous increase in the height of the maxima. The behavior is similarfor both isotopic chains here considered and is in agreement with the resultsfound in previous studies of charge form factors on various isotopic chains,which were carried out with different mean-field models.

The theoretical results for elastic electron scattering on the N = 28, 20 and14 isotonic chains [4] at ǫ = 850 MeV are shown in Figs. 3.6, 3.7, and 3.8.Many of these nuclei lie in the region of the nuclear chart that is likely to beexplored in future electron-scattering experiments.

The differential cross sections are presented in panel (a) of Figs. 3.6, 3.7,and 3.8. These cross sections have been calculated at ε = 850 MeV in theDWBA and with the self-consistent relativistic ground-state charge densities.Although Coulomb distortion is included in the calculations, the elastic crosssections are still related to the behavior of the corresponding proton chargedensity as in Eq. (3.1).

With increasing proton number along the chain the positions of the diffrac-tion minima usually shift toward smaller scattering angles, i.e., smaller valuesof the momentum transfer q. The shift of the minima is also accompaniedby a simultaneous increase in the height of the corresponding maxima of thecross sections. The differential cross section provides information on the pro-ton density distribution and we can look for possible correlations between thecross sections that can be directly related to the behavior of the proton dis-tributions. In panel (b) of Figs. 3.6, 3.7, and 3.8 we plot the evolution of theposition of the first minimum of the elastic cross sections for each isotone inthe chain. For the N = 28 isotonic chain in panel (b) of Fig. 3.6 there is an

40


10 20 30 40 50θ [deg]

1×10-11

1×10-9

1×10-7

1×10-5

1×10-3

1×10-1

1×101

dσ/d

Ω [

mb/

sr]

40Mg

42Si

44S

46Ar

48Ca

50Ti

52Cr

54Fe

56Ni

850 MeV

(a)

14.5 15 15.5 16 16.5 17 17.5θ

min [deg]

1×10-4

1×10-3

1×10-2

1×10-1

dσ/d

Ω [

mb/

sr]

N=28 first minimum

40Mg

42Si

44S

46Ar

48Ca

50Ti

52Cr

54Fe

56Ni

slope = -1.43

slope = -2.51

(b)

Figure 3.6: Panel (a): differential cross sections along the N = 28 isotonicchain for elastic electron scattering at ε = 850 MeV as functions of θ. Panel(b): evolution of the first minimum of the differential cross section with thescattering angle θ.

41


10 20 30 40 50

θ [deg]1×10

-10

1×10-8

1×10-6

1×10-4

1×10-2

1×100

1×102

dσ/d

Ω [

mb/

sr]

28O

30Ne

32Mg

34Si

36S

38Ar

40Ca

42Ti

44Cr

46Fe

850 MeV

(a)

15 16 17 18 19 20θ

min [deg]

1×10-5

1×10-4

1×10-3

1×10-2

1×10-1

dσ/d

Ω [

mb/

sr]

N=20 first minimum

28O

30Ne

32Mg

34Si

36S

38Ar

40Ca

42Ti

44Cr

46Fe

slope = - 1.17

slope = -1.72

(b)

Figure 3.7: The same as in Fig. 3.6 but for the N = 20 isotonic chain.

42


10 20 30 40 50θ [deg]

1×10-10

1×10-8

1×10-6

1×10-4

1×10-2

1×100

dσ/d

Ω [

mb/

sr]

22O

24Ne

26Mg

28Si

30S

32Ar

34Ca

850 MeV

(a)

17 18 19 20 21 22θ

min [deg]

1×10-5

1×10-4

1×10-3

1×10-2

dσ/d

Ω [

mb/

sr]

N=14 first minimum

22O

24Ne

26Mg

28Si

30S

32Ar

34Ca

slope = - 1.08

slope = - 1.35

(b)

Figure 3.8: The same as in Fig. 3.6 but for the N = 14 isotonic chain.

43


evident transition between proton-poor and proton-rich isotones and it is notpossible to fit the positions of the first minima with a straight line. In addi-tion, the minimum for 48Ca occurs at a larger scattering angle than for 46Ar.However, it is still possible to draw a straight line that connects the minimafor the lighter isotones up to 46Ar and another line for the heavier isotones.The slope of the two lines are different. The isotones from 48Ca up to 56Nihave similar proton densities at small r and present only small differences atlarge r, which are ascribed to the filling of the 1f7/2 shell. In the case of thelighter isotones, we see that the 46Ar and 44S minima do not lie on the fittinglines. This is related to the non-negligible contribution of the proton in the2s1/2 shell, which has occupation number 0.330 for 46Ar and 0.153 for 44S. Thedensitiy of 40Mg gets some small contribution from the 2s1/2 and the 1d3/2

shells but the minimum of its cross section is aligned with that of 42Si.For the N = 14 isotonic chain the evolution of the position of the first

minimum of the differential cross section for each isotone in the chain, whichis shown in panel (b) of Fig. 3.8, can adequately be described by the twodifferent straight lines drawn in the figure, one connecting the minima of thelighter isotones up to 28Si and the other one connecting the minima of theheavier isotones.

3.5 Parity-Violating Elastic Scattering

An accurate description of matter distribution in nuclei is a longstanding prob-lem in modern nuclear physics that has a wide impact on our understandingof nuclear structure. Whereas the charge distribution has been measured withhigh accuracy using electron-nucleus elastic scattering, so that the charge radiiare usually known with uncertainties lower than 1% [125, 126], our knowledgeof neutron distribution is considerably less precise. Several experiments ofneutron radius have been carried out in recent years [127–129], but the use ofhadronic probes produces uncertainties in the experimental results due to theassumptions of the models required to deal with the complexity of the stronginteraction.

When a photon is exchanged between two charged particles a Z0 boson isalso exchanged. At the energies of interest in electron scattering the strengthof the weak process mediated by the Z0 boson is negligible compared withthe electromagnetic strength. The role played by the Z0 exchange is thereforenot significant unless an experiment is set up to measure a parity-violatingobservable. While the electromagnetic interaction conserves parity, the weakinteraction does not and this is how we are sensitive to Z0 exchange in electronscattering.

The degree of parity violation can be measured by the parity-violatingasymmetry Apv, or helicity asymmetry, which is defined as the difference be-tween the cross sections for the scattering of electrons longitudinally polarizedparallel and antiparallel to their momentum. This difference arises from the in-

44

3.5. Parity-Violating Elastic Scattering

terference of photon and Z0 exchange. As it has been shown in Refs. [100,130],the asymmetry in the parity-violating elastic polarized electron scattering rep-resents an almost direct measurement of the Fourier transform of the neutrondensity.

3.5.1 Theoretical Framework

The electron spinor for elastic scattering on a spin-zero nucleus can be writtenas the solution of a Dirac equation with total potential

U(r) = V (r) + γ5A(r) , (3.6)

where V (r) is the Coulomb potential and A(r) is the axial potential thatresults from the weak neutral current amplitude and which depends on theFermi constant GF ≃ 1.16639 × 10−11 MeV−2, i.e.,

A(r) =GF

2√

2ρW (r) . (3.7)

The weak charge density ρW is related to the neutron density and it is defined

ρW (r) =

∫

dr′GE(|r − r′|)[

− ρn(r′) + (1 − 4 sin2 ΘW )ρp(r′)]

, (3.8)

where ρn and ρp are point neutron and proton densities, respectively, GE(r)is the electric form factor defined in Eq. (3.5), and sin2 ΘW ≃ 0.23 is theWeinberg angle. The axial potential of Eq. (3.7) is much smaller than thevector potential and, because 1 − 4 sin2 ΘW ≪ 1, it depends mainly on theneutron distribution ρn(r).

In the limit of vanishing electron mass, the helicity states

Ψ± =1

2(1 ± γ5) Ψ (3.9)

satisfy the Dirac equation

[α · p + U±(r)] Ψ± = EΨ± , (3.10)

with

U±(r) = V (r) ± A(r) . (3.11)

The parity-violating asymmetry Apv, or helicity asymmetry, is defined

Apv =

dσ+dΩ

− dσ−dΩ

dσ+dΩ

+dσ−dΩ

, (3.12)

45


0 5 10 15 20 25θ [deg]

0

1×10-5

2×10-5

Apv

14O

16O

18O

20O

22O

24O

26O

28O

Figure 3.9: Parity violating asymmetry parameter Apv for elastic electronscattering at ε = 850 MeV as function of the scattering angle θ on 14−28O.

where +/− refers to the elastic scattering on the potential U±(r). In Bornapproximation, neglecting strangeness contributions and the electric neutronform factor, the parity-violating asymmetry can be rewritten as [131, 132]

Apv =GF Q

2

4√

2πα

[

4 sin2 ΘW − 1 +Fn(q)

Fp(q)

]

. (3.13)

Because 4 sin2 ΘW − 1 is small and Fp(q) is known, we see that Apv providesa practical method to measure the neutron form factor Fn(q) and hence theneutron radius. For these reasons parity-violating electron scattering (PVES)has been suggested as a clean and powerful tool for measuring the spatialdistribution of neutrons in nuclei.

3.5.2 Results for Oxygen and Calcium Isotopic Chains

In the following we present the theoretical results for oxygen and calcium iso-topic chains. The calculation starts with the self-consistent relativistic groundstate proton and neutron densities. The charge and weak densities are calcu-lated by folding the point proton and neutron densities (see Eq. (3.8)). Theresulting Coulomb potential V (r) and weak potential A(r) (see Eq. (3.7))are used to construct U±(r). The cross sections for elastic electron scatter-ing are obtained from the numerical solution of the Dirac equation for elec-tron scattering in the U±(r) potential and includes Coulomb distortion ef-fects [89, 102, 133, 134]. The cross sections for positive and negative helicity

46


0 5 10 15 20θ [deg]

0

5×10-6

1×10-5

Apv

36Ca

38Ca

40Ca

42Ca

44Ca

46Ca

48Ca

50Ca

52Ca

54Ca

56Ca

Figure 3.10: The same as in Fig. 3.9, but for 36−56Ca. The black arrowemphasizes the evolution of Apv as a function of the neutron number N .

0 0.5 1 1.5

∆10

15

20

25

30

θ min

[de

g]

Calcium

850 MeVOxygen

Figure 3.11: First minimum positions of the asymmetry parameter Apv asfunctions of ∆ = (N − Z)/Z for 14−28O and 36−56Ca. The black dashed linesrepresent the best linear fit.

47


0 2 4 6r [fm]

0

0.05

0.1ρ n [

fm-3

]22

O24

O22

O (no gap)

10 20 30θ [deg]

0

1×10-5

2×10-5

Apv

850 MeV

(b)

(a)

Figure 3.12: Panel (a): neutron density distributions for some selected nuclei(22O and 24O) that could be interpreted as candidates for bubble structure[92]. Panel (b): parity violating asymmetry parameter Apv for elastic electronscattering at ε = 850 MeV as a function of the scattering angle θ.

electron states are calculated and the resulting asymmetry parameter Apv isplotted as a function of the scattering angle.

In Figs. 3.9 and 3.10 we plot the parity-violating asymmetry parametersApv for 14−28O and 36−56Ca nuclei for elastic electron scattering at ε = 850MeV. At ε = 850 MeV the values of Apv are of the order of 10−5, with lowervalues for smaller angles and larger values for larger angles.

As suggested in Ref. [100] the asymmetry parameter Apv provides a directmeasurement of the Fourier transform of the neutron density. This relationhas been tested and confirmed in Ref. [130] comparing asymmetries and thesquares of the Fourier transforms of the neutron densities. Another way torelate Apv to neutron distributions of finite nuclei is by looking at possiblelinear correlations between the asymmetry parameter and some well definedobservables. We suggest to use the first minima positions θmin and the neutronexcess ∆ = (N −Z)/Z, i.e., how the minima of Apv evolve from neutron-poorto neutron-rich nuclei. In Fig. 3.11, θmin is plotted as a function of ∆ foroxygen and calcium isotopes. The dashed lines suggest that for both isotopechains the evolution of Apv as function of ∆ is well approximated by a linearfit with a very similar slope. To test the robustness of this correlation it isinteresting to study if Apv is affected by density distribution oscillations atsmall radii that could appear in some selected cases. In Ref. [92], 22O and

48


24O isotopes have been studied as possible candidates for bubble nuclei, i.e.,nuclear systems with a strong depleted central density. In Fig. 3.12 we plotthe asymmetry parameters Apv for these nuclei. Neutron density profiles showlarge differences at small distances. No appreciable effects are obtained in thecorresponding asymmetries up to θ ≃ 20o, then for larger scattering anglesthe asymmetries are sensitive to the differences in the density distributionsand are significantly different. Therefore, Apv is still a reliable observable tostudy neutron radii even if we include pairing correlations, but we must limitto angles smaller than the first minimum position.

3.5.3 Neutron Density of 208Pb

In addition to predictions about oxygen and calcium isotopic chains we alsoprovide calculations [135] for recent measurements on 208Pb and future ex-periments. The PREX Collaboration [103] at JLab used PVES to study theneutron distribution of 208Pb and provided us with the first determination ofthe neutron radius through an electroweak probe that gives Rskin = 0.33+0.16

−0.18

fm for the neutron skin thickness. Although the total error is large, the PREXmethod is very interesting and future higher statistics data are expected toreduce the uncertainty [105]. The CREX Collaboration at JLab has made asuccessful proposal to measure the neutron radius of 48Ca using PVES with agoal of ±0.02 fm in accuracy [107]. Recently, in a measurement of the coherentπ0 photoproduction from 208Pb at Mainz [106], the shape of the neutron dis-tribution has been found to be 20% more diffuse than the charge distributionand the neutron skin thickness is Rskin = 0.15 ± 0.03 (stat) +0.01

−0.03 (syst) fm.This value is compatible with previous independent measurements, i.e., protonelastic scattering [129,136], x-ray cascade of antiprotonic atoms [127,128], anti-analog giant dipole resonances [137–139], giant quadrupole resonances [140],pigmy dipole resonance [141–144], electric dipole polarizability [145–147] orpionic probes [148].

The neutron skin of 208Pb has important implications for astrophysics [13,149,150], owing to its strong correlation with the pressure of neutron matter atdensities near 0.1 fm−3. The larger the pressure of neutron matter, the thickeris the skin as neutrons are pushed out against surface tension. The samepressure supports neutron stars against gravity, therefore correlations betweenneutron skins of neutron-rich nuclei and various neutron star properties arenaturally expected [151, 152]. In addition, the magnitude of Rskin in heavynuclei provides very interesting information on the nature of 3-body forces innuclei, nuclear drip lines and collective nuclear excitations, as well as heavy-ioncollisions. A recent review of experimental measurements of Rskin and theirtheoretical implications can be found in Ref. [153].

In this section we present and discuss numerical predictions for the neu-tron density distribution of 208Pb. In Ref. [3] we have already compared ourcalculations for the asymmetry parameter Apv using the relativistic DDME2interaction with the results of the first run of PREX on 208Pb and we have

49


provided numerical predictions for the future experiment CREX on 48Ca. InRef. [135] we have extended the work undertaken in Ref. [3] comparing resultsobtained with different non-relativistic and relativistic model interactions. Ourresults are compared with the recent (γ, π0) data from Mainz and with the dataof the PREX experiment.

The best description of heavy nuclei, at the moment, relies on energy den-sity functionals in terms of effective interactions calibrated on the bulk prop-erties of a limited set of nuclei. The isoscalar part of the interaction is usuallyconstrained by reproducing binding energies and charge radii (208Pb is usuallyincluded in fit protocol) where the isospin-dependent part of the interactionis mainly constrained reproducing some ab-initio equation of state (EOS) forneutron matter, like the Akmal-Friedmann-Pandharipande EOS [154], or theempirical value of the asymmetry energy at the saturation point. So far, the-oretical calculations based on realistic potentials are limited to medium-lightnuclei, even if new approaches based on renormalization group potentials lookvery promising [155]. Here we consider different non-relativistic and RMFmodels and compare their predictions for the neutron distribution of 208Pb.The details of the mean-field approaches we have adopted in our investigationare presented in various publications, for instance in [5, 8, 21, 26, 36, 156–165].We do not repeat here the derivation of the various expressions used in ourcalculations but we refer the readers to the original papers. Our strategy isto explore all variants of density functional approaches in terms of covariant(Walecka type) vs. non-covariant (Skyrme and Gogny) descriptions, finiterange vs. contact interactions and non-linear vs. density dependent couplings.

We have checked that the different forces adopted for our calculations givesome differences in the neutron single-particle levels around the Fermi surfacein 208Pb, but do not produce significant inversions in the energy levels. Thelevels above the N = 126 shell closure are unoccupied.

Generally, the nucleon distributions are parameterized as a single sym-metrised two-parameter Fermi distribution (2pF) [166] with half-height radiusR and diffuseness a. The analysis of the (γ, π0) cross sections data from Mainzgives R = 6.70 ± 0.03 fm and a = 0.55 ± 0.01 (stat) +0.02

−0.03 (syst) fm [106] andsuggests that the neutron distribution of 208Pb is ≈ 20% more diffuse than thecharge distribution and that the neutron skin of lead is of partial halo type.

We have computed the half-height radius and diffuseness parameter of the2pF neutron density distributions extracted from the different models. Neitherthe non-relativistic Gogny and Skyrme nor the RMF models are able to simul-taneously reproduce the experimental data for both R and a; similar findingsare obtained with the RMF models where the zero-range point-coupling in-teraction is used instead of the meson exchange or the relativistic functionalswith density-dependent vertex functions [167, 168].

To obtain a simple model of the neutron density distributions, we haveevaluated the weighted average of the parameters of the 2pF profiles extractedfrom the results of the different models and have obtained Rmean = 6.82±0.02

50


0 1 2 3 4 5 6 7 8 9 10 11 12r [fm]

0

0.02

0.04

0.06

0.08

0.1

- ρ W

eak [

fm-3

]NL3DDME1PCF1SLY4NL-SHSIIIMEAN

208Pb

Figure 3.13: Theoretical weak charge density in comparison with the exper-imental error band as determined in Ref. [104] for 208Pb with the kinematicsof the PREX experiment. The results have been obtained using Eq. 3.8 whereρn and ρp have been computed with the different mean-field models.

fm and amean = 0.56 ± 0.01 fm: the surface diffuseness is in fair agreementwith the Mainz data but the radius is a bit larger, however we are aware thatthe mean value of the half-height radius in 208Pb may strongly depend on theset of models that have been used. The errors in Rmean and amean have beenestimated from the standard deviation of the numerical values forR and a givenby the different models that we have considered. To be more confident, wehave checked that the 2pF profile of the charge distribution obtained with thisprocedure is able to satisfactorily reproduce the experimental data of elasticelectron scattering cross sections off 208Pb.

In Fig. 3.13 we show our theoretical predictions for the weak charge density(−ρW ) that has been deduced from the weak charge form factor [104,169]. Theorange error band represents the incoherent sum of experimental and modelerrors. Our predictions for different interactions are in rather good agreementwith the empirical data. In addition, the weak distribution evaluated using the2pF functions for the proton and neutron density distributions with weightedaverage parameters is also in good agreement with the data.

The PREX Collaboration measured the parity-violating asymmetry param-

51


5.5 5.55 5.6 5.65 5.7 5.75 5.8 5.85 5.9 5.95 6R

rms neutron [fm]

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

Apv

[pp

m]

DDME2

DDME1

L2NL-S

H

NL3-II

NL3

SIII (m

od)

SIIISLY5 (mod

)

SKM*

SLY5

SLY4

PC

MEAN

D1S

PKDD

DDPC1

Figure 3.14: Parity-violating asymmetry at the kinematics of the PREX ex-periments versus the neutron rms radius for 208Pb. The dashed orange line isa linear fit of the correlation between the neutron rms radius and Apv. Thered square shows the experimental data from PREX [103] with statistical andsystematic errors.

eter Apv averaged over the experimental acceptance function ǫ(θ) [170]

〈Apv〉 =

∫

dθ sin θApv(θ)dσ

dΩǫ(θ)

∫

dθ sin θdσ

dΩǫ(θ)

, (3.14)

where Apv(θ) and dσ/dΩ are the asymmetry and the differential cross section atthe scattering angle θ. The charge radius of 208Pb is very well known [125,126];therefore, the empirical estimate Apv = 0.656±0.060 (stat) ±0.014 (syst) ppmcan be related to the neutron radius and the neutron rms radius results Rn =5.78+0.16

−0.18 fm that implies that the neutron skin thickness is Rskin = 0.33+0.16−0.18

fm.The results for the parity-violating asymmetry Apv versus the neutron rms

radius for different models are displayed in Fig. 3.14. The result with the 2pFfunctions for the density distributions with averaged parameters is also in goodagreement with the data. An analysis in Ref. [171] shows that there is a verysignificant linear correlation between Apv and Rn. A comprehensive investi-gation in [172], performed before the actual results of PREX were published,confirms the correlation and, taking advantage of results of many different non-relativistic and RMF models, demonstrates that an accuracy of ∼ 3% in Apv

corresponds to ∼ 1% accuracy in Rn. The theoretical predictions in Fig. 3.14

52


0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5neutron skin [fm]

0.55

0.6

0.65

0.7

0.75

Apv

[pp

m]

DDME2

DDME1

NL-SH

L2

NL3NL3-II

PC

SIIISIII

(mod

)

SLY4SLY5

SLY5 (mod

) SKM*

MAINZ ( )γ,π0

ΜΕΑΝ

OSAKA-RCNP ( p, p )

D1S

PKDD

DDPC1

Figure 3.15: Parity-violating asymmetry at the kinematics of the PREX exper-iments versus the neutron skin for 208Pb. The red square shows the experimen-tal data from PREX [103] with statistical and systematic errors. The dashedorange line is a linear fit of the correlation between the neutron skin and Apv.The vertical solid green lines show the constraints on Rskin from Mainz (γ, π0)measurements [106]. The vertical purple dashed lines show the constraints onRskin from Osaka polarized proton elastic scattering measurements [129].

are in agreement with the data but they all predict a smaller radius than thecentral value of 5.78 fm. To obtain a significantly larger Rn and a smaller Apv

the Lagrangian density should contain also the mixed isoscalar-isovector cou-pling term as described in Ref. [173]. We observe, however, that a large neutronradius is not in agreement with other experimental measurements [153].

In Fig. 3.15 we present the results for the parity-violating asymmetry ver-sus the neutron skin predicted by the different models. Owing to the fact thatthe neutron skin is highly correlated with the neutron radius, these results aresimilar to those in Fig. 3.14. The constraints on the neutron skin from Mainz(γ, π0) [106], as well as those from Osaka polarized proton elastic scatteringmeasurements at proton energy ε = 295 MeV [129], are displayed for a com-parison. Although all the predictions in Figs. 3.14 and 3.15 are compatiblewith the PREX results, the large error bars prevent us from discriminatingamong some of them. Other Rskin measurements are more precise and seem torule out models with either very small or very large neutron skins. However,a careful analysis of all available data in Ref. [174] demonstrates that it is stillpremature to rule out the existence of a thick neutron skin in 208Pb.

53


3.6 Inclusive Quasi-Elastic Scattering

In the QE kinematic region the nuclear response to an electromagnetic probeis dominated by one-nucleon processes, where the scattering occurs with onlyone nucleon, which is then emitted by a direct knockout mechanism, and theremaining nucleons of the target behave as spectators. In electron scatter-ing experiments the emitted nucleon can be detected in coincidence with thescattered electron. Kinematic situations can be envisaged where the residualnucleus is left in a discrete eigenstate and the final state is completely deter-mined. This is the exclusive one-nucleon knockout and the (e, e′p) reaction hasbeen widely investigated [42]. If only the scattered electron is detected, thefinal nuclear state is not determined and the measured cross section includesall the available final states. This is the inclusive (e, e′) scattering.

Within the QE kinematic domain, electron scattering can adequately bedescribed by a model based on the non-relativistic or relativistic impulse ap-proximation (IA or RIA). For the exclusive scattering the IA assumes that theinteraction occurs through a one-body current only with a quasi-free nucleonwhich is then knocked out of the nucleus. For the inclusive scattering, the IAassumes that the cross section is given by the incoherent sum of one-nucleonknockout processes due to the interaction of the probe with all the nucleonsof the nucleus. With this assumption, we have the problem to describe theFSI between the emitted nucleon and the residual nucleus that are an essentialingredient for the comparison with data.

In the exclusive (e, e′p) reaction FSI are usually described in the distorted-wave impulse approximation (DWIA) by a complex optical potential where theimaginary part gives an absorption that reduces the calculated cross section.This reduction is essential to reproduce (e, e′p) data. Models based on a non-relativistic DWIA or a relativistic DWIA (RDWIA) are indeed able to give anexcellent description of (e, e′p) data [42, 58, 66, 70, 71, 175–179].

In the inclusive scattering a model based on the DWIA or the RDWIA,where the cross section is given by the sum of all integrated one-nucleon knock-out processes and FSI are described by a complex optical potential with animaginary absorptive part, is conceptually wrong. The optical potential de-scribes elastic NA scattering and its imaginary part accounts for the fact that,if other channels are open besides the elastic one, part of the incident flux islost in the elastically scattered beam and goes to the inelastic channels whichare open. In the exclusive reaction, where only one channel is considered, itis correct to account for the flux lost in the selected channel. In the inclusivescattering all the final-state channels are included, the flux lost in a channelmust be recovered in the other channels, and in the sum over all the channelsthe flux can be redistributed but must be conserved. The DWIA and RDWIAdo not conserve the flux.

Different approaches have been used to describe FSI in the RIA calculationsfor the inclusive QE electron- and neutrino-nucleus scattering [109–114, 116,180–189]. In the relativistic plane-wave impulse approximation (RPWIA) the

54

3.6. Inclusive Quasi-Elastic Scattering

plane-wave approximation is assumed for the emitted nucleon wave functionand FSI are simply neglected. In other models FSI are incorporated in theemitted nucleon states by using real potentials, either retaining only the realpart of the relativistic optical potential (rROP) [188, 189] or using the sameenergy-independent RMF potential that describes the initial nucleon state [112,180, 184, 190]. Both the rROP and RMF conserve the flux, but the rROP isconceptually wrong because the optical potential has to be complex, owingto the presence of open inelastic channels. Its real and imaginary parts arerelated by dispersion relations and a model that arbitrarily omits a part isconceptually wrong. We note that the RMF fulfills the dispersion relationsand maintains the continuity equation.

For the calculations presented in this section we have employed the RGFmodel, in which FSI are described in the inclusive process consistently with theexclusive scattering by the same complex optical potential, but the imaginarypart is used in the two cases in a different way and in the inclusive scatteringit redistributes the flux in all the channels and the total flux is conserved.Detailed discussions of the RGF model can be found in Refs. [109–116,191,192].In the next subsection we give only the main features of the model. We notethat the model assumes that the ground state of the nucleus |Ψi〉 is non-degenerate; this is a suitable approximation for the even isotopes of oxygenand calcium considered in this work, with spin and parity 0+ [193].

3.6.1 Relativistic Green’s Function Model

In the one-photon exchange approximation the inclusive differential cross sec-tion for the QE (e, e′) scattering on a nucleus is obtained from the contractionbetween the lepton tensor Lµν and the hadron tensor W µν as

dσ

dǫ dΩ= σM

[

vLRL + vTRT

]

. (3.15)

The coefficients v come from the lepton tensor whose components are givenby products of the matrix elements of the electron current between initial andfinal electron states and that, under the assumption of the plane-wave approx-imation for the electron wave functions, depend only on the lepton kinematics,

vL =

( |Q2||q|2

)2

, vT = tan2 θ

2− |Q2|

2|q|2. (3.16)

All nuclear structure information is contained in the longitudinal and trans-verse response functions RL and RT , expressed by

RL(q, ω) = W 00(q, ω) , RT (q, ω) = W 11(q, ω) +W 22(q, ω) , (3.17)

in terms of the diagonal components of the hadron tensor, which is given bybilinear products of the transition matrix elements of the nuclear electromag-netic many-body current operator Jµ between the initial state of the nucleus

55


|Ψi〉, of energy Ei, and the final states |Ψf〉, of energy Ef , both eigenstates ofthe nuclear Hamiltonian H , as

W µν(q, ω) =∑

i

∑

∫

f

〈Ψf |Jµ(q)|Ψi〉〈Ψi|Jν†(q)|Ψf〉 δ(Ei + ω − Ef ) , (3.18)

involving an average over the initial states and a sum over the undetected finalstates. The sum runs over the scattering states corresponding to all of theallowed asymptotic configurations and includes possible discrete states [42,58].For the inclusive scattering the diagonal components of the hadron tensor canequivalently be expressed as

W µµ(q, ω) = −1

πIm 〈Ψi|Jµ†(q)G(Ef)Jµ(q)|Ψi〉 , (3.19)

where Ef = Ei+ω and G(Ef ) is the Green’s function, the full A-body propaga-tor, which is related to the many body nuclear Hamiltonian. A similar but morecumbersome expression is obtained for the non-diagonal components [110].

The A-body Green’s function in Eq. (3.19) defies a practical evaluation.Some approximations are required to reduce the problem to a tractable form.With suitable approximations, which are basically related to the IA, the com-ponents of hadron tensor can be written in terms of the single particle optical-model Green’s function. This result has been derived by arguments basedon multiple scattering theory [194] or by means of projection operators tech-niques [109, 110, 191, 192]. In the latter framework, the matrix element inEq. (3.19) is decomposed into the sum

Im 〈Ψi|Jµ†GJµ|Ψi〉 ∼ A∑

n

Im 〈Ψi|Jµ†GnJµ|Ψi〉 , (3.20)

where

Gn = PnGPn , Pn =

∫

dr |r;n〉〈n; r| (3.21)

is the projection of the full Green’s function onto the channel subspace spannedby the orthonormalized set of states |r;n〉, corresponding to a nucleon at thepoint r and the residual nucleus in the state |n〉. The n-th operator Gn is thethe Green’s function associated with the optical potential Hamiltonian whichdescribes the elastic scattering of a nucleon by the (A−1)-nucleus in the state|n〉. The sum over n includes discrete eigenstates and resonances embedded inthe continuum [109, 191].

The approximation in Eq. (3.20) has been derived neglecting non-diagonalterms PnGPm and retaining only the one-body part Jµ of the current operator.It is basically a single-particle approach where one assumes that Jµ connectsthe initial state |Ψi〉 only with states in the channel subspace spanned by thevectors |r;n〉, i.e., only with states asymptotically corresponding to single-nucleon knockout. However, not only these states, but all the allowed final

56


states are included in the inclusive response, as Gn in Eq. (3.21) contains thefull propagator G: all the allowed final states are taken into account by theoptical potential, in particular by its imaginary part.

The complexity of an explicit calculation of Gn can be avoided by meansof its spectral representation, which is based on a biorthogonal expansion interms of the eigenfunctions of the non-Hermitian optical potential and of itsHermitian conjugate [109, 191]. In the single-particle representation one ob-tains

W µµ(q, ω) =∑

n

[

Re T µµn (Ef − εn, Ef − εn)

− 1

πP∫ ∞

M

dE 1

Ef − εn − E ImT µµn (E , Ef − εn)

]

,

(3.22)

where P denotes the principal value of the integral, n is the eigenstate of theresidual nucleus with energy ǫn, and

T µµn (E , E) = λn 〈ϕn|jµ†(q)

√

1 − V ′(E)|χ(−)E (E)〉

× 〈χ(−)E (E)|

√

1 − V ′(E)jµ(q)|ϕn〉 .(3.23)

In Eq. (3.23) χ(−)E and χ

(−)E are eigenfunctions, belonging to the eigenvalue

E , of the single-particle optical potential and of its Hermitian conjugate, ϕn

is the overlap between |Ψi〉 and |n〉, i.e., a single-particle bound state, andthe spectroscopic factor λn is the norm of the overlap function. The factor√

1 − V ′(E), where V ′(E) is the energy derivative of the optical potential,accounts for interference effects between different channels and justifies thereplacement in the calculations of the Feshbach optical potential V by thelocal phenomenological one.

Disregarding the square root correction, the matrix elements in Eq. (3.23)are the transition amplitudes of the usual RDWIA model for the exclusive(e, e′p) reaction, but both eigenfunctions χ

(−)E and χ

(−)E of V(E) and of V†(E)

are considered. In the exclusive scattering the imaginary part of the opticalpotential accounts for the flux lost in the channel n towards the channels dif-ferent from n, which are not included in the exclusive process. In the inclusiveresponse, where all the channels are included, this loss is compensated by acorresponding gain of flux due to the flux lost, towards the channel n, in theother final states asymptotically originated by the channels different from n.This compensation is performed by the first matrix element in the right handside of Eq. (3.23), which involves the eigenfunction χ

(−)E of the Hermitian con-

jugate optical potential, where the imaginary part has an opposite sign andhas the effect of increasing the strength. Therefore, in the RGF model theimaginary part of the optical potential redistributes the flux lost in a channelin the other channels, and in the sum over n the total flux is conserved. TheRGF model allows to recover the contribution of non-elastic channels starting

57


50 100 150 200 250 300 3500

2

4

6

8

10

12

14

[n

b/(

MeV

sr)

]σ

[MeV]ω

O(e,e’)16

(a)

O data16

RGF

50 100 150 200 250 3000

2

4

6

8

10

12

14

16

[n

b/(

MeV

sr)

]σ

[MeV]ω

Ca(e,e’)40

(b)

Ca data40

RGF

Figure 3.16: Differential cross section of the reactions 16O(e, e′), panel (a),and 40Ca(e, e′), panel (b), for different beam energies and electron scatteringangles, ε = 1080 MeV and θ = 32o for 16O(e, e′) and ε = 841 MeV andθ = 45.5o for 40Ca(e, e′), as a function of the energy transfer ω. The RGFresults are compared with the experimental data from [195] (16O(e, e′)) and[196] (40Ca(e, e′)).

from the complex optical potential that describes elastic NA scattering dataand provides a consistent treatment of FSI in the exclusive and in the inclusivescattering.

3.6.2 Results

The theoretical results for isotopic and isotonic chains are presented and dis-cussed in the following. Some results obtained in the RPWIA are also presentedfor a comparison. In the calculations of the matrix elements in Eq. (3.23) thesingle-particle bound nucleon states are obtained from the RMF model withdensity-dependent meson-nucleon vertices and the DDME2 parametrization asdescribed in Chapter 2. The single-particle scattering states are eigenfunctionsof the energy-dependent and A−dependent (A is the mass number) parame-

58


50 100 150 200 250 300 35002468

1012141618

[n

b/(

MeV

sr)

]σ

[MeV]ω

RPWIA

(a)

O14

O16

O 18

O 22

O28

50 100 150 200 250 300 35002468

1012141618

[n

b/(

MeV

sr)

]σ

[MeV]ω

RPWIA

(b)

O proton14

O proton16

O proton18

O proton22

O proton 28

O neutron 14

O neutron 16

O neutron 18

O neutron22

O neutron 28

Figure 3.17: In panel (a) the differential RPWIA cross section for the inclusiveQE (e, e′) reaction on 14−28O at ε = 1080 MeV and θ = 32o is shown as afunction of ω. In panel (b) the separate contributions of protons (thick lines)and neutrons (thin lines) are displayed.

terization for the relativistic optical potential of Ref. [197], which is fitted toproton elastic scattering data on several nuclei in an energy range up to 1040MeV. The different number of neutrons along the O and Ca isotopic chainsproduces different optical potentials (see Ref. [197] for more details). For thesingle-nucleon current we have used the relativistic free nucleon expressiondenoted as CC2 [109, 198].

The predictions of the RGF model have been compared with experimentaldata for QE electron- and neutrino-nucleus scattering in a series of papers [109–116,182,186], where the calculations have been performed with different RMFmodels for the bound states and different parameterizations of the relativisticoptical potential.

In Fig. 3.16 our RGF results are compared with the experimental (e, e′)cross sections for two different kinematics on 16O and 40Ca target nuclei [195,196]. The agreement with the data is satisfactory, at least in the energy re-gion of the QE peak. Above the QE peak, at larger energy transfer, other

59


50 100 150 200 250 300 35002468

1012141618

[n

b/(

MeV

sr)

]σ

[MeV]ω

RGFO14

O16

O 18

O 22

O28

Figure 3.18: Differential RGF cross section for the inclusive QE (e, e′) reactionon 14−28O in the same kinematics as in Fig. 3.17.

50 100 150 200 25002468

101214161820

[n

b/(

MeV

sr)

]σ

[MeV]ω

RPWIACa

36

Ca40

Ca44

Ca 48

Ca 52

Ca56

Figure 3.19: Differential RPWIA cross section for the inclusive QE (e, e′)reaction on 36−56Ca at ε = 560 MeV and θ = 60o as a function of ω.

contributions of not QE type come into play and RGF results underpredictthe experimental data. The RGF model was developed to describe FSI in in-clusive QE electron scattering and is in general able to give a reasonable andeven good description of QE data. For energy regions below and above the QEpeak other contributions, not included in the RGF model, can be important.Even in the QE region, the relevance of contributions like meson exchange cur-rents and ∆ effects should be carefully evaluated before definite conclusionscan be drawn about the comparison with data [199–201]. Such contributionsmay be significant even in the QE region, in particular in kinematics where thetransverse component of the nuclear response plays a major role in the crosssection.

The cross section of the inclusive QE (e, e′) reaction on 14−28O isotopes atε = 1080 MeV and θ = 32o are shown in Fig. 3.17. In a first approximation, we

60


50 100 150 200 25002468

10121416182022

[n

b/(

MeV

sr)

]σ

[MeV]ω

RGFCa

36

Ca40

Ca44

Ca 48

Ca 52

Ca56

Figure 3.20: The same as in Fig. 3.19, but in the RGF model.

have neglected FSI and calculations have been performed in the RPWIA. Inthis case, the differences between the results for the various isotopes are entirelydue to the differences in the single particle bound state wave functions of eachisotope. While only the charge proton density distribution contributes to thecross section of elastic electron scattering, the cross section of QE electronscattering is obtained from the sum of all the integrated exclusive one-nucleonknockout processes, due to the interaction of the probe with all the individualnucleons, protons and neutrons, of the nucleus and contains information onthe dynamics of the initial nuclear ground state. The separate contributionsfrom protons and neutrons are also shown in the lower panel of Fig. 3.17. Inan usual experiment where only the scattered electron is detected these twoquantities cannot be separated experimentally, but their comparison can giveuseful information on the different role of protons and neutrons in the inclusiveQE cross section. The main role is played by protons, which give most of thecontribution. Increasing the neutron number it is quite natural to understandthe proportional increase of the neutron contribution. No significant increase isfound in the proton contribution. Thus, the increase of the cross section in theupper panel of the figure is due to the increase of the neutron contribution. Theshift of the proton contribution towards higher values of ω seen in the figure ismainly related to the increase of the proton separation energy with increasingneutron number (increasing the neutron number the protons experience morebinding and their separation energies increase) than to changes in the protonwave functions. A different and opposite shift can be seen in the case of theneutron contribution and, therefore, the final effect is that the shift is stronglyreduced in the QE cross section shown in the upper panel of Fig. 3.17.

In Fig. 3.18 we show the QE (e, e′) cross sections calculated for oxygenisotopes with the RGF model and in the same kinematics as in Fig. 3.17. Thegeneral trend of the cross sections, their magnitude, and their evolution withrespect to the change of the neutron number are generally similar in RPWIA

61


50 100 150 200 250 300 350 4000

5

10

15

20

25

30

35

40

45

50

[nb/(MeV sr)]

σ

[MeV]ω

RPWIA

(a)

Mg 40

Si 42

S 44

Ar 46

Ca 48

Ti 50

Cr 52

Fe 54

Ni 56

50 100 150 200 250 300 350 4000

5

10

15

20

25

30

35

40

45

[nb/(MeV sr)]

σ

[MeV]ω

RPWIA

(b)

O 28

Ne 30

Mg 32

Si 34

S 36

Ar 38

Ca 40

Ti 42

Cr 44

Fe 46

50 100 150 200 250 300 350 4000

5

10

15

20

25

30

35

[nb/(MeV sr)]

σ

[MeV]ω

RPWIA

(c)

O 22

Ne 24

Mg 26

Si 28

S 30

Ar 32

Ca 34

Figure 3.21: Differential RPWIA cross sections for the inclusive QE (e, e′)reaction on isotones with N = 28, 20, and 14 in panels (a), (b), and (c) atε = 1080 MeV and θ = 32o as functions of the energy transfer ω.

62


50 100 150 200 250 300 350 4000

10

20

30

40

50

[nb/(MeV sr)]

σ

[MeV]ω

RGF

(a)

Mg 40

Si 42

S 44

Ar 46

Ca 48

Ni 56

50 100 150 200 250 300 350 4000

5

10

15

20

25

30

35

40

45

[nb/(MeV sr)]

σ

[MeV]ω

RGF

(b)

O 28

Si 34

S 36

Ar 38

Ca 40

Fe 46

50 100 150 200 250 300 350 4000

5

10

15

20

25

30

35

[nb/(MeV sr)]

σ

[MeV]ω

RGF

(c)

O 22

Ne 24

Mg 26

Si 28

S 30

Ar 32

Ca 34

Figure 3.22: Differential RGF cross sections for the inclusive QE (e, e′) reactionon some selected isotones with N = 28, 20, and 14 in panels (a), (b), and (c)at ε = 1080 MeV and θ = 32o as functions of ω.

63


and RGF. The FSI effects in the RGF calculations produce, however, somedifferences which can be seen in the low energy transferred region, where thecross sections for 14,16,18O are enhanced with respect to those for 22O and 28O.In addition, the shift towards higher ω is more significant than in the RPWIAcase, but for 28O.

The cross section of the inclusive QE (e, e′) reaction on 36−56Ca isotopesat ε = 560 MeV and θ = 60o calculated in the RPWIA and in the RGFare shown in Figs. 3.19 and 3.20, respectively. The general behavior of thecross sections and their evolution with increasing neutron number is similarfor calcium and oxygen isotopes. The magnitude increases with the neutronnumber, but FSI effects are somewhat more visible for calcium isotopes. TheRGF cross sections on 36,40,44,48Ca in Fig. 3.20 are enlarged over a wide range ofω and are slightly reduced with respect to the RPWIA results in Fig. 3.19. Thisis particularly visible for 48Ca and produces an apparently large gap betweenthe cross sections of 48Ca and 52Ca.

The cross section of the inclusive QE (e, e′) reaction along the N = 28, 20,and 14 isotonic chains at ε = 1080 MeV and θ = 32o are shown in Fig. 3.21.Also in this case, in a first approximation we have neglected FSI and calcu-lations have been performed in the RPWIA. In this approach the differencesbetween the results for the various isotones are entirely due to the differencesin the single particle bound state wave functions of each isotone. The QE crosssection contains information on the dynamics of the initial nuclear ground stateand it is obtained suming all the integrated exclusive one-nucleon knockoutprocesses due to the interaction of the probe with all the individual nucleonsof the nucleus. The main role is played by protons, which give most of thecontribution. Increasing the proton number along each chain, owing to theenhancement of the proton contribution, there is a proportional increase of theQE cross sections. In contrast, no increase is found in the neutron contribu-tion, which is less significant than the proton one. We can see in panel (b)of Fig. 3.21 that, even if the central density of 34Si is reduced, the QE crosssections of 34Si and 36S do not show any significant difference that could beconsidered as a signal of bubble nuclei.

In Fig. 3.22 we show the QE (e, e′) cross sections calculated in the RGFmodel for selected isotones of the N = 28, 20, and 14 chains in the samekinematics as in Fig. 3.21. The general trend of the cross sections, their mag-nitude, and their evolution with respect to the change of the proton numberalong each chain are generally similar in RPWIA and RGF. The RGF resultsare, however, somewhat larger. The FSI effects in the RGF calculations pro-duce visible distortion effects: the RGF cross sections are not as symmetricalas the corresponding RPWIA ones and show a tail toward large values of theenergy transferred ω that is related to the description of FSI with a complexenergy-dependent relativistic optical potential.

64

3.7. Summary and Conclusions

3.7 Summary and Conclusions

In this chapter we have presented and discussed numerical predictions for thecross section and the parity-violating asymmetry in elastic and QE electronscattering on oxygen and calcium isotopic chains and on the N = 14, 20, and28 isotonic ones with the aim to investigate their evolution with increasingneutron and proton numbers.

The understanding of the properties of exotic nuclei is one of the majortopic of interest in modern nuclear physics. Large efforts in this directions havebeen done over last years and are planned for the future. The use of electronsas probe provides a powerful tool to achieve this goal. The RIB facilities indifferent laboratories have opened the possibility to give insight into nuclearstructures which are not available in nature, as they are not stable, but whichare important in astrophysics and had a relevant role in the nucleosynthesis.

Electron scattering is well fitted for studying nuclear properties, as its inter-action is well known and relatively weak with respect to the hadron force andcan therefore more adequately explore the details of inner nuclear structures.As a consequence of this weakness, the cross sections become very small andmore difficult experiments have to be performed. Electron scattering experi-ments off exotic nuclei have been proposed in the ELISe experiment at FAIRand in the SCRIT project at RIKEN. We hope that our theoretical predictionswill be useful for clarifying the different aspects of the measurements, givinginformation on the order of magnitude of the measurable quantities and there-fore making possible a more precise evaluation of the experimental difficulties.Moreover, a theoretical investigation can be helpful to envisage the most in-teresting quantities to be measured in order to explore the properties of exoticnuclear structures.

In this work, both elastic and inclusive QE electron scattering have beenconsidered. The elastic scattering can give information on the global propertiesof nuclei and, in particular, on the different behavior of proton and neutrondensity distributions. The inclusive QE scattering is affected by the dynami-cal properties, being the integral of the spectral density function over all theavailable final states, and, due to the reaction mechanism, preferably exploitsthe single-particle aspects of the nucleus. In addition, when combined withthe exclusive (e, e′p) scattering, it is able to explore the evolution of the single-particle model with increasing asymmetry between the number of neutronsand protons. Many interesting phenomena are predicted in this situation: inparticular, the modification of the shell model magic numbers. A definite re-sponse can be obtained from the comparison with experimental data, whichwill discriminate between the different theoretical models, mainly referring toRMF approaches.

As case studies for the present investigation we have selected oxygen andcalcium isotope chains and N = 14, 20, and 28 isotonic ones. The calculationshave been carried out within the framework of the RMF model. The nuclearwave functions are obtained considering a system of nucleons coupled to the

65


exchange mesons and the electromagnetic field through an effective Lagrangian.The calculated cross sections include both the hadronic and Coulomb FSI. Theinclusive QE scattering is calculated with the RGF model, which conserves theglobal particle flux in all the final state channels, as it is required in an inclusivereaction.

First, the models have been compared with experimental data already avail-able on stable isotopes in order to check their reliability. Then, the same modelshave been used to calculate elastic and inclusive QE cross sections on exoticisotopic and isotonic chains. The possible disagreement of the experimentalfindings from the theoretical predictions will be a clear indication of the insur-gence of new phenomena related to the proton to neutron asymmetry.

For the isotopic chains our results show an evolution of the calculated quan-tities without discontinuities. The increase of the neutron number essentiallyproduces an increase of the nuclear and proton densities and a flattening ofthe charge density.

For the isotonic chains our results show that the evolution of some specificobservables can be useful to test shell effects related to the filling of single-particle orbits. The increase of the proton number along each chain essentiallyproduces an enhancement and an extension of the proton densities. The densi-ties of the proton-rich isotones are significantly extended toward larger r withrespect to those of the proton-deficient ones. Pronounced shell effects are visi-ble in the nuclear interior. The differential cross sections calculated for elasticelectron scattering show that increasing proton number along an isotonic chainthe positions of the diffraction minima generally shift toward smaller scatter-ing angles, corresponding to lower values of the momentum transfer. The shiftis accompanied by a simultaneous increase in the height of the correspondingmaxima of the cross sections. A plot of the evolution of the position of thefirst minimum for each isotone in the chain shows a transition between proton-poor and proton-rich isotones. It is not possible to fit the positions of the firstminima with a straight line, but it is possible to draw two straight lines, con-necting the minima of the lighter and heavier isotones, with a different slope.This behavior has been found for all the three isotonic chains.

Finally, the parity-violating asymmetry parameter has been calculated inorder to investigate the neutron skin, as the weak current is essentially obtainedfrom the interaction with neutrons. Numerical predictions have been presentedand discussed for isotopic and isotonic chains. Moreover, we have presentedand discussed numerical predictions for the neutron density distribution of208Pb. The determination of the neutron distribution in nuclei has proven tobe a serious challange to our understanding of nuclear structure and it is oneof the major topics of interest in nuclear physics. Great experimental andtheoretical efforts have been devoted over the last years to achieve this goal.In the next years several experiments are planned, in different laboratoriesworldwide, to measure the neutron skin thickness, i.e., the difference betweenthe neutron and proton distributions, as accurately as possible.

66

3.7. Summary and Conclusions

PVES is an accurate and model-independent tool for probing neutron prop-erties as it is directly related to the Fourier transform of the neutron density.Starting from various different theoretical models for nuclear structure, we haveextracted the 2pF parameters for the neutron distribution and we have com-pared them with the very recent (γ, π0) data from Mainz. We have then ana-lyzed the linear correlation between the neutron radius and the parity-violatingasimmetry. The PREX data at average momentum transfer q = 0.475 fm−1

have unfortunately a too much large experimental uncertainty to discriminateamong the models and only the future run PREX-II will help to rule out someof them.

67


68

Chapter 4Theoretical Foundations forProton Scattering

4.1 Introduction

Elastic scattering is the predominant event in the interactions of nucleons withnuclei. This process has been extensively studied over many decades bothexperimentally and theoretically and there now exists extensive data on thescattering of nucleons from stable nuclei. The nuclear optical model potentialrepresents an essential tool for investigating nuclear reactions and permits todescribe NA scattering across wide regions of the nuclear landscape. With thistheoretical instrument we are able to compute the scattering observables suchas the differential cross section, the analyzing power, and the spin rotationfor elastic scattering reactions on target nuclei close to the valley of stability.We can also gain further information about the distortion of the scatteredwaves that is an important input for the study of inelastic processes and forcalculations on a wide variety of nuclear reactions. In particular, the opticalpotential plays an important role in the area of electron scattering, where, aswe stated in Chapter 3, it is the basic ingredient to take into account FSI inthe inclusive QE scattering within the RGF model and in the exclusive QE(e, e′p) reaction within the DWIA.

All formulations of the nuclear optical model for elastic scattering have incommon an allowance of flux loss from the incident beam to non-elastic chan-nels when the beam energy is above inelastic and reaction thresholds. Thesemodel formulations range from strong geometric forms to ones based uponcomplex potential representations. The geometric approaches [202] remainvalid and appropriate for high energies, typically above 1 GeV, while the useof complex potential models is most appropriate for energies below that value.Those complex potential models broadly fall into two classes, the former beingphenomenological and the latter microscopic.

The concept of a complex optical potential as a single particle representa-

69

4. Theoretical Foundations for Proton Scattering

tion of NA interactions dates at least to the study by Bethe [203] of neutron-nucleus cross sections. At early stages, optical potentials were all phenomeno-logical and studies of that phenomenology culminated in attempts to pre-scribe global forms for all target masses and for projectile energies typicallyup to 40 MeV. Tabulations of those potential parameters have been made inRef. [204]. Likewise there has been a number of reviews of the topic of whichthose of Refs. [205–207] are a useful selection. Phenomenological and semi-phenomenological optical potentials are still used to interpret elastic scatteringdata as well as to define the distorted waves required in DWBA analyses of non-elastic processes. Likewise the semi-phenomenological approach has reached avery sophisticated stage and it was possible to successfully analyze [208] datafor many nuclei for an energy range from keV to GeV. In general, phenomeno-logical potentials are based on the existence of many adjustable parameters,setted up fitting a large amount of experimental data, that describe the shapeof the density distribution of the nuclear target and vary smoothly with theenergy of the projectile and the mass number of the target.

With the advent of the next generation of radioactive-beam facilities newregions of the nuclear chart will be explored and a new wealth of nuclearphenomena will be discovered. These new areas of the nuclear landscape arecharacterized by nuclei with neutron and proton excess far from the β-stability.The study of such systems will be thus performed employing both electron- andNA scattering reactions. In Chapter 3 we presented the theoretical frameworkand the numerical results obtained studying these new exotic systems withelectron scattering. In this chapter we focus our attention on proton-nucleuselastic scatering and on the importance of the framework of nuclear opticalmodel potentials in describing such reactions. Due to the dependence of phe-nomenological potentials on parameters which have been determined through afit of experimental data on stable nuclei, it is reasonable to think that they willnot be able to give a reliable description of the scattering observables of thesenew exotic nuclei. For this reason it is more appealing the idea of a microscopicoptical potential which is free from the dependence on several phenomenolog-ical constants and it is essentially based on the existence of two quantities:the nuclear matter distribution of the target nucleus and the NN interaction,described by the NN t matrix. The developments of the last decades and theprogresses reached in the microscopic optical model made the nucleon-basedmicroscopic theory of NA scattering able to predict the experimental scatter-ing observables with a high level of accuracy. All these models based on theNN interaction are non-relativistic and a review of these models can be foundin Ref. [209].

The theoretical justification for the NA optical potential built in termsof underlying NN scattering amplitudes has been given more than 60 yearsago by Chew [210] and Watson et al. [211, 212]. It was supposed that, forsufficiently high incident energies, those NA interactions would be ascertainedfrom free NN scattering and in Ref. [213] Bethe showed that the cross section

70

4.1. Introduction

and polarization from the scattering of 310 MeV protons from 12C at forwardscattering angles were consistent with that conjecture. After this first step,Kerman, McManus, and Thaler (KMT) [214] developed the Watson multiplescattering approach expressing the NA optical potential by a series expansionin terms of the free NN scattering amplitudes. Those formulations were basedon the definition of an effective interaction between the projectile and the targetnucleons. Lucid reviews of these theories have been given by Feshbach [206]and Adhikari and Kowalski [215]. They also gave many details of the NNscattering amplitudes and t matrices. In spite of this, for some time, partlydue to a lack of knowledge of the underlying NN scattering amplitudes, it wasnot possible to reach an adequate numerical implementation of those theories.This partially contributed to analyze the NN phase shifts [216, 217] and todevelop the NN potentials.

A breakthrough in the studies of NN and NA scattering happened about50 years ago, when, from the one side reliable NN scattering amplitude andphase shift analyses [218] to the pion threshold were made and, from the otherside, the Nijmegen [219], Paris [220], and later Bonn [221] NN potentialswere developed to fit those NN amplitudes and phase shifts. In light of this,the theories of the microscopic optical potential were reviewed at the seminalworkshop in Hamburg in 1978 [222]. Finally, an incentive to implement thesetheories was also given by the different experimental programs that producedmany and varied high quality data sets for energies up to 1 GeV [223].

In almost all studies of nucleon-induced reactions [205–207], the opticalmodel for scattering has a central role and, for this reason, different reviewswere published. A comprehensive one on non-relativistic and relativistic opticalmodel formalisms for energies below 1 GeV is that of Ray, Hoffmann, andCoker [224]. In the theory of the optical potential the non-relativistic multiplescattering theory is used with the NN scattering amplitudes modified fromthe free NN values and this is designated as an effective interaction. Themodifications of the free NN amplitudes are due to Pauli blocking and meanfield effects for both projectile and bound state nucleons. In fact, the twointeracting nucleons are not free but their interaction takes place within thenuclear medium. In addition, there are other effects due to the convolutionof the NN scattering amplitudes with target structure that require off theenergy shell scattering amplitudes. Finally, another important feature is thecomplete antisymmetrization of the A + 1 nucleon scattering system whichleads to direct and knockout exchange amplitudes for NA scattering. Theeffect of such exchange amplitudes is not small at any energy and they are asource of non-locality. All non-relativistic multiple scattering theories sharethe underlying principle that the fundamental dynamics leading to the NNpotential is unaltered although medium effects vary the NN amplitudes fromthe free ones.

If only the free NN t matrix on the energy shell is necessary in calculationsof the optical potential, the experimental NN amplitudes can be used. But

71


if both the on- and off-shell properties of the t matrix are important, suitablerepresentations of those properties are required. One of such representationsfor the effective interaction used a local superposition of Gaussians or Yukawasin coordinate space [225, 226]. At the same time, Love and Franey [227] alsodefined an effective t matrix comprising a sum of central, spin-orbit, and tensorcomponents. However, their parameter values were chosen to match the thenexistent NN amplitudes [218] in an energy range between 50 MeV and 1 GeV.So their force was controlled solely by the on-shell NN amplitudes.

In parallel with the coordinate space investigations, there has been an on-going interest in finding a rigorous treatment of momentum space multiplescattering theory for NA scattering, the most recent of which are presentedin Refs. [228–232]. The NN potential is an essential ingredient in the NAscattering theory and its off-shell properties play an important role. To obtaina good description of these properties we rely on the NN scattering theorythat involves one-boson-exchange (OBE) potentials that are termed realisticbecause they reproduce the experimental NN phase shifts. The most com-monly used realistic potentials are those given by groups from Nijmegen [233],Paris [220], Bonn [234], Argonne [235], and Hamburg [236]. In particular, inall the calculations presented in this chapter we adopt the CD-Bonn poten-tial [237], that is an evolution of the original Bonn potential [234], and twodifferent versions of Chiral potentials, that are not OBE potentials, developedby Entem and Machleidt [238] and Epelbaum, Glockle, and Meißner [239].The second important element of the NA scattering theory is the microscopicstructure of the nuclear target, for which we use the RMF description depictedin Chapter 2.

In Section 4.2 we introduce the general scattering problem represented bythe LS equation for the entire system composed by the projectile and thetarget nucleus. This equation is separated into a simple one-body equation forthe transition matrix and a complicated one for the optical potential. Aftersome different approximations the expression of the optical potential is thenreduced to a simple form in which the NN t matrix and the nuclear densityare factorized, but the optical potential still mantains its off-shell character.In Section 4.3 we give some general remarks about the NN potentials usedto compute the NN t matrix. The novelty in our calculations is the use ofChiral potentials [238, 239] as basic ingredient to calculate the microscopicoptical potential and then the scattering observables. We also spend a fewwords about the CD-Bonn potential [237] which has been used from otherauthors to perform this kind of calculations. In Section 4.4 we give the explicitformulae to compute the central and the spin-orbit parts of the NN t matrixthat are used to calculate the optical potential and are proportional to theNN Wolfenstein amplitudes. Finally, in Section 4.5 we present and discussthe theoretical results for the NN amplitudes obtained with the different NNpotentials. The results for the NA scattering observables, obtained with theoptical potential developed in Section 4.2 are shown in Chapter 5, where we

72

4.2. The First-Order Optical Potential

give the details of the treatment of the NA transition matrix.

4.2 The First-Order Optical Potential

4.2.1 The Spectator Expansion

The basic motivation behind the spectator expansion [228] is that the solutionof the full many-body problem is beyond present capabilities, hence an expan-sion series is constructed for multiple scattering theory predicated upon thenumber of target nucleons interacting directly with the projectile. Hence theexpansion involves terms where the projectile interacts directly with one targetnucleon plus a second order term where the projectile interacts directly withtwo target nucleons, and so on to third and subsequent orders. The separationof these terms with respect to these categories of interactions is not completelyfixed due to the nature of the complicated (A+1)-body propagator, hence somepossible choices, detailed in Ref. [228], must be differentiated. At the heartof the standard approach to the elastic scattering of a single projectile froma target of A particles is the separation of the LS equation for the transitionoperator T

T = V + V G0(E)T . (4.1)

into two parts, i.e. an integral equation for T

T = U + UG0(E)PT , (4.2)

where U is the optical potential operator, and an integral equation for U

U = V + V G0(E)QU . (4.3)

In the above equations the operator V represents the external interaction, suchthat the Hamiltonian for the entire (A+ 1)-particle system is given by

HA+1 = H0 + V . (4.4)

Asymptotically, the system is in an eigenstate of H0, and the free propagatorG0(E) for the projectile plus target nucleus system is

G0(E) =1

E −H0 + iǫ. (4.5)

The operators P and Q are projection operators with

P +Q = 1 , (4.6)

and P is defined such that Eq. (4.2) is solvable. In this case P is conventionallytaken to project onto the elastic channel, such that, among other properties,we have

[G0, P ] = 0 . (4.7)

73


For the scattering of a single particle projectile from an A-particle target thefree Hamiltonian is given by

H0 = h0 +HA , (4.8)

where h0 is the kinetic energy operator for the projectile and HA stands forthe target Hamiltonian. Thus the projector P can be defined as

P =|ΦA〉〈ΦA|〈ΦA|ΦA〉

, (4.9)

where |ΦA〉 corresponds to the ground state of the target, satisfying the con-dition given in Eq. (4.7), and fulfilling

HA |ΦA〉 = EA |ΦA〉 . (4.10)

With these definitions the transition operator for elastic scattering may bedefined as Tel = PTP , in which case Eq. (4.2) can be written as

Tel = PUP + PUPG0(E)Tel . (4.11)

Thus the transition operator for elastic scattering is given by a straightforwardone-body integral equation, which requires, of course, the knowledge of the op-erator PUP . The theoretical treatment which follows consists of a formulationof the many-body equation, Eq. (4.3), where expressions for U are derived suchthat PUP can be calculated accurately without having to solve the completemany-body problem.

In this dissertation, the presence of two-body forces only is assumed. Theextension to many-body forces is straightforward. With this assumption theoperator U for the optical potential can be expressed as

U =

A∑

i=1

Ui (4.12)

where Ui is given by

Ui = v0i + v0iG0(E)Q

A∑

j=1

Uj , (4.13)

provided that

V =

A∑

i=1

v0i . (4.14)

The two-body potential v0i acts between the projectile and the ith target nu-cleon. Through the introduction of an operator τi which satisfies

τi = v0i + v0iG0(E)Qτi , (4.15)

74


we can rearrange Eq. (4.13) as

Ui = τi + τiG0(E)Q∑

j 6=i

Uj . (4.16)

This rearrangement process can be continued for all A target particles, so thatthe operator for the optical potential can be expanded in a series of A termsof the form

U =A∑

i=1

τi +A∑

i,j 6=i

τij +A∑

i,j 6=i,k 6=i,j

τijk + · · · . (4.17)

This is the spectator expansion, where each term is treated in turn. Theseparation of the interactions according to the number of interacting nucleonsis not unique, due to the many-body nature of G0(E). The finite series given inEq. (4.17) together with the definitions of τi, τij , ... given above constitute oneform of the spectator expansion in multiple scattering theory. Various otherforms could also be found [240]. Differences between one form or another resideprimarily in the treatment of the many-body propagator G0(E). The spectatorexpansion derives its name from the underlying idea that in lowest order alltarget constituents but the initially struck one (particle i) are “passive”. In thenext order all target constituents but the ith and jth particle are passive, andso on.

4.2.2 The Impulse Approximation

The first order term in the spectator expansion, τi as given by Eq. (4.15), is nowexamined. Since for elastic scattering only PτiP , or equivalently 〈ΦA|τi|ΦA〉,need to be considered, Eq. (4.15) can be reexpressed, with this in mind, as

τi = v0i + v0iG0(E)τi − v0i + v0iG0(E)Pτi = τi − τiG0(E)Pτi , (4.18)

or

〈ΦA|τi|ΦA〉 = 〈ΦA|τi|ΦA〉 − 〈ΦA|τi|ΦA〉1

(E − EA) − h0 + iǫ〈ΦA|τi|ΦA〉 ,

(4.19)where τi is defined as the solution of

τi = v0i + v0iG0(E)τi . (4.20)

The combination of Eqs. (4.18) and (4.2) corresponds to the first order Watsonscattering expansion [211, 212]. If the projectile-target nucleon interactionis assumed to be the same for all target nucleons and if isospin effects areneglected then the KMT scattering integral equation [214] can be derived fromthe first order Watson scattering expansion.

Since Eq. (4.19) is a simple one-body integral equation, the principal prob-lem is to find a solution of Eq. (4.20). Of course, due to the many-body

75


character of G0(E), Eq. (4.20) is a many-body integral equation, and, in fact,it can be solved not more easily than the original equation Eq. (4.1). However,G0(E) may be written as

G0(E) =1

E − h0 −HA + iǫ=

1

E − h0 − hi −Wi −H i + iǫ(4.21)

where h0 is the kinetic energy of the projectile and hi the kinetic energy of theith target particle,

Wi =∑

j 6=i

vij , (4.22)

andH i = HA − hi −Wi . (4.23)

Since H i has no explicit dependence on the ith particle, then Eq. (4.20) maybe simplified by the replacement of H i by an average energy Ei. This is notnecessarily an approximation since G0(E) might be regarded as

G0(E) =1

(E − Ei) − h0 − hi −Wi − (H i − Ei) + iǫ(4.24)

and (H i−Ei) could be set aside to be treated in the next order of the expansionof the propagator G0(E). Thus, we consider now G0(E) to be Gi(E), where

Gi(E) =1

(E − Ei) − h0 − hi −Wi + iǫ, (4.25)

so that τi = τi + (higher-order corrections), and Eq. (4.20) reduces to

τi = v0i + v0iGi(E)τi . (4.26)

Equation (4.26) can also be reexpressed as

τi = t0i + t0igiWiGi(E)τi , (4.27)

where the operators t0i and gi are defined as

t0i = v0i + v0igit0i (4.28)

and

gi =1

(E − Ei) − h0 − hi + iǫ. (4.29)

The quantity Wi represents the coupling of the struck target nucleon to theresidual nucleus. In practical calculations, Wi has been taken to be an averageone-body potential independent of the particle label i. It has been also usedan average value of the energy Ei, which has taken to be zero.

At this point, one could think that a proper consideration of this quantity isnot of first order, and that it should be put together with the next higher orderin the spectator expansion. In that case one would obtain the so called IA tothe optical potential, which can be viewed as τ ≈ τ ≈ t0i, where the operatort0i can be identified with the free NN t matrix. In the case of the IA, onenever needs to solve any integral equation for more than two particles. Thishas made the IA very practical in intermediate energy nuclear physics and hasover many years led to a large body of work based upon this approximation.

76


4.2.3 The KMT Multiple Scattering Theory

Up to now we have introduced the general scattering problem in momentumspace and we have developed the formalism to treat the LS equation for theelastic scattering amplitude and the spectator expansion for the optical po-tential operator. In particular, we have analyzed the first order term of thespectator expansion and we have seen how this term can be simply treated inthe IA.

Now we want to give explicit expressions to the previous equations devel-oping a theoretical framework in which we can compute the optical potentialand the transition amplitude for the elastic scattering observables. In orderto do it we follow the path outlined in Ref. [241], that is based on the KMTmultiple scattering theory and treats the optical potential at the first orderof the spectator expansion in the IA. In the momentum space representation,Eq. (4.11) is the elastic scattering equation for the transition amplitude. Thisis the LS integral equation, which is

Tel(k′,k;E) = Uel(k

′,k;E) +

∫

d3pUel(k

′,p;E) Tel(p,k;E)

E(k0) −E(p) + iǫ, (4.30)

where k0 is the initial on-shell momentum and E(k0) is the correspondinginitial energy of the system in the NA frame. To compute the scatteringobservables we only need the on-shell term Tel(k0,k0;E) of the transition ma-trix T , but in the present treatment we consider the full off-shell matrix withthe general initial and final momenta k and k′, respectively. The principaladvantage of performing the calculation in this manner is that it avoids thenecessity of devising approximate treatments of the microscopic theory of theoptical potential which lead to manageable, but questionable, forms for U inthe coordinate space representation. Additional advantages include the simpleincorporation of relativistic kinematics and recoil.

As we said we restrict to the first-order term of the optical potential. Thematrix element 〈ΦA|τi|ΦA〉 given in Eq. (4.19) represents the full-folding opticalpotential and is given explicitly as

〈k′,ΦA|PUP |k,ΦA〉 ≡ Uel(k′,k;ω) = A 〈k′,ΦA|τ(ω)|k,ΦA〉 , (4.31)

where τ represents any one of the τi evaluated at the energy ω, and |k,ΦA〉stands for |k〉 ⊗ |ΦA〉. The first-order KMT prescription is equivalent to theIA τi ≈ t0i with τi calculated from Eq. (4.18) and U(k′,k;ω) from Eq. (4.31),without further approximation. The target ground state matrix elements ofτi do not have to be calculated explicitly, since U(k′,k;ω) is the solution ofthe sum of the one-body integral equations represented by Eq. (4.19) and it issufficient to consider the driving term. In this way, Eq. (4.18) takes on the sameform as the integral equation for the elastic scattering amplitude Tel(k

′,k;E)given in Eq. (4.30). The combination of the three equations (4.18), (4.30),and (4.31) yields the familiar first-order KMT formulation in which the elastic

77


scattering amplitude is given by

Tel(k′,k;E) =

A

A− 1T (k′,k;E) , (4.32)

where the auxiliary elastic amplitude is determined by the solution of theintegral equation

T (k′,k;E) = U(k′,k;ω) +

∫

d3pU(k′,p;ω) T (p,k;E)

E(k0) −E(p) + iǫ, (4.33)

and the auxiliary first-order optical potential is defined by

U(k′,k;ω) = (A− 1) 〈k′,ΦA|τ(ω)|k,ΦA〉 , (4.34)

where τ is any one of the τi. Since the operator τi contains no restriction(other than the antisymmetry of the target nucleons) on the allowed inter-mediate states, the auxiliary optical potential U contains contributions fromintermediate states in which the target propagates in its ground state. Weemphasize that these contributions are not spurious since they are not presentin the true (Watson) optical potential which is the equivalent of the abovementioned KMT formulation and which is given by Eq. (4.31). We also notethat, although an approximation of τi in Eq. (4.34) by the free NN t matrixt0i ignores nuclear medium modifications of the NN scattering amplitude con-tained in the definition of the auxiliary KMT optical potential U , it does notignore nuclear medium modifications of the effective NN scattering operator(τ) used in describing the equivalent physical (Watson) optical potential. Inorder to simplify our model we make another approximation that consits inreplacing the matrix τi in Eq. (4.34) with the free NN t0i matrix.

Our problem is then described in the zero momentum frame of the NAsystem by Eq. (4.33), where

U(k′,k;ω) = (A− 1) 〈k′,ΦA|t(ω)|k,ΦA〉 , (4.35)

and t is any one of the free NN t0i matrices. In Eq. (4.35) the integrationsover the initial and final momenta of the struck nucleon of the target are re-duced to one momentum integration through conservation of total two-nucleonmomentum obeyed by the operator t. That is, through the use of the relation

〈k′,p′|t(ω)|k,p〉 = δ(k′ + p′ − k − p) 〈k′,p′|t(ω)|k,p〉 , (4.36)

with the introduction of the variables

q ≡ k′ − k , K ≡ 1

2(k′ + k) , p ≡ p− 1

2q , (4.37)

78


Eq. (4.35), when expressed with kinematics that maintain time reversal invari-ance, becomes

U(k′,k;ω) = (A− 1)

∫

d3p∑

s,i

⟨

k′, p− 1

2q, s, i

∣

∣

∣

∣

t(ω)

∣

∣

∣

∣

k, p +1

2q, s, i

⟩

× ρs,iint

[

p− A− 1

A

q

2+

K

A, p +

A− 1

A

q

2+

K

A

]

.

(4.38)

In Eq. (4.38), s and i are the spin and isospin projections of the struck nucleon,which must be conserved for elastic scattering, and U(k′,k;ω) and the matrixelement of t are operators in the spin and isospin space of the projectile nucleon.The quantity ρs,iint is the one-nucleon density matrix of the target for a nucleon ofspin and isospin projections s and i, respectively. To simplify this expression,we assume that we are dealing with a spin-saturated nucleus, i.e., ρs,iint = ρs

′,iint =

ρiint. With this assumption, the spin trace will eliminate those components ofthe t matrix which depend (linearly) on the spin of the struck nucleon. Theeliminated terms are the tensor term, and the σ1·σ2 and σ1·σ2τ1·τ2 parts of thecentral term. The remaining terms are the spin-independent central term andpart of the spin-orbit term for each of the proton-proton and neutron-protont matrices. We denote these reduced t matrices by t′ and the correspondingdensity matrix by ρNint, where N = n, p.

Thus we have for a proton projectile

U(k′,k;ω) =A− 1

A

∑

N=n,p

∫

d3p

⟨

k′, p− 1

2q

∣

∣

∣

∣

t′pN(ω)

∣

∣

∣

∣

k, p +1

2q

⟩

× ρNint

[

p− A− 1

A

q

2+

K

A, p +

A− 1

A

q

2+

K

A

]

,

(4.39)

where the density matrices are normalized to N for neutrons and Z for protons.It is convenient to make a change of integration variable from p to P = p +(K/A), so that Eq. (4.39) becomes

U(k′,k;ω) =A− 1

A

∑

N=n,p

∫

d3P

⟨

k′,P − q

2− K

A

∣

∣

∣

∣

t′pN(ω)

∣

∣

∣

∣

k,P +q

2− K

A

⟩

× ρNint

[

P − A− 1

A

q

2,P +

A− 1

A

q

2

]

.

(4.40)

We note that the density matrix is related to the momentum space densityprofile ρN(q) of the nucleus by

ρN(q) ≡ ρNint

[

A− 1

Aq

]

=

∫

d3P ρNint

[

P − A− 1

A

q

2,P +

A− 1

A

q

2

]

, (4.41)

79


where ρN (q) is the Fourier transform of the position space point neutron orpoint proton density profile defined in a system of coordinates with the originat the center of mass of the nucleus. The normalization is such that ρN(q =0) = N or Z. If q is the momentum transfer taken up by the degree of freedomdescribed by the position R of the struck nucleon relative to the center of massof the target, then [(A − 1)/A]q is the momentum transfer taken up by thedegree of freedom described by the position r = [A/(A − 1)]R of the strucknucleon relative to the center of mass of the core. We will deal with the densityρN (q), since it can be obtained from relativistic model calculations for nuclearstructure.

The NN t-matrix element in Eq. (4.40) is evaluated in the zero-momentumframe of the NA system. The relationship with the corresponding t-matrixelement evaluated in the zero-momentum frame of the two nucleons is assumedto be

⟨

k′,P − q

2− K

A

∣

∣

∣

∣

t′pN(ω)

∣

∣

∣

∣

k,P +q

2− K

A

⟩

NA

= η(P , q,K) 〈κ′|t′pN(ω)|κ〉NN

,

(4.42)where the initial nucleon momentum κ and the final nucleon momentum κ′

in the zero momentum frame of the NN system are taken to be given by thenon-relativistic definitions

κ =1

2

[

k −(

P +q

2− K

A

)]

, κ′ =1

2

[

k′ −(

P − q

2− K

A

)]

. (4.43)

The first factor η in Eq. (4.42) is the Møller factor for the frame transformationand is given by

η(P , q,K) =

[

Eproj(κ′)Eproj(−κ′)Eproj(κ)Eproj(−κ)

Eproj(k′)Eproj

(

P − q

2− K

A

)

Eproj(k)Eproj

(

P + q

2− K

A

)

]1

2

,

(4.44)where Eproj(k) is the relativistic energy of a nucleon of momentum k. Thisfactor imposes the Lorentz invariance of flux. The NN t matrix on the right-hand side of Eq. (4.42) can be written in the form

〈κ′|t′pN(ω)|κ〉NN

≡ t′pN(κ′,κ;ω) = t′pN(q,KNN ;ω) , (4.45)

where

q = κ′ − κ = k′ − k (4.46)

and

KNN = κ′ + κ =A+ 1

AK − P . (4.47)

This form is useful since the roles of the variables q and KNN are reversed ina spatial exchange operation. The model of the NN t matrix that we employ

80


is specified in terms of its dependence on q and KNN . With these notations,Eq. (4.40) becomes

U(q,K;ω) =A− 1

A

∑

N=n,p

∫

d3P η(P , q,K) t′pN(q,KNN ;ω)

× ρNint

[

P − A− 1

A

q

2,P +

A− 1

A

q

2

]

.

(4.48)

This is the full-folding expression for the first-order KMT optial potential.

4.2.4 Fixed Beam Energy Approximation

Now we want to explicitly consider the dependence of the t′pN matrices inEq. (4.48) on the energy ω. In our calculations we adopt the relativistic def-inition of energy to calculate the energy E in the NA center-of-mass system.We can define ω from the definition of τi in Eq. (4.20), where we see that Eis completely determined by the propagator G0(E). In the IA and consideringt′pN in the NA frame the energy argument for the propagator becomes

ω = E −

(

A−1A

K + P)2

4m. (4.49)

Here E is the total energy in the NA system. Since t′pN is a slowly varyingfunction of energy [242], for the evaluation of ω we also neglect the couplingwith the integration variable P and we fix the energy of the transition ampli-tude at one half the beam energy in the laboratory frame which is equivalentto the on-shell center-of-mass energy for the two nucleons. Quantitatively

ω =Tlab2

=1

2

k2lab2m

, (4.50)

where klab is the on-shell momentum in the laboratory system. This is thefixed beam energy approximation, which is an hystoric choice performed in allcalculations based on KMT formulation. A review of this type of calculationscan be found in Ref. [224]. This approximation can be justified as follows:the energy ω is the energy which would be correct if the struck target nucleonhad zero momentum before and after the collision. This favors the forwardscattering process, and it was argued that, due to the narrow peaking of thedensity in momentum space, the whole scattering process is dominated byforward scattering.

4.2.5 The Optimum Factorization Approximation

The expression for the optical potential given in Eq. (4.48) is numerically quitedifficult to compute and for this reason we adopt the optimum factorization

81


approximation. Since the nuclear size is significantly larger than the range ofthe NN interaction, and therefore also of the t matrix (if the energy parameteris fixed), the most slowly varying factor in Eq. (4.48) is ηt′pN . The methodof optimum factorization is to expand ηt′pN in a Taylor series in P about afixed value P0, which is chosen by requiring that the contribution of the firstderivative term is minimized. The expansion

η(P , q,K) t′pN(q,KNN(P );ω) = η(P0, q,K) t′pN(q,KNN(P0);ω)

+ (P − P0) ∂P0

[

η(P0, q,K) t′pN(q,KNN(P0);ω)]

+ · · ·(4.51)

when used in Eq. (4.48) gives

U(q,K;ω) =A− 1

Aη(P0, q,K)

∑

N=n,p

t′pN(q,KNN(P0);ω) ρN(q)

+ ∆U + · · · ,(4.52)

where

∆U =A− 1

A

∑

N=n,p

∂P0

[


×∫

d3P (P −P0) ρNint

[

P − A− 1

A

q

2,P +

A− 1

A

q

2

]

,

(4.53)

and ρN(q) is given by Eq. (4.41). The time-reversal invariance property of theground state density matrix for even-even nuclei leads to

∫

d3P P ρNint

[

P − A− 1

A

q

2,P +

A− 1

A

q

2

]

= 0 , (4.54)

that inserted in Eq. (4.53) gives

∆U = −A− 1

A

∑

N=n,p

∂P0

[


× P0

∫

d3P ρNint

[

P − A− 1

A

q

2,P +

A− 1

A

q

2

]

.

(4.55)

If we choose P0 = 0, ∆U = 0, KNN = [(A + 1)/A]K and the optimallyfactorized optical potential is

U(q,K;ω) =A− 1

Aη(q,K)

∑

N=n,p

t′pN

[

q,A + 1

AK;ω

]

ρN(q) , (4.56)

where η(q,K) shall be used to stand for η(P = 0, q,K). Referring back toEq. (4.42), we see that the optimum choice for factorization, P = 0, selects

82


the initial momentum of the struck nucleon to be (q/2)− (K/A) and the finalmomentum to be −(q/2)− (K/A) in the frame of zero total momentum of thesystem. In the limit of a single nucleon for the target (A = 1), these momentabecome −k and −k′, which are the correct values for NN scattering. Also inthis limit, η → 1 and ρN(q) → 1. In the general case, the recoil effects for afinite mass target are included through the kinematics employed in Eq. (4.56).

The optical potential obtained so far is an operator in the spin space ofthe projectile. To make the spin dependence explicit we write the t matrix t′pN(which has been averaged over the spin of the struck nucleon) in the form

t′pN(q,KNN ;ω) = tcpN(q,KNN ;ω) +i

4σ · q ×KNN t

lspN(q,KNN ;ω) . (4.57)

The first term corresponds to the central spin-independent contribution, whilethe second term corresponds to the spin-orbit contribution. In the latter termthe usual total Pauli spin operator of the NN system is replaced by justthe projectile’s Pauli spin operator, the other having been eliminated by thetrace over the spin of the struck nucleon. With an on-shell constraint, tcpN forN = n, p is proportional to the NN Wolfenstein spin-independent scatteringamplitudes apn(q) and app(q), respectively, while tlspN , for N = n, p, is propor-tional to the NN Wolfenstein spin-dependent scattering amplitudes cpn(q) andcpp(q), respectively. The substitution of Eq. (4.57) into Eq. (4.56) gives theoptical potential as

U(q,K;ω) = U c(q,K;ω) +i

2σ · q ×K U ls(q,K;ω) , (4.58)

where the central term is given by

U c(q,K;ω) =A− 1

Aη(q,K)

∑

N=n,p

tcpN

[

q,A+ 1

AK;ω

]

ρN (q) , (4.59)

and the spin-orbit term is given by

U ls(q,K;ω) =A− 1

Aη(q,K)

(

A+ 1

2A

)

∑

N=n,p

tlspN

[

q,A+ 1

AK;ω

]

ρN (q) .

(4.60)Eqs. (4.58)-(4.60) exhibit the nonlocality and off-shell effects present in theoptimally factorized optical potential. The nonlocality is evident in the de-pendence of η and tpN upon K. The factor (i/2)σ · (q × K) comes fromthe momentum representation of the spin-orbit operator L · S and is alwaysincluded in optical potentials which are referred to as local.

4.2.6 Kinematical Variables

Now we want to give some details about the different kinematical variablesused in the different frames. In the NA center-of-mass frame we can use the

83


two variables (k′,k) = (k′, k, θ), or, in alternative, the two variables (q,K) =(q,K, ϑ). All these variables lay in the scattering plane and they are linked bythe following direct and inverse relations:

q = k′ − k

K =1

2(k′ + k)

,

k′ = K +q

2

k = K − q

2

. (4.61)

For the angle η between q and K we have:

q ·K =1

2

(

k′ 2 − k2)

,

q =√k′ 2 + k2 − 2k′k cos θ ,

K =1

2

√k′ 2 + k2 + 2k′k cos θ ,

cos ϑ =q ·KqK

=k′ 2 − k2

√

k′ 4 + k4 + 2k′ 2k2(1 − 2 cos2 θ).

(4.62)

In theNN center-of-mass frame we can use the two variables (κ′,κ) = (κ′, κ, φ),or, in alternative, the two variables (qNN ,KNN) = (qNN , KNN , ϕ). All thesevariables are linked by the following relation:

qNN = κ′ − κ

KNN = κ′ + κ,

κ′ =1

2

(

KNN + qNN

)

κ =1

2

(

KNN − qNN

)

. (4.63)

For the angle ϕ between qNN and KNN we have:

qNN ·KNN = κ′ 2 − κ2 ,

qNN =√

κ′ 2 + κ2 − 2κ′κ cosφ ,

KNN =√

κ′ 2 + κ2 + 2κ′κ cosφ ,

cosϕ =qNN ·KNN

qNNKNN=

κ′ 2 − κ2√

κ′ 4 + κ4 + 2κ′ 2κ2(1 − 2 cos2 φ).

(4.64)

Moreover, Eq. (4.43) gives us the relation between (k′,k) and (κ′,κ) variables

κ =1

2

[

k −(

P +q

2− K

A

)]

, κ′ =1

2

[

k′ −(

P − q

2− K

A

)]

, (4.65)

and from these we obtain:

qNN = q , KNN =A + 1

AK − P . (4.66)

In the optimum factorization approximation the relation between (k′,k) and(κ′,κ) variables becomes

κ =1

2

[

k − q

2+

K

A

]

, κ′ =1

2

[

k′ +q

2+

K

A

]

, (4.67)

84

4.3. The Nucleon-Nucleon Potential

and

KNN =A+ 1

AK , ϑ = ϕ . (4.68)

From the last relation we also have

κ =1

2

(

A + 1

AK − q

)

, κ′ =1

2

(

A + 1

AK + q

)

. (4.69)

4.3 The Nucleon-Nucleon Potential

Before discussing the NN t matrices and their calculation, we want to spend afew words about the NN potentials used to generate the NN interaction thatpermits us to compute the LS equation in all NN partial waves. As stated inthe introduction we adopt two kinds of potentials: the CD-Bonn [237] poten-tial, developed by Machleidt, and two different versions of the Chiral potentialat the fourth order (N3LO) developed by Entem and Machleidt [238] and Epel-baum, Glockle, and Meißner [239], respectively. The CD-Bonn potential hasbeen already adopted [243] to compute a theoretical optical potential and thescattering observables and we use it to check the correctness and the reliabilityof our calculations. These results will then be compared with the ones obtainedusing different versions of the Chiral potential, which, so far, has never beenemployed to calculate a theoretical optical potential. In the rest of this sec-tion we will give some general concepts about these potentials with particularattention to the chiral potential and its derivation from Chiral PerturbationTheory (ChPT). For more details we refer to Refs. [237–239] and referencestherein.

4.3.1 The CD-Bonn Potential

The CD-Bonn NN potential is based upon the same philosophy as the Bonnfull potential [234] outlined in Ref. [237] and on the OBE model, which includesonly single-meson exchanges between the two interacting nucleons. It is energyindependent and it is defined in the framework of the usual non-relativistic LSequation. Thus, it can be applied in the same way as any other conventionalNN potential. However, the crucial point is that it reproduces importantpredictions by the Bonn full model avoiding the problems that the Bonn fullmodel creates in applications. The charge dependence (CD) predicted by theBonn full model is reproduced accurately by the new potential, which is whyit is called the CD-Bonn potential. The off-shell behavior of CD-Bonn is basedupon the relativistic Feynman amplitudes for meson exchange. Therefore, theCD-Bonn potential differs off-shell from conventional NN potentials, a factthat has attractive consequences in nuclear structure applications.

The CD-Bonn potential is based upon meson exchange and all mesons withmasses below the nucleon mass are included, i.e., π, η, ρ (770), and ω (782).In addition to this, two scalar-isoscalar σ bosons are introduced. However,

85


for the η (with a mass of 547.3 MeV), it is assumed a vanishing coupling tothe nucleon, which implies to drop the η. This assumption is supported bysemi-empirical evidence from various sources.

The model starts from the following Lagrangians that describe the couplingof the mesons of interest to nucleons:

Lπ0NN = −gπ0ψiγ5τ3ψϕ(π0) ,

Lπ±NN = −√

2gπ±ψiγ5τ±ψϕ(π±) ,

LσNN = −gσψψϕ(σ) ,

LωNN = −gωψγµψϕ(ω)µ ,

LρNN = −gρψγµτψ ·ϕ(ρ)µ − fp

4Mpψσµντψ · (∂µϕ

(ρ)ν − ∂νϕ

(ρ)µ ) ,

(4.70)

where ψ denotes nucleon fields, ϕ meson fields, and τ3,± are standard defini-tions of Pauli matrices and combinations thereof for isospin 1/2. Mp is theproton mass which is used as scaling mass in the ρNN Lagrangian to makefp dimensionless. To avoid the creation of unmotivated charge dependence,the scaling mass Mp is used in the ρNN vertex no matter what nucleons areinvolved.

In the center-of-mass system of the two interacting nucleons, the OBEFeynman amplitude generated by meson α is

−iVα(q′, q) =u1(q

′)Γ(α)1 u1(q)Pαu2(−q′)Γ

(α)2 u2(−q)

(q′ − q)2 −m2α

, (4.71)

where Γ(α)i (i = 1, 2) are vertices derived from the above Lagrangians, ui Dirac

spinors representing the interacting nucleons, and q and q′ their relative four-momenta in the initial and final states, respectively; Pα divided by the denom-inator is the appropriate meson propagator.

The OBE potential is defined by (i times) the sum over the OBE Feynmanamplitudes of the mesons included in the model, i.e.,

V (q′, q) =

√

M

E ′

√

M

E

∑

α=π0,π±,ρ,ω,σ1,σ2

Vα(q′, q) F2α(q′, q; Λα) . (4.72)

As customary, E ′ =√

M2 + q′ 2, E =√

M2 + q2, M is the nucleon mass, andF2

α(q′, q; Λα) are the form factors applied to the meson-nucleon vertices. Thesquare root factors make it possible to cast the unitarizing, relativistic, three-dimensional Blankenbecler-Sugar equation for the scattering amplitude, thatis a reduced version of the four-dimensional Bethe-Salpeter (BS) equation, intothe following form:

T (q′, q) = V (q′, q) +

∫

d3k V (q′,k)M

q2 − k2 + iǫT (k, q) . (4.73)

86


This is the familiar non-relativistic LS equation and, thus, Eq. (4.72) definesa relativistic potential which can be consistently applied in conventional, non-relativistic nuclear structure, in the usual way. The form factors in Eq. (4.72)regularize the amplitudes for large momenta (short distances) and account forthe extended structure of nucleons in a phenomenological way.

The Feynman amplitudes, Eq. (4.71), are in general non-local expressionsand the square root factors in Eq. (4.72) create additional nonlocality. Whilefor heavy vector-meson exchange nonlocality appears quite plausible, we haveto stress here that even the one-pion-exchange Feynman amplitude is non-local. This fact is often overlooked. It is important because the pion createsthe dominant part of the tensor force which plays a crucial role in nuclearstructure.

In summary, one characteristic point of the CD-Bonn potential is that ituses the Feynman amplitudes of meson exchange in its original form; localapproximations are not applied. This has impact on the off-shell behavior ofthe potential, particularly, the off-shell tensor potential. It is well known thatthe off-shell behavior of an NN potential is an important factor in microscopicnuclear structure calculations. Therefore, the predictions by the CD-Bonnpotential for nuclear structure problems differ in a characteristic way from theones obtained with local NN potentials.

4.3.2 The Chiral Potential

QCD is the theory of strong interactions. It deals with quarks, gluons andtheir interactions and is part of the Standard Model of Particle Physics. QCDis a non-Abelian gauge field theory with color SU(3) the underlying gaugegroup. The non-Abelian nature of the theory has dramatic consequences.While the interaction between colored objects is weak at short distances orhigh-momentum transfer (“asymptotic freedom”), it is strong at long distances(1 fm) or low energies, leading to the confinement of quarks into colorless ob-jects, the hadrons. Consequently, QCD allows for a perturbative analysis atlarge energies, whereas it is highly nonperturbative in the low-energy regime.Nuclear physics resides at low energies and the force between nucleons is aresidual color interaction similar to the van der Waals force between neutralmolecules. Therefore, in terms of quarks and gluons, the nuclear force is a verycomplicated problem that, nevertheless, can be attacked with brute comput-ing power on a discretized Euclidean space-time lattice, known as lattice QCD.Advanced lattice QCD calculations are under way and continuously improved.However, since these calculations are very time-consuming and expensive, theycan only be used to check a few representative key issues. For everyday nuclearstructure physics, a more efficient approach is needed.

The efficient approach is an EFT. For the development of an EFT it iscrucial to identify a separation of scales. In the hadron spectrum, a large gapbetween the masses of the pions and the masses of the vector mesons, like ρ(770) and ω (782), can clearly be identified. Thus, it is natural to assume that

87


the pion mass sets the soft scale, Q ∼ mπ, and the rho mass the hard scale,Λχ ∼ mρ, also known as the chiral symmetry breaking scale. This is suggestiveof considering an expansion in terms of the soft scale over the hard scale, Q/Λχ.Concerning the relevant degrees of freedom, we noticed already that, for theground state and the low-energy excitation spectrum of an atomic nucleus aswell as for conventional nuclear reactions, quarks and gluons are ineffectivedegrees of freedom, while nucleons and pions are the appropriate ones. Tomake sure that this EFT is not just another phenomenology, it must have afirm link with QCD. The link is established by having the EFT observe allrelevant symmetries of the underlying theory.

Another important step for the development of an EFT is to identify therelevant symmetries of low-energy QCD and investigate if and how they arebroken. In this case the relevant symmetry we have to consider is the chi-ral symmetry and its breaking configurations. In the limit of vanishing quarkmasses this symmetry tells us that left- and right-handed components of mass-less quarks do not mix. In presence of finite quark masses this symmetry isexplicitly broken, but, since the up and down quark masses are small as com-pared to the typical hadronic mass scale of ∼ 1 GeV, the explicit chiral sym-metry breaking due to non-vanishing quark masses is very small. There is alsoan evidence that the (approximate) chiral symmetry of the QCD Lagrangian isspontaneously broken for dynamical reasons of nonperturbative origin whichare not fully understood at this time. A (continuous) symmetry is said tobe spontaneously broken if a symmetry of the Lagrangian is not realized inthe ground state of the system. A spontaneously broken global symmetry im-plies the existence of massless Goldstone bosons with the quantum numbersof the broken generators. In this case the broken generators are pseudoscalar.The Goldstone bosons are thus identified with the isospin triplet of the (pseu-doscalar) pions, which explains why pions are so light. The pion masses are notexactly zero because the up and down quark masses are not exactly zero either(explicit symmetry breaking). Thus, pions are a truly remarkable species: theyreflect spontaneous as well as explicit symmetry breaking. Goldstone bosonsinteract weakly at low energy. They are degenerate with the vacuum and,therefore, interactions between them must vanish at zero momentum and inthe chiral limit (mπ → 0).

Pion-Exchange Contributions in ChPT

After the discussion of the soft and hard scales, and the identification of therelevant symmetries of low-energy QCD, we are ready to construct the mostgeneral Lagrangian consistent with the chiral symmetry and its breaking con-figurations. The relevant degrees of freedom are pions (Goldstone bosons) andnucleons. Since the interactions of Goldstone bosons must vanish at zero mo-mentum transfer and in the chiral limit (mπ → 0), the low-energy expansion ofthe Lagrangian is arranged in powers of derivatives and pion masses. The hardscale is the chiral symmetry breaking scale, Λχ ≈ 1 GeV. Thus, the expansion

88


is in terms of powers of Q/Λχ where Q is a (small) momentum or pion mass.This is the ChPT.

The effective Lagrangian needed to derive nuclear forces can formally bewritten as,

Leff = Lππ + LπN + LNN + · · · , (4.74)

where Lππ deals with the dynamics among pions, LπN describes the interactionbetween pions and a nucleon, and the ellipsis stands for terms that involve twonucleons plus pions and three or more nucleons with or without pions, relevantfor nuclear many-body forces. The individual Lagrangians are organized asfollows:

Lππ = L(2)ππ + L(4)

ππ + · · · (4.75)

andLπN = L(1)

πN + L(2)πN + L(3)

πN + · · · , (4.76)

where the superscript refers to the number of derivatives or pion-mass inser-tions (chiral dimension) and the ellipsis stands for terms of higher dimensions.The two-nucleon contact interaction term LNN consists of four-nucleon fields(four-nucleon legs) and no meson fields. Such terms are needed to renormal-ize loop integrals, to make results reasonably independent of regulators, andto parametrize the unresolved short-distance dynamics of the nuclear force.Because of parity, nucleon contact interactions come only in even powers ofderivatives as

LNN = L(0)NN + L(2)

NN + L(4)NN + · · · . (4.77)

Based upon the effective pion Lagrangians discussed above we can derive thepion-exchange contributions to the NN interaction order by order. There areinfinitely many pion-exchange contributions to the N interaction and, thus,we need to get organized. First, we arrange the various pion-exchange contri-butions according to the number of pions being exchanged between the twonucleons,

Vπ = V1π + V2π + V3π + · · · , (4.78)

where the meaning of the subscripts is obvious and the ellipsis represents 4πand higher pion exchanges. Second, for each of the above terms, we assume alow-momentum expansion:

V1π = V(0)1π + V

(2)1π + V

(3)1π + V

(4)1π + · · · ,

V2π = V(2)2π + V

(3)2π + V

(4)2π + · · · ,

V3π = V(4)3π + · · · ,

(4.79)

where the superscript denotes the order ν and the ellipses stand for contribu-tions of fifth and higher orders. Due to parity and time reversal, there are nofirst order contributions in Eq. (4.79). The order ν is given by

ν = 2L+∑

i

∆i , (4.80)

89


Figure 4.1: Hierarchy of nuclear forces in ChPT. Solid lines represent nucleonsand dashed lines pions. Small dots, large solid dots, solid squares, and soliddiamonds denote vertices of index ∆ = 0, 1, 2, and 4, respectively. Furtherexplanations can be found in Ref. [244].

where L is the number of loops in the Feynman diagram and

∆i ≡ di +ni

2− 2 . (4.81)

In the above expression di is the number of derivatives or pion-mass insertionsand ni is the number of nucleon fields (nucleon legs) involved in vertex i; thesum runs over all vertices i contained in the diagram under consideration.Moreover, since n pion create L = n − 1 loops, the leading order for n-pionexchange occurs at ν = 2n − 2. For the explicit evaluation of V1π, V2π, andV3π, one by one and order by order, see Ref. [244].

The Hierarchy of Nuclear Forces

The chief point of the ChPT expansion is that, at a given order ν, there existsonly a finite number of graphs. This is what makes the theory calculable.The expression (Q/Λχ)ν+1 provides a rough estimate of the relative size of thecontributions left out and, thus, of the accuracy at order ν. In this sense, thetheory can be calculated to any desired accuracy and has predictive power.

90


ChPT and power counting imply that nuclear forces emerge as a hierarchycontrolled by the power ν as displayed in Fig. 4.1.

In lowest order, better known as leading order (LO, ν = 0), the NN am-plitude is made up by two momentum-independent contact terms (∼ Q0),represented by a four-nucleon-leg graph with a small-dot vertex and a staticone-pion exchange (1PE) graph. This is, of course, a rather crude approxima-tion to the two-nucleon forces (2NF), but accounts already for some impor-tant features. The 1PE provides the tensor force, necessary to describe thedeuteron, and it explains NN scattering in peripheral partial waves of veryhigh orbital angular momentum. At this order, the two contacts which con-tribute only in S waves provide the short- and intermediate-range interactionwhich is somewhat crude.

In the next order, ν = 1, all contributions vanish due to parity and time-reversal invariance.

Therefore, the next-to-leading order (NLO) is ν = 2. Two-pion exchange(2PE) occurs for the first time (“leading 2PE”) and, thus, the creation of amore sophisticated description of the intermediate-range interaction is startinghere. Since the loop involved in each pion diagram implies already ν = 2, thevertices must have ∆i = 0. Therefore, at this order, only the lowest orderπNN and ππNN vertices are allowed which is why the leading 2PE is ratherweak. Furthermore, there are seven contact terms of O(Q2) which contributein S and P waves. The operator structure of these contacts include a spin-orbit term besides central, spin-spin, and tensor terms. Thus, essentially allspin-isospin structures necessary to describe the 2NF phenomenologically havebeen generated at this order. The main deficiency at this stage of developmentis an insufficient intermediate-range attraction.

This problem is finally fixed at order three (ν = 3), next-to-next-to-leadingorder (NNLO). The 2PE involves now the two derivative ππNN vertices thatrepresent correlated 2PE as well as intermediate ∆(1232)-isobar contributions.It is well known from the meson phenomenology of nuclear forces that thesetwo contributions are crucial for a realistic and quantitative 2PE model. Con-sequently, the 2PE now assumes a realistic size and describes the intermediate-range attraction of the nuclear force about right. Moreover, first relativisticcorrections come into play at this order and there are no new contacts.

The reason why we talk of a hierarchy of nuclear forces is that two- andmany-nucleon forces are created on an equal footing and emerge in increasingnumber as we go to higher and higher orders. At NNLO, the first set of non-vanishing three-nucleon forces (3NF) occurs. In fact, at the previous order,NLO, irreducible three-nucleon graphs appear already, however, it has beenshown by Weinberg and others that these diagrams all cancel. Since non-vanishing 3NF contributions happen first at order (Q/Λχ)3, they are very weakas compared to 2NF which start at (Q/Λχ)0.

More 2PE is produced at ν = 4, next-to-next-to-next-to-leading order(N3LO). Two-loop 2PE graphs show up for the first time and so does three-

91


pion exchange (3PE) which necessarily involves two loops. 3PE was found tobe negligible at this order. Most importantly, 15 new contact terms ∼ Q4

arise. They include a quadratic spin-orbit term and contribute up to D waves.Mainly due to the increased number of contact terms, a quantitative descrip-tion of the two-nucleon interaction up to about 300 MeV laboratory energyis possible at N3LO. Besides further 3NF, four-nucleon forces (4NF) start atthis order. Since the leading 4NF come into existence one order higher thanthe leading 3NF, 4NF are weaker than 3NF. Thus, ChPT provides a straight-forward explanation for the empirically known fact that 2NF ≫ 3NF ≫ 4NFand so on.

Definition of the Nucleon-Nucleon Potential

Now we have everything we need for a realistic nuclear force: long, intermedi-ate, and short-ranged components. Since the irreducible diagrams that makeup the potential are calculated using covariant perturbation theory, it is consis-tent to start from the covariant BS equation describing two-nucleon scattering.In operator notation, the BS equation reads

T = V + VGT , (4.82)

where T denotes the invariant T -matrix for the two-nucleon scattering process,V the sum of all connected two-particle irreducible diagrams, and G is (−i)times the relativistic two-nucleon propagator. The BS equation is equivalentto a set of two equations

T = V + V gT ,

V = V + V (G − g)V = V + V1π (G − g)V1π + · · · , (4.83)

where g is a covariant three-dimensional propagator which preserves relativisticelastic unitarity and the ellipsis stands for terms of irreducible 3π and higherpion exchanges which we neglect.

When we speak of covariance in conjunction with (heavy baryon) ChPT,we are not referring to manifest covariance. Relativity and relativistic off-shelleffects are accounted for in terms of a Q/M expansion up to the given orderand up to the number of pions we take into consideration. Thus, Eq. (4.83)for V is evaluated, after some approximations, in the following way,

V ≈ V1π + V ′2π , (4.84)

where V1π is the on-shell 1PE and V ′2π subsumes all 2π exchanges without the

iterated V1π and is, therefore, also known as the irreducible 2PE. Since allcontributions are calculated on-shell, the potential has no energy dependence.Adding the short-range contact terms Vct to the above, yields the full NNpotential:

V = V1π + V ′2π + Vct . (4.85)

92


The irreducible 2PE V ′2π is organized according to increasing orders,

V ′2π = V

′ (2)2π + V

′ (3)2π + V

′ (4)2π + · · · , (4.86)

while the contact potential comes in even orders as

Vct = V(0)ct + V

(2)ct + V

(4)ct + · · · . (4.87)

In summary, the NN potential V , calculated to certain orders, is given by:

VLO = V1π + V(0)ct ,

VNLO = VLO + V′ (2)2π + V

(2)ct ,

VNNLO = VNLO + V′ (3)2π ,

VN3LO = VNNLO + V′ (4)2π + V

(4)ct .

(4.88)

The potential V satisfies the relativistic BS equation, Eq. (4.83), which readsexplicitly as

T (q′, q) = V (q′, q) +

∫

d3k

(2π)3V (q′,k)

M2

Ek

1

q2 − k2 + iǫT (k, q) , (4.89)

where Ek =√M2 + k2. The advantage of using a relativistic scattering equa-

tion is that it automatically includes relativistic corrections to all orders. Thus,in the scattering equation, no propagator modifications are necessary whenraising the order to which the calculation is conducted. Defining

V (q′, q) ≡ 1

(2π)3

√

M

Eq′V (q′, q)

√

M

Eq

,

T (q′, q) ≡ 1

(2π)3

√

M

Eq′T (q′, q)

√

M

Eq

,

(4.90)

where the factor 1/(2π)3 is added for convenience, the BS equation collapsesinto the usual non-relativistic LS equation,

T (q′, q) = V (q′, q) +

∫

d3k V (q′,k)M

q2 − k2 + iǫT (k, q) . (4.91)

Since V satisfies Eq. (4.91), it can be used like a usual non-relativistic potential,and T may be perceived as the conventional non-relativistic T -matrix.

Iteration of V in the LS equation, Eq. (4.91), requires cutting V off for highmomenta to avoid infinities. This is consistent with the fact that ChPT is alow-momentum expansion which is valid only for momenta Q≪ Λχ ≈ 1 GeV.

Therefore, the potential V is multiplied with the regulator function f(q′, q),

V (q′, q) 7→ V (q′, q) f(q′, q) (4.92)

93


withf(q′, q) = exp

[

−(q′/Λ)2n − (q/Λ)2n

]

, (4.93)

such that

V (q′, q) f(q′, q) ≈ V (q′, q)

1 −[

(

q′

Λ

)2n

+( q

Λ

)2n]

+ · · ·

. (4.94)

Typical choices for the cutoff parameter Λ that appears in the regulator areΛ ≈ 0.5 GeV ≪ Λχ ≈ 1 GeV. Eq. (4.94) provides an indication of the factthat the exponential cutoff does not necessarily affect the given order at whichthe calculation is conducted. For sufficiently large n, the regulator introducescontributions that are beyond the given order. Assuming a good rate of conver-gence of the chiral expansion, such orders are small as compared to the givenorder and, thus, do not affect the accuracy at the given order. In calculations,one uses, of course, the exponential form, Eq. (4.93), and not the expansion,Eq. (4.94). On a similar note, we also do not expand the square-root factors inEqs. (4.90) because they are kinematical factors which guarantee relativisticelastic unitarity.

It is pretty obvious that results for the T -matrix may depend sensitivelyon the regulator and its cutoff parameter. This is acceptable if one wishesto build models, however, the EFT approach wishes to be fundamental innature and not just another model. In field theories, divergent integrals arenot uncommon and methods have been developed to deal with them. Oneregulates the integrals and then removes the dependence on the regularizationparameters (scales and cutoffs) by renormalization. In the end, the theoryand its predictions do not depend on cutoffs or renormalization scales. As afinal comment, we simply observe that so-called renormalizable quantum fieldtheories, like QED, have essentially one set of prescriptions that takes care ofrenormalization through all orders, while, in contrast, EFTs are renormalizedorder by order.

4.4 The Nucleon-Nucleon Transition Matrix

The NN elastic scattering amplitude for scattering from relative momentumκ to κ′, denoted with M(κ′,κ, ω), is related to the antisymmetrized transitionmatrix elements by the usual relation (~ = 1)

M(κ′,κ, ω) = 〈κ′|M(ω)|κ〉 = −4π2µ 〈κ′|t(ω)|κ〉 , (4.95)

with µ the NN reduced mass. The most general form of this amplitude,consistent with invariance under rotation, time reversal and parity is [245]

M = a+ c(σ1 + σ2) · n +m(σ1 · n)(σ2 · n)

+ (g + h)(σ1 · l)(σ2 · l) + (g − h)(σ1 · m)(σ2 · m) ,(4.96)

94

4.4. The Nucleon-Nucleon Transition Matrix

where

l =κ′ + κ

|κ′ + κ| , m =κ′ − κ

|κ′ − κ| , n =κ× κ′

|κ× κ′| , (4.97)

are the unit vectors defined by the NN scattering plane. The amplitudes a, c,m, g, and h can be expressed as complex functions of the relative energy ω, andinitial and final relative momenta κ and κ′ of the NN pair in their center-of-mass frame. The amplitudes of Eq. (4.96) are given in the Hoshizaki notation;there are different ways to define them. A survey of the other decompositionscan be found in Refs. [246–248] and references therein. We also may note thatfor an even-even nucleus with J = 0, terms linear in the spin of the targetnucleons average to zero; only a and c amplitudes survive and these ones areconnected to the central and spin-orbit part of the NN t matrix.

For ease of calculation the NN amplitudes are expressed in terms of thedecomposition of the scattering amplitude into components describing spinsinglet (S = 0) and spin triplet (S = 1) scattering, MS

ν′ν , where ν and ν ′

refer to the incident and final spin projections in the triplet state. In therepresentation in which these projections are referred to an axis of quantizationalong the incident beam direction (κ) we have

a =1

4(2M1

11 +M100 +M0

00) ,

c =i

2√

2(M1

10 −M101) ,

m =1

4(M1

00 − 2M11−1 −M0

00) ,

g =1

4(M1

11 −M000 +M1

1−1) ,

h =1

4 cosφ(M1

11 −M100 −M1

1−1) .

(4.98)

The amplitudes MSν′ν = 〈κSν ′|M(ω)|κSν〉 and hence the amplitudes a through

h, are obtained [249, 250] in terms of the partial wave components of the NNamplitude, MJS

L′L(κ′, κ;ω), defined according to:

M(κ′,κ;ω) =2

π

∑

JLL′SM

iL−L′YL′SJM (κ′)MJS

L′L(κ′, κ;ω)YLS †JM (κ) , (4.99)

where YLSJM is the spin-angle function

YLSJM(κ) =

∑

Λν

(LΛSν|JM) Y ΛL (κ) ⊗ χSν , (4.100)

and Y ΛL and χSν are the spherical harmonic and spin wave function of the NN

pair, respectively. Explicitly,

MSν′ν =

2

π

∑

JMLL′ΛΛ′

iL−L′

(L′Λ′Sν ′|JM)(LΛSν|JM)

× Y Λ′

L′ (κ′)Y Λ ∗L (κ)MJS

L′L(κ′, κ;ω) .

(4.101)

95


tS=0,T=1L,LL : 1S0,

1D2,1G4,

1I6,1K8 tS=0,T=0

L,LL : 1P1,1F3,

1H5,1J7

tS=1,T=1L−1,LL : 3P0,

3F2,3H4,

3J6,3L8 tS=1,T=0

L−1,LL : 3D1,3G3,

3I5,3K7

tS=1,T=1L,LL : 3P1,

3F3,3H5,

3J7 tS=1,T=0L,LL : 3D2,

3G4,3I6,

3K8

tS=1,T=1L+1,LL : 3P2,

3F4,3H6,

3J8 tS=1,T=0L+1,LL : 3S1,

3D3,3G5,

3I7

Table 4.1: Partial waves of the NN potential used to construct the three-dimensional NN t matrix tpN(κ′,κ;ω).

Detailed formulas for the required MSν′ν amplitudes in terms of the partial

wave amplitudes MJSL′L(κ′, κ;ω) for a quantization axis along the incident beam

direction are collected in App. A. According to those formulas we are nowable to give the explicit expressions for the proton-proton and proton-neutroncentral and spin-orbit parts of the NN t matrix. The optical potential obtainedin the previous section is an operator in the spin space of the projectile. Tomake the spin dependence explicit we write the t matrix in the form (N = p, n):

tpN(κ′,κ;ω) = tcpN(κ′,κ;ω) +i

2σ · κ′ × κ tlspN(κ′,κ;ω) . (4.102)

In terms of the partial wave components tSTJLL(κ′, κ;ω) we have the followingresuls for the central part:

tcpp(κ′,κ;ω) =

1

4π2

∞∑

L=0

PL(cos φ)[

(2L+ 1) tS=0,T=1L,LL (κ′, κ;ω)

+ (2L+ 1) tS=1,T=1L,LL (κ′, κ;ω) + (2L− 1) tS=1,T=1

L−1,LL (κ′, κ;ω)

+ (2L+ 3) tS=1,T=1L+1,LL (κ′, κ;ω)

]

,

tcpn(κ′,κ;ω) =1

8π2

∞∑

L=0

PL(cos φ)[

(2L+ 1) tS=0,T=0L,LL (κ′, κ;ω)

+ (2L+ 1) tS=1,T=0L,LL (κ′, κ;ω) + (2L− 1) tS=1,T=0

L−1,LL (κ′, κ;ω)

+ (2L+ 3) tS=1,T=0L+1,LL (κ′, κ;ω) + (2L+ 1) tS=0,T=1

L,LL (κ′, κ;ω)

+ (2L+ 1) tS=1,T=1L,LL (κ′, κ;ω) + (2L− 1) tS=1,T=1

L−1,LL (κ′, κ;ω)

+ (2L+ 3) tS=1,T=1L+1,LL (κ′, κ;ω)

]

.

96

4.5. Theoretical Results for Nucleon-Nucleon Amplitudes

The spin-orbit part is given by:

tlspp(κ′,κ;ω) = − 1

2π2

∞∑

L=1

d PL(cosφ)

d cosφ

1

κ′κ

[

− 2L− 1

LtS=1,T=1L−1,LL (κ′, κ;ω)

− 2L+ 1

L(L + 1)tS=1,T=1L,LL (κ′, κ;ω) +

2L+ 3

L + 1tS=1,T=1L+1,LL (κ′, κ;ω)

]

,

tlspn(κ′,κ;ω) = − 1

4π2

∞∑

L=1

d PL(cosφ)

d cosφ

1

κ′κ

[

− 2L− 1

LtS=1,T=0L−1,LL (κ′, κ;ω)

− 2L+ 1

L(L + 1)tS=1,T=0L,LL (κ′, κ;ω) +

2L+ 3

L + 1tS=1,T=0L+1,LL (κ′, κ;ω)

− 2L− 1

LtS=1,T=1L−1,LL (κ′, κ;ω) − 2L+ 1

L(L + 1)tS=1,T=1L,LL (κ′, κ;ω)

+2L + 3

L+ 1tS=1,T=1L+1,LL (κ′, κ;ω)

]

.

The partial wave components tSTJLL(κ′, κ;ω) are computed in the NN center-of-mass frame, starting from a NN interaction computed with a NN potentiallike the CD-Bonn or the Chiral one. The tSTJLL(κ′, κ;ω) matrices are computedin each partial wave up to J = 8 using the matrix-inversion method (seeApp. B.1 for the details) and are collected in the Tab. 4.1.

4.5 Theoretical Results for Nucleon-Nucleon

Amplitudes

In this section we present and discuss the theoretical results for the proton-proton and proton-neutron a and c Wolfenstein amplitudes used to computethe central and spin-orbit parts of the three-dimensional NN t matrix. Cal-culations are performed using three different potentials to generate the NNinteraction: the CD-Bonn potential by Machleidt and two different versions ofthe Chiral potential at the fourth order (N3LO) based on works of Entem andMachleidt (Ch-N3LO) and Epelbaum, Glockle, and Meißner (Ch-N3LO-500MeV, Ch-N3LO-600 MeV, and Ch-N3LO-700 MeV).

This Chiral potential consists of 1PE, 2PE, and a string of contact inter-actions with an increasing number of derivatives (zero, two, four) that param-eterize the shorter ranged components of the nuclear force. Such an approachhas, in general, a variety of advantages over more conventional schemes orphenomenological models. First, it offers a systematic method to improve cal-culations by going to ever increasing orders in the power counting and it allowsto give theoretical uncertainties. Second, one can consistently derive 2NF and3NF, which has never been achieved before and, third, nucleon and nuclearproperties can be calculated from one effective Lagrangian.

In Section 4.3 we have described the main characteristics of CD-Bonn andChiral potentials that we use to generate the NN potential and to compute the

97


0.10.20.30.40.50.6

a pp [

fm]

CD-Bonn

Ch-N3LO

Ch-N3LO - 500 MeV

-0.1

0

0.1

0.2

0.3

-0.005

0

0.005

0.01

0.015

c pp [

fm]

0

0.1

0.2

0.3

0.4

0.4

0.6

0.8

1

a pn [

fm]

0.2

0.3

0.4

0.5

0.6

0 30 60 90 120 150 180φ [deg]

-0.06-0.04-0.02

00.020.040.06

c pn [

fm]

0 30 60 90 120 150 180φ [deg]

0

0.05

0.1

0.15

0.2

Re Im

Figure 4.2: Real (left panel) and imaginary (right panel) parts of proton-proton and proton-neutron a and c Wolfenstein amplitudes as functions of thecenter-of-mass NN angle φ. All the amplitudes are computed at 100 MeVusing the CD-Bonn (red), the Ch-N3LO (green), and the Ch-N3LO-500 MeVpotentials. Data are taken from Ref. [251].

tSTJLL(κ′, κ;ω) matrices in each partial wave. However there are some differencesbetween the two codes that permit to generate the Chiral potential and beforediscussing the numerical results we outline these differences.

Entem and Machleidt, who first presented a Chiral potential at the fourthorder, use in their computer code the 2PE contributions based on dimen-sional regularization, which has a very singular short-range behavior. In con-trast, Epelbaum, Glockle, and Meißner employ spectral function regularization,which allows for a better separation between the long- and short-distance con-tributions and allows to significantly improve the convergence of chiral EFTfor the two-nucleon system. Second, Entem and Machleidt present results onlyfor one choice of the cut-off necessary to regulate the high-momentum com-ponents in the LS equation to generate the scattering and the bound states.In contrast, Epelbaum et al. perform systematic variations of this cut-off andother parameters which allow to give not only central values but also theo-retical uncertainties. Third, Epelbaum et al. employ a relativistic version

98


0.10.20.30.40.50.6

a pp [

fm]

Ch-N3LO - 700 MeV

Ch-N3LO - 600 MeV

-0.1

0

0.1

0.2

0.3

-0.008-0.006-0.004-0.002

00.0020.004

c pp [

fm]

0

0.1

0.2

0.3

0.4

0.2

0.4

0.6

0.8

a pn [

fm]

0.2

0.3

0.4

0.5

0.6

0 30 60 90 120 150 180φ [deg]

-0.06-0.04-0.02

00.020.04

c pn [

fm]

0 30 60 90 120 150 180φ [deg]

0

0.05

0.1

0.15

0.2

Re Im

Figure 4.3: Real (left panel) and imaginary (right panel) parts of proton-proton and proton-neutron a and c Wolfenstein amplitudes as functions of thecenter-of-massNN angle φ. All the amplitudes are computed at 100 MeV usingthe Ch-N3LO-600 MeV (red) and the Ch-N3LO-700 MeV (green) potentials.Data are taken from Ref. [251].

of the Schrodinger equation, which allows to calculate consistently relativisticcorrections also in three and four nucleon systems. In addition to these maindifferences, there are also a different treatment of the isospin breaking effectsand other less significant differences.

For our purposes the key points concern the possibility to vary the cut-offin the LS equation and the regularization scheme. So, in the computer codeby Epelbaum et al. we can choose the value of these parameters: the first one(LS cut-off) is the cut-off used in the LS equation and the second one (SFRcut-off) is the cut-off used in the spectral-function representation of the 2PEpotential. The SFR cut-off is varied between 500 and 700 MeV. The LS cut-offis the largest possible value which still leads to low-energy constants (LECs)of natural size and it depends on the chiral order. The LECs are related topion-nucleon vertices and are taken consistently from studies of pion-nucleonscattering in ChPT. The two cut-offs cannot be varied independently, but foreach value of the SFR cut-off we have one or two values of the LS cut-off for

99


0

0.1

0.2

0.3

0.4a pp

[fm

]

CD-Bonn

Ch-N3LO

Ch-N3LO - 500 MeV

-0.2-0.1

00.10.20.30.4

0

0.01

0.02

0.03

0.04

c pp [

fm]

0

0.1

0.2

0.3

0.4

0

0.2

0.4

0.6

0.8

1

a pn [

fm]

00.10.20.30.40.5

0 30 60 90 120 150 180φ [deg]

-0.05

0

0.05

c pn [

fm]

0 30 60 90 120 150 180φ [deg]

00.05

0.10.150.2

0.25

Re Im

Figure 4.4: The same as in Fig. 4.2 but for an energy of 200 MeV. Data aretaken from Ref. [251].

a total number of combinations of five. The available choices are:

1. LS cut-off = 450 MeV, SFR cut-off = 500 MeV,




5. LS cut-off = 600 MeV, SFR cut-off = 700 MeV.

There are three different values of the SFR cut-off: 500, 600, and 700 MeV. Forthe first one there is only one value for the LS cut-off, while for the other twothere are two possible values for the LS cut-off. Thus, the theoretical resultsobtained with the Epelbaum potential are indicated using the value of theSFR cut-off: for a value of 500 MeV we use a line because we have only one LScut-off, while for the other two, 600 and 700 MeV, we use a band representingthe calculations performed with both LS cut-offs.

In Fig. 4.2 the theoretical results for the real and imaginary parts of proton-proton and proton-neutron a and c Wolfenstein amplitudes computed at an

100


0

0.1

0.2

0.3

0.4a pp

[fm

]

Ch-N3LO - 600 MeV

Ch-N3LO - 700 MeV

-0.2-0.1

00.10.20.30.4

0

0.01

0.02

0.03

c pp [

fm]

0

0.1

0.2

0.3

0.4

0

0.2

0.4

0.6

0.8

1

a pn [

fm]

00.10.20.30.40.5

0 30 60 90 120 150 180φ [deg]

-0.05

0

0.05

c pn [

fm]

0 30 60 90 120 150 180φ [deg]

00.05

0.10.150.2

0.25

Re Im

Figure 4.5: The same as is Fig. 4.3 but for an energy of 200 MeV. Data aretaken from Ref. [251].

energy of 100 MeV are shown as functions of the center-of-mass NN angle φ.The calculations are performed with the CD-Bonn, Ch-N3LO, and Ch-N3LO-500 MeV potentials. The experimental data are globally well reproduced bythe three potentials with the only exception of the real part of the proton-proton c amplitude that is overestimated. However, we remark that this is asmall quantity, two orders of magnitude smaller than the respective imaginarypart, and it will have a little contribution in the expression for the optical po-tential. Concerning the other amplitudes, some small discrepancies are foundin comparison with data: the Ch-N3LO-500 MeV potential underestimatesthe minum around 135 degrees of the real cpn amplitude and overestimatesthe maximum of the imaginary cpp amplitude, while the CD-Bonn potentialslightly underestimates the app amplitude at small angles and the imaginarypart of the apn amplitude.

In Fig. 4.3 we show the results at 100 MeV obtained with the Ch-N3LO-600MeV and Ch-N3LO-700 MeV potentials. Also in this case the two potentialsare in good agreement with the experimental data with the only exception ofthe real part of the cpp amplitude. In particular, they show very similar resultsand in many cases the red and the green bands are superimposed. Their trends

101


are also very close to the ones obtained with the Ch-N3LO-500 MeV potentialshowed in Fig. 4.2 and they display the same discrepancy in comparison withthe experimental data in the real part of the cpn amplitude and in the peak ofthe imaginary part of the cpp amplitude.

Since ChPT is a low-momentum expansion of QCD, we expect that thebehavior of Chiral potentials, based on it, will become worse with the increasingof the energy. In Figs. 4.4 and 4.5 we present the results corresponding to theones shown in Figs. 4.2 and 4.3 but for an energy of 200 MeV. All the potentialsare not able to reproduce the experimental data of the real part of the cppamplitude, but the CD-Bonn and the Ch-N3LO potentials give satisfactoryresults in agreement with the data for all the other amplitudes, while the Ch-N3LO-500 MeV potential begins to fail. In particular, it overestimates the realcpn amplitude, it underestimates the imaginary part of the cpp amplitude, andit displays a little discrepancy in the imaginary part of the cpn amplitude forangles larger than 90 degrees. Moreover, the real part of app is completelywrong and, due to the magnitude of this amplitude, we expect a bad result forthe optical potential.

In Fig. 4.5 the increasing of the cut-off helps to enhance the results. Alsoin this case the potentials are not able to reproduce the experimental dataof the real cpp amplitude, but they show a global good agreement with thedata for all the other amplitudes, and, as shown in Fig. 4.3, in many cases thetwo bands are superimposed. The main difference with respect to the resultsobtained with the Ch-N3LO-500 MeV potential shown in Fig. 4.4 is given bythe results for the real app amplitude, where the increasing of the cut-offs makesthe potential able to reproduce the experimental data with better accuracy.

102

Chapter 5Relativistic Kinematics and theScattering Observables

In Chapter 4 we have introduced the proton-nucleus scattering problem inmomentum space which consists in solving two different equations: the firstone for the transition matrix and the second one for the optical potential.After some approximations we have reduced the complicated expression for theoptical potential to a simple one in which the potential is given in a factorizedform of the NN t matrix and the nuclear density. We have also seen how theNN t matrix can be separated into a central and a spin-orbit part and howthe two parts can be computed in the partial wave representation.

In this chapter, starting from the optical potential calculated in Chapter 4,we will focus on the solution of the relativistic LS equation that permits tocompute the NA transition amplitude and calculate the physical observables.In Section 5.1 we introduce the variables and the relativistic kinematics usedto pass from the NN frame, in which we evaluate the NN t matrix, to theNA frame, in which we evaluate the transition amplitude. In Section 5.2 weshow how to explicitly compute the NA transition amplitude in the partialwave representation in the NA center-of-mass frame and in Section 5.3 we usethese amplitudes to obtain the expressions for the scattering observables. InSection 5.4 we give the details of the procedure used to insert the Coulombinteraction in the previous scheme. Finally, in Section 5.5 we present anddiscuss the theoretical results for the scattering observables computed with andwithout the Coulomb interaction. In particular, we first compare our results for16O and 40Ca with experimental data and, successively, we present and discussour numerical predictions, obtained with both CD-Bonn and Chiral potentials,for the two oxygen and calcium isotopic chains. Finally, in Section 5.6 wesummarize our results and present our conclusions.

103

5. Relativistic Kinematics and the Scattering Observables

5.1 Kinematics

The collision in the NA reference frame occurs in a scattering plane where thebeam is located along the z axis. The target is a spherical spin 0 nucleus whichwhen struck will undergo recoil. The projectile is a spin 1/2 nucleon, either aneutron or a proton, but in the following we will only consider the latter case.To compute the elastic scattering cross section and the other observables weemploy the momentum space representation of the elastic scattering equationfor the transition amplitude. The problem at hand is now described in thezero momentum frame of the NA system by Eq. (4.33), where the input is theoptical potential U(k′,k;ω) given in Chapter 4 by Eqs. (4.59) and (4.60) andobtained from a NN potential.

The variables k and k′ are the momentum of the projectile before andafter the collision in the NA frame. The momentum transfer q and the totalmomentum K are defined in Eq. (4.37). These vectors are used throughoutthis chapter because they make the computation of the optical potential moreconvenient. In three dimensions, the vectors q and K define the scatteringplane. The vector normal to this plane is given by

N =k′ × k

|k′| |k| . (5.1)

The vectors q, K, and N span the entire momentum space and thus all func-tions in this chapter can be represented as functions of this set of vectors.

5.1.1 Relativistic Kinematics

To calculate energies that enter Eq. (4.33) and the NN equations we use therelativistic definition. In the following we also assume c = 1. The center ofmass energy in the NA system is given by

E = Eproj + Etarg =√

k20 +m2proj +

√

k20 +m2targ , (5.2)

where m is the rest mass and k0 is the on-shell momentum in the NA centerof mass frame. To obtain k0 we use the invariant mass S, which is given by

S =√

m2targ +m2

proj + 2mtargElab , (5.3)

where Elab is the total projectile energy given by

Elab = Tlab +mproj =√

k2lab +m2proj . (5.4)

Here Tlab is the laboratory kinetic energy of the projectile. Then, we use thefact that E is equal to the invariant mass S and from the previous equationswe obtain:

k0 =klabmtarg

S. (5.5)

104

5.1. Kinematics

The same derivation is carried out for the momenta and energies in the center-of-mass frame for the NN system where the NN t matrix is defined. Aninvariant mass is here defined by using the mass of the projectile and target

SNN =√

2m2proj + 2mprojElab , (5.6)

and analogous to the NA system we find the on-shell NN center-of-mass mo-mentum as

kNN =klabmproj

SNN. (5.7)

In the NN frame the total energy is defined as

ENN = 2√

k2NN +m2proj . (5.8)

Finally, from previous equations we obtain

T 2lab +

(

2mproj −2k2NN

mproj

)

Tlab − 4k2NN = 0 , (5.9)

with the solution

Tlab =2k2NN

mproj. (5.10)

We point out that the optical potential calculation is a non-relativistic cal-culation in which the non-relativistic momentum is conserved. However, asdescribed above, the NA scattering kinematics is treated relativistically.

5.1.2 The Møller Factor

In Chapter 4 we derived an expression for the central and spin-orbit partsof the optimally factorized optical potential given by Eqs. (4.58)-(4.60). Inparticular, we introduced the Møller factor in order to impose the Lorentzinvariance of the flux when we pass from the NA system to NN system. Nowwe are able to explicitly evaluate it. For an implementation it is better touse NA variables. Starting from Eq. (4.44) we set P = 0, and adopting therelativistic kinematics we obtain

η(q,K) =

[

E2proj

[

12

(

A+1A

K − q)]

E2proj

[

12

(

A+1A

K + q)]

Eproj

(

K − q

2

)

Eproj

(

q

2− K

A

)

Eproj

(

K + q

2

)

Eproj

(

−q

2− K

A

)

] 1

2

,

(5.11)

105


where

E2proj

[

1

2

(

A+ 1

AK ± q

)]

=1

4

[(

A+ 1

A

)2

K2 + q2

± 2(A+ 1)

AqK cos η

]

+m2proj ,

Eproj

(

K ± q

2

)

=

√

K2 +q2

4± qK cos η +m2

proj ,

Eproj

(

±q

2− K

A

)

=

√

q2

4+K2

A2∓ qK

Acos η +m2

proj .

(5.12)

5.2 The Transition Amplitude in Partial Wave

Representation

The optimally factorized first-order KMT optical potential as an operator inthe spin space of the projectile is given in Eq. (4.58) as

U(k′,k;ω) = U c(k′,k;ω) +i

2σ · k′ × k U ls(k′,k;ω) . (5.13)

From the conservation of total angular momentum and parity, this spin oper-ator can be expanded as

U(k′,k;ω) =2

π

∑

JLM

YL 1

2

JM(k′) ULJ(k′, k;ω)YL 1

2†

JM (k) , (5.14)

where J = L± 1/2 and YL 1

2

JM is the standard spin-angle function

YL 1

2

JM(k) =∑

Λν

(LΛ12ν|JM) Y Λ

L (k) ⊗ χ 1

2ν . (5.15)

When the expansion in Eq. (5.14) is substituted into the Eq. (4.33) and we usethe same decomposition for the T matrix

T (k′,k;E) =2

π

∑

JLM

YL 1

2

JM(k′) TLJ(k′, k;E)YL 1

2†

JM (k) , (5.16)

the partial wave components of the resulting transition operator for elasticscattering are given by

TLJ(k′, k;E) = ULJ(k′, k;ω) +2

π

∫ ∞

0

dp p2ULJ(k′, p;ω) TLJ(p, k;E)

E(k0) −E(p) + iǫ, (5.17)

where

E(k0) = Eproj(k0) + Etarg(k0) =√

k20 +m2proj +

√

k20 +m2targ ,

E(p) = Eproj(p) + Etarg(p) =√

p2 +m2proj +

√

p2 +m2targ ,

(5.18)

106

5.3. The Scattering Observables

and mproj and mtarg are the projectile and the target mass, respectively. The

details of the calculation of the partial wave components ULJ of the opticalpotential are provided in Appendix C. We summarize here the importantformulae in two steps. In terms of the partial wave components of the quantitiesU c(k′,k;ω) and U ls(k′,k;ω), we have

ULJ(k′, k;ω) = U cL(k′, k;ω) + CLJ V

lsL (k′, k;ω) ,

CLJ =1

2

[

J(J + 1) − L(L + 1) − 3

4

]

,

V lsL (k′, k;ω) =

k′k

2L+ 1

[

U lsL+1(k

′, k;ω) − U lsL−1(k

′, k;ω)]

.

(5.19)

To obtain these results, the quantities U c(k′,k;ω) and U ls(k′,k;ω) are ex-panded in a manner similar to Eq. (5.14) with the difference that the partialwave components, U c

L and U lsL , are independent of J . The partial wave com-

ponents of U c(k′,k;ω) and U ls(k′,k;ω) can be calculated in terms of the NNt-matrix components and nuclear densities from Eqs. (4.59) and (4.60). Theprojection can be performed numerically by evaluating the integrals

U cL(k′, k;ω) = π2

∫ +1

−1

dxPL(x)U c(k′,k;ω) = π2

∫ +1

−1

dxPL(x)U c(k′, k, x;ω) ,

U lsL (k′, k;ω) = π2

∫ +1

−1

dxPL(x)U ls(k′,k;ω) = π2

∫ +1

−1

dxPL(x)U ls(k′, k, x;ω) ,

(5.20)

where x = cos θ and the potentials in terms of k′ and k are obtained fromEqs. (4.59) and (4.60) with

q(x) =√k′ 2 + k2 − 2k′kx ,

K(x) =1

2

√k′ 2 + k2 + 2k′kx ,

q ·K =1

2

(

k′ 2 − k2)

.

(5.21)

The one-dimensional integral equation, Eq. (5.17), for the partial wave ele-ments TLJ is solved for the complex potentials ULJ . In actual calculations, thenumber of L values needed to represent the nuclear optical potential at thelevel of accuracy required through the partial wave components ULJ(k′,k;ω)can be as large as 30 for a 16O target at 200 MeV and 40 for a 40Ca target atthe same energy.

5.3 The Scattering Observables

The most general form of the scattering amplitude for the scattering of spin1/2 projectiles from a spin 0 target is given by

〈k′χ 1

2

ν ′|M(E)|kχ 1

2

ν〉 = −4π2µ 〈k′χ 1

2

ν ′|T (E)|kχ 1

2

ν〉 , (5.22)

107


where χ 1

2

are the Pauli spinors. For elastic scattering we have k′ = k = k0.

The projection of the spin state on the axis of quantization is given by ν and ν ′,and the reduced mass µ is relativistically defined. The matrix M of Eq. (5.22)is an element in the spin space which is composed of the Pauli spin matricesσx, σy, σz and the unit matrix 1. Under the assumptions of parity conservationand rotational invariance the most general form of M is given by

M(k0, θ) = A(k0, θ) + σ · N C(k0, θ) . (5.23)

The first term A(k0, θ) cannot induce any change of the spin, while C(k0, θ)does and for this reason it is sometimes called the spin-flip amplitude. Theamplitudes A(k0, θ) and C(k0, θ) are obtained from the partial wave solutionsof Eq. (5.17) as

A(k0, θ) =1

2π2

∞∑

L=0

[

(L+ 1)F+L (k0) + LF−

L (k0)]

PL(cos θ) ,

C(k0, θ) =i

2π2

∞∑

L=1

[

F+L (k0) − F−

L (k0)]

P 1L(cos θ) ,

(5.24)

where

P 1L(x) =

√1 − x2

d

dxPL(x) . (5.25)

The functions F±L denotes FLJ for J = L ± 1/2, respectively, and they are

given by

FLJ(k0) = − A

A− 14π2µ(k0)TLJ(k0, k0;E) , (5.26)

where the relativistic reduced mass is

µ(k0) =Eproj(k0)Etarg(k0)

Eproj(k0) + Etarg(k0). (5.27)

We now derive the expressions for the scattering observables which can beobtained in spin 1/2 - spin 0 scattering. We start from Eq. (5.23), and choosea coordinate system such that the normal vector N points in the y direction.Thus one only has to consider σ · N = σy. This means that one obtains thescattering amplitude for the scattering of nucleons of a given initial spin stateto a given final spin state by placing the operator A+ Cσy between the Paulispinors for these polarisation directions. The corresponding cross section isthen the absolute value of this amplitude squared. In the usual representationof the spin matrices, where σz is diagonal, the Pauli spinors are:

χ±x =1√2

(

1±1

)

, χ±y =1√2

(

1±i

)

, χ+z =

(

10

)

, χ−z =

(

01

)

.

(5.28)

108

5.3. The Scattering Observables

Now we consider the cross section for +y → +y scattering; the polarization isout of the scattering plane and the cross section is given by

dσ

dΩ(θ,+y → +y) =

∣

∣χ†+y(A+ Cσy)χ+y

∣

∣

2

=

∣

∣

∣

∣

1√2

(1,−i)[

A+ C

(

0 −ii 0

)]

1√2

(

1i

) ∣

∣

∣

∣

2

=∣

∣A+ C∣

∣

2.

(5.29)

For −y → −y the cross section is given by

dσ

dΩ(θ,−y → −y) =

∣

∣χ†−y(A + Cσy)χ−y

∣

∣

2

=

∣

∣

∣

∣

1√2

(1, i)

[

A+ C

(

0 −ii 0

)]

1√2

(

1−i

) ∣

∣

∣

∣

2

=∣

∣A− C∣

∣

2,

(5.30)

while for ±y → ∓y we obtain

dσ

dΩ(θ,+y → −y) =

∣

∣χ†+y(A+ Cσy)χ−y

∣

∣

2

=

∣

∣

∣

∣

1√2

(1,−i)[

A+ C

(

0 −ii 0

)]

1√2

(

1−i

)∣

∣

∣

∣

2

= |0|2 ,

(5.31)

and, similarly,dσ

dΩ(θ,−y → +y) = |0|2 . (5.32)

These relations show that the operator A+Cσy can rotate spins about the yaxis, but cannot change +y into −y. The unpolarised differential cross sectiondσdΩ

(θ) is obtained from the sum over the final spin states and the average overthe initial spin states. If we define the cross section for an average of initialstates as

dσ

dΩ(θ, i→ +y) ≡ dσ

dΩ(θ,+y → +y) +

dσ

dΩ(θ,−y → +y) ,

dσ

dΩ(θ, i→ −y) ≡ dσ

dΩ(θ,+y → −y) +

dσ

dΩ(θ,−y → −y) ,

(5.33)

we can then write the unpolarised cross section as a combination of the twoequations (all initial states to all final states)

dσ

dΩ(θ) =

1

2

[

dσ

dΩ(θ, i→ +y) +

dσ

dΩ(θ, i→ −y)

]

, (5.34)

which using Eqs. (5.29)-(5.30) becomes

dσ

dΩ(θ) =

1

2

[

|A(θ) + C(θ)|2 + |0|2 + |0|2 + |A(θ) − C(θ)|2]

= |A(θ)|2 + |C(θ)|2 ,(5.35)

109


where an implicit dependence on the elastic momentum k0 is assumed.In order to obtain the analyzing power, the spins of the outgoing projectiles

are measured, while the incident beam may be unpolarised. If the differencebetween the +y and −y cross section is taken and the result divided by theunpolarised cross section, we obtain the analyzing power Ay

Ay(θ) =dσdΩ

(θ, i→ +y) − dσdΩ

(θ, i→ −y)dσdΩ

(θ, i→ +y) + dσdΩ

(θ, i→ −y). (5.36)

By using Eqs. (5.29)-(5.30) we can write this as

Ay(θ) =|A(θ) + C(θ)|2 − |A(θ) − C(θ)|2

|A(θ) + C(θ)|2 + |A(θ) − C(θ)|2

=A∗(θ)C(θ) + A(θ)C∗(θ)

|A(θ)|2 + |C(θ)|2

=2Re[A∗(θ)C(θ)]

|A(θ)|2 + |C(θ)|2.

(5.37)

The last independent measurement involves the rotation of the spin vectorin the scattering plane, i.e. protons polarised along the +x axis have a finiteprobability of having the spin polarised along the z axis after the collision.Consider an incident polarised beam along +x and a vector which describesthe polarisation in the z-direction of the scattered protons. The observabledescribing this “rotation” of the spin in the scattering plane is called the spinrotation parameter Q, and is defined as the difference of the cross sections for+z and −z states, divided by the sum

Q(θ) =dσdΩ

(θ,+x→ +z) − dσdΩ

(θ,+x→ −z)dσdΩ

(θ,+x→ +z) + dσdΩ

(θ,+x→ −z). (5.38)

As done earlier, we can explicitly calculate the different terms in Eq. (5.38):

dσ

dΩ(θ,+x→ +z) =

∣

∣χ†+x(A+ Cσy)χ+z

∣

∣

2

=

∣

∣

∣

∣

1√2

(1, 1)

[

A + C

(

0 −ii 0

)](

10

) ∣

∣

∣

∣

2

=1

2

∣

∣A+ iC∣

∣

2,

(5.39)

and

dσ

dΩ(θ,+x→ −z) =

∣

∣χ†+x(A+ Cσy)χ−z

∣

∣

2

=

∣

∣

∣

∣

1√2

(1, 1)

[

A+ C

(

0 −ii 0

)](

01

) ∣

∣

∣

∣

2

=1

2

∣

∣A− iC∣

∣

2.

(5.40)

110

5.4. Treatment of the Coulomb Potential

Using the results of Eqs. (5.39) and (5.40), Eq. (5.38) becomes

Q(θ) =12

∣

∣A+ iC∣

∣

2 − 12

∣

∣A− iC∣

∣

2

12

∣

∣A+ iC∣

∣

2+ 1

2

∣

∣A− iC∣

∣

2

=i[C(θ)A∗(θ) − A(θ)C∗(θ)]

|A(θ)|2 + |C(θ)|2

=2Im[A(θ)C∗(θ)]

|A(θ)|2 + |C(θ)|2.

(5.41)

We note that Ay and Q do complement each other: Ay is a measure of anyspin dependence out of the scattering plane, while Q is a measure of spindependence in the plane. The spin observables are a tool used in probing thenuclear structure and force. Since they are normalized with the cross sectionthey only vary from −1 to 1 (no units), while the cross section is measured inbarns which is 10−28 m2.

5.4 Treatment of the Coulomb Potential

Up to now we have considered the theoretical framework for computing a mi-croscopic optical potential and the scattering observables starting from a NNpotential. In this section we want to include in this framework the Coulombinteraction between the incoming proton with charge e and the spin 0 targetwith charge Ze.

The interest and necessity to develop an accurate treatment of the Coulombinteraction in momentum space has been further motivated in intermediate-energy proton-nucleus scattering, since it is now well established that the non-locality of the first-order optical potential for proton-nucleus scattering must betreated accurately in order to reproduce experimental data for proton-nucleuselastic scattering at intermediate energies. It has been clearly shown thatintermediate-energy calculations of proton-nucleus elastic scattering are highlysensitive to the details of the non-locality of the optical potentials, especiallynear interference minima or at larger scattering angles. Since the non-localityof the proton-nucleus optical potentials has its origin in the non-locality of theelementary proton-nucleon interaction and because the on-shell proton-nucleonamplitude is related to the proton-nucleon scattering cross sections, it followsthat formulating optical potentials directly in the momentum space representsa more natural description of the physics. For this reason, it is of great interestto be able to treat the Coulomb interaction in momentum space in an exact,numerically reliable way.

The problem of scattering by the long-range Coulomb force is theoreticallywell understood and an established technology is available for the calculation ofscattering observables for strongly interacting charged particles. These proce-dures are defined and carried out in coordinate space, whereas, for the reasons

111


stated above, many calculations in nuclear physics are performed in momen-tum space. It may appear strange that a problem which has a well definedsolution in coordinate space should occasion difficulties in momentum space.The problem is that the Fourier transform of the so called coordinate-spaceCoulomb wavefunction does not exist in a functional sense. The logarithmicsingularity due to the long range of the Coulomb force, that can be treatedeasily in coordinate space, is far more intractable in momentum space.

The algorithm [252,253] we use involves the application of the well knowntwo-potential formula to the scattering of a charged particle from a chargedtarget. The interaction is separated into the sum of two parts: the “point”Coulomb interaction and the short-ranged residuum which is given by thesum of the nuclear potential and the short-range Coulomb interaction dueto the finite dimension of the nucleus. Since the Coulomb T -matrix is knownanalytically, we need only to compute the residual Coulomb-modified transitionmatrix. It is known that this latter problem can be transformed into theproblem of solving a LS equation, for which the input potential is modifiedthrough the introduction of the Coulomb distortion.

In this approach the KMT first-order optical potential is given by

U ′(k′,k;ω) = V c(q) + U(k′,k;ω) , (5.42)

where U is the nuclear potential we have already discussed, V c(q) is the dis-tributed Coulomb potential given by vc(q) ρch(q), where vc(q) is the momentum-space Coulomb potential between the projectile and a proton, and ρch(q) isthe nuclear charge form factor, that, for simplicity, is taken as a uniformallycharged sphere. Now we separate the distributed Coulomb potential as

V c(q) = V cpt(q) + V c

s (q) , (5.43)

where V cpt(q) is the Coulomb potential due to a point charge Ze, and V c

s (q) isthe remainder, which is of short range. From Eq. (5.42) the total first-orderoptical potentia1 is written as

U ′(k′,k;ω) = V cpt(q) + U(k′,k;ω) , (5.44)

where the total short-range piece is

U(k′,k;ω) = V cs (q) + U(k′,k;ω) . (5.45)

The total scattering amplitude due to the potential U ′(k′,k;ω) can be writtenin the standard way as

M(k0, θ) = A(k0, θ) + σ · N C(k0, θ) , (5.46)

where now instead of Eqs. (5.24) we have

A(k0, θ) = F cpt(k0, θ) +

1

2π2

∞∑

L=0

e2iσL[

(L + 1)F+L (k0) + LF−

L (k0)]

PL(cos θ) ,

C(k0, θ) =i

2π2

∞∑

L=1

e2iσL[

F+L (k0) − F−

L (k0)]

P 1L(cos θ) .

(5.47)

112

5.4. Treatment of the Coulomb Potential

In Eqs. (5.47) F cpt(k0, θ) is the Coulomb scattering amplitude due to a point

charge [254]

F cpt(k0, θ) =

−η(k0) exp[

2iσ0 − iη(k0) ln(1 − cos θ)]

k0(1 − cos θ), (5.48)

where

η(k) =µZα

k(5.49)

is the Sommerfeld parameter, µ is the reduced mass of Eq. (5.27), and α is thefine structure constant. The Coulomb phase shifts σL are given by

σL = arg Γ[

L + 1 + iη(k0)]

. (5.50)

The partial wave scattering amplitudes F±L are obtained from the solution

of the Coulomb distorted T matrix

T (k′,k;E) = U(k′,k;ω) +

∫

d3pU(k′,p;ω) T (p,k;E)

E(k0) −E(p) + iǫ, (5.51)

where

T (k′,k;E) = 〈k′|T (E)|k〉 = 〈ψ(+)c (k′)|T (E)|ψ(+)

c (k)〉U(k′,k;ω) = 〈k′|U(ω)|k〉 = 〈ψ(+)

c (k′)|U(ω)|ψ(+)c (k)〉 ,

(5.52)

and ψ(+)c (k) is the Coulomb distorted wave function. In order to solve Eq. (5.51),

we need to be able to generate the momentum space matrix element U(k′,k;ω)as given in Eqs. (5.52). We begin with the potential U(k′,k;ω), discussedin Chapter 4, and we transform it into coordinate space through the doubleFourier transform

U(r′, r;ω) =

∫

d3k′d3k 〈r′|k′〉 U(k′,k;ω) 〈k|r〉 (5.53)

and then we construct the matrix element of Eqs. (5.52) by folding U(r′, r;ω)with coordinate space Coulomb wave functions

U(k′,k;ω) =

∫

d3r′d3r 〈ψ(+)c (k′)|r′〉 U(r′, r;ω) 〈r|ψ(+)

c (k)〉 . (5.54)

The reason for applying this procedure is that the Coulomb wave functionsare well defined in coordinate space, whereas their counterparts in momentumspace do not exist in a functional sense. In this way U(k′,k;ω) is the totalshort-range potential given by the sum of the nuclear potential and the short-range Coulomb potential due to the finite size of the nucleus. The long-rangepoint-like part of the Coulomb interaction is included analitically in Eqs. (5.47).

113


For an implementation it is better to treat the previous equations in thepartial wave representation. If we use for U c

L(k′, k;ω) and U lsL (k′, k;ω) the same

partial wave decomposition adopted in App. C we obtain (a = c, ls)

Ua(k′,k;ω) =2

π

∑

JLM

YL 1

2

JM(k′) UaL(k′, k;ω)YL 1

2†

JM (k)

=2

π

∑

LΛ

Y ΛL (k′) Ua

L(k′, k;ω) Y Λ ∗L (k) ,

(5.55)

and

Ua(r′, r;ω) =∑

JLM

YL 1

2

JM(r′) UaL(r′, r;ω)YL 1

2†

JM (r)

=∑

LΛ

Y ΛL (r′) Ua

L(r′, r;ω) Y Λ ∗L (r) .

(5.56)

for the momentum and coordinate space, respectively. Moreover, using thestandard expansion for the plane waves

〈r′|k′〉 =4π

(2π)3/2

∑

LΛ

iLjL(k′r′)Y ΛL (r′)Y Λ ∗

L (k′) ,

〈k|r〉 =4π

(2π)3/2

∑

LΛ

(−i)LjL(kr)Y ΛL (k)Y Λ ∗

L (r) ,(5.57)

we can expand Eq. (5.53) for the central and the spin-orbit parts as

UaL(r′, r;ω) =

4

π2

∫ ∞

0

dk′ k′ 2∫ ∞

0

dk k2jL(k′r′)UaL(k′, k;ω)jL(kr) , (5.58)

where jL(kr) are the spherical Bessel functions. Expanding in partial wavesthe Coulomb functions as

〈ψ(+)c (k′)|r′〉 =

4π

(2π)3/2

∑

LΛ

(−i)LFL(η, k′r′)

k′r′Y ΛL (k′)Y Λ ∗

L (r′) ,

〈r|ψ(+)c (k)〉 =

4π

(2π)3/2

∑

LΛ

iLFL(η, kr)

krY ΛL (r)Y Λ ∗

L (k) ,

(5.59)

where η = η(k) is the Sommerfeld parameter of Eq. (5.49), we can also rewriteEq. (5.54) for the central and the spin-orbit parts as

UaL(k′, k;ω) =

1

k′k

∫ ∞

0

dr′ r′∫ ∞

0

dr rFL(η, k′r′)UaL(r′, r;ω)FL(η, kr) . (5.60)

In Eq. (5.60) FL is the regular Coulomb function given by

FL(η, kr) = CL(η)(kr)L+1eikr1F1(L+ 1 + iη; 2L+ 2;−i2kr) ,

CL(η) =2Le−

πη

2 |Γ(1 + L + iη)|(2L+ 1)!

,(5.61)

114

5.5. Results for Scattering Observables

where 1F1 is the confluent hypergeometric function 1F1(a; b; z).The potential U(k′,k;ω) can be expanded in partial waves as in Eq. (5.14)

U(k′,k;ω) =2

π

∑

JLM

YL 1

2


2†

JM (k) , (5.62)

where


lsL (k′, k;ω) ,

CLJ =1

2

[

J(J + 1) − L(L + 1) − 3

4

]

,


k′k

2L+ 1

[

U lsL+1(k

′, k;ω) − U lsL−1(k

′, k;ω)]

,

(5.63)

and in the same way we can expand the T matrix in Eq. (5.51) as

T (k′,k;E) =2

π

∑

JLM

YL 1

2

JM(k′) TLJ(k′, k;E)YL 1

2†

JM (k) , (5.64)

where the partial wave components are

TLJ(k′, k;E) = ULJ(k′, k;ω) +2

π

∫ ∞

0

dp p2ULJ(k′, p;ω) TLJ(p, k;E)

E(k0) −E(p) + iǫ. (5.65)

The partial wave scattering amplitudes F±L that enter Eq. (5.47) are given by

FLJ(k0) = − A

A− 14π2µ(k0)TLJ(k0, k0;E) . (5.66)

Thus the Coulomb interaction is included in the model using Eqs. (5.47). Start-ing with U c

L(k′, k;ω) and U lsL (k′, k;ω) and using Eqs. (5.58) and (5.60) we com-

pute the U cL(k′, k;ω) and U ls

L (k′, k;ω) potentials. These potentials are thenused to compute ULJ(k′, k;ω), Eq. (5.63), and inserting this one in Eq. (5.65)we obtain the TLJ matrix that we need to calculate Eq. (5.66) and finally theamplitudes F±

L , where ± stands for J = L± 1/2.

5.5 Results for Scattering Observables

In this section we present and discuss our numerical results for the NA elasticscattering observables obtained with the optical potential developed in Chap-ter 4 and the calculational framework discussed in the previous sections of thischapter. First we show the results obtained for 16O and 40Ca with all the con-sidered NN potentials and without Coulomb distortion. In this way we intendto investigate the sensitivity of the results to the choice of the NN potentialand to discern and discriminate among the results obtained with the Chiralpotential by Epelbaum with the three SFR cut-off values. Then, the results

115


calculated for 16O and 40Ca at different energies and including the Coulombdistortion are compared with available data. Finally, we present and discussour numerical predictions for the scattering observables along oxygen (16−28O)and calcium (40−56Ca) isotopic chains, computed with the CD-Bonn and Chiralpotentials.

Due to the possibility to use five different sets of LS and SFR cut-offs in theEpelbaum code, first of all we present the results obtained with all NN poten-tials: CD-Bonn, Ch-N3LO, and the three Ch-N3LO potentials by Epelbaumcorresponding to the three values of the SFR cut-off. Our motivations are:1) to check the correctness of our calculations comparing our results obtainedwith the CD-Bonn potential with the corresponding results obtained by otherauthors with the same NN potential; 2) to check the reliability of our newresults obtained with chiral potentials from the comparison with the corre-sponding results obtained with the CD-Bonn potential; 3) to identify the bestset of values for the LS and SFR cut-offs of the Epelbaum code. Since in thisfirst step our aim is to check the sensitivity of the results to the different NNpotentials, in all these calculations we have neglected the Coulomb interactionbetween the proton and the target nucleus. Moreover, in order to emphasizeas much as possible the differences between the different NN potentials andalso on the basis of the results obtained in Chapter 4 for the NN Wolfensteinamplitudes a and c, the scattering observables have been calculated at 200MeV in the laboratory frame. Chiral potentials are based upon ChPT thatis a low-momenutm expansion of QCD and this energy may be considered alimit for the use of such potentials.

In panel (a) of Fig. 5.1 we show the differential cross sections for elasticproton scattering on 16O as functions of the scattering angle θ calculated withall the different NN potentials. All results exhibit only one minimum at 25degrees, but the curve calculated with the Ch-N3LO-500 MeV potential is abit lower than the other ones. In panel (b) of Fig. 5.1 we display the sameresults but for 40Ca. Here we note that the value of the cross section at smallerangles is larger compared with the 16O one and that there is a more pronouncedminimum around 16 degrees and another one, less pronounced, between 30 and35 degrees. Also in this case the result obtained with the Ch-N3LO-500 MeVis lower than the results given by the other potentials, which are very close toone another.

In panel (a) of Fig. 5.2 we show the results for the analyzing power on 16Ocalculated with the different NN potentials. All results display two maximaaround 20 and 33 degrees and two minima around 25 and 45 degrees. It isinteresting to note that the first minimum occurs at the same angle as in thecross section. This polarization observable is more sensitive than the crosssection to the choice of the NN potential in the calculation of the opticalpotential and can help us to determine the best choice. The Ch-N3LO-500MeV potential underestimates the result obtained with the CD-Bonn potential,especially around the first minimum. In contrast, the result obtained with the

116

5.5. Results for Scattering Observables

Ch-N3LO-700 MeV potential exhibits the opposite behavior and overestimatesboth minima. The Ch-N3LO-600 MeV potential shows a trend close to thosegiven by the CD-Bonn and the Ch-N3LO potentials. A similar behavior isshown in panel (b) of Fig. 5.2 by the corresponding results for 40Ca. In thiscase there are more maxima and minima, but the results obtained with theCD-Bonn potential are underestimated by the Ch-N3LO-500 MeV potentialand overestimated by the Ch-N3LO-700 MeV one.

The results for the spin rotation Q on 16O calculated with the differentNN potentials are shown in panel (a) of Fig. 5.3. In this case, the resultsobtained with the CD-Bonn potential are overestimated around 15 degreesand underestimated for angles larger than 50 degrees when the Ch-N3LO-500MeV potential is used in the calculations. The result obtained with the Ch-N3LO-700 MeV potential are lower than the results obtained with CD-Bonnaround the first minimum. The same situation occurs in panel (b) of Fig. 5.3in which we display the corresponding results for 40Ca.

From the results displayed in Figs. 5.1, 5.2, 5.3, we can see that our cal-culations with the CD-Bonn potential are in good agreement with the resultsobtained with the same model and with the same NN potential in Ref. [257]This agreement can be considered a proof of the correctness of our numericalcalculations. The reliability of the new optical potentials obtained with theChiral potentials is substantially demonstrated by the comparison with theCD-Bonn results. The scattering observables calculated with CD-Bonn andwith the Ch-N3LO potentials are in reasonable agreement. Concerning thebest set of values for the LS and SFR cut-offs in the Epelbaum code we canconclude that 600 MeV for the SFR cut-off is the best value because it givesthe results closest to those obtained with the CD-Bonn potential. Therefore,in the rest of this section we present results obtained with only three poten-tials: the CD-Bonn, the Ch-N3LO, and the Ch-N3LO-600 MeV potentials. Inaddition, all the calculations include the Coulomb interaction between the in-cident proton and the target nucleus that can have significant effects on thescattering observables and therefore on the comparison with data. In fact, inthe following the cross sections and analyzing powers calculated for the elasticproton scattering on 16O and 40Ca at different energies are also compared withthe experimental data.

In Fig. 5.4 we show the scattering observables for 16O at 100 MeV. At thisenergy all the three potentials give very similar results for all scattering ob-servables except for Ay and Q for scattering angles greater than 50 degreesand around the maximum of Q. The cross section is well reproduced by allpotentials also in the range of the minimum between 30 and 35 degrees. Theeffects of Coulomb distortion can be seen from the comparison with the cor-responding results in panel (a) of Fig. 5.1. The polarization observables aremore sensitive to the differences in the potentials and to the ingredients andapproximations of the model and are therefore more difficult to reproduce. Inparticular, all potentials overestimate the experimental data for Ay up to the

117


maximum, then they display a downward trend.

A similar result is obtained in Fig. 5.5 where we display the scattering ob-servables for 16O at 135 MeV. In this case all potentials reproduce very wellthe experimental cross section and globally describe the shape of Ay, but areunable to reproduce its magnitude for angles larger than 20 degrees. For an-gles lower than 20 degrees the results obtained using the CD-Bonn potentialand the Ch-N3LO-600 MeV potential give similar values and are able to repro-duce the data, while the calculation performed with the Ch-N3LO potentialunderestimates them.

In Fig. 5.6 we plot the results for 16O obtained at 200 MeV. At this energythe potentials reproduce the cross section and give better results also for theanalyzing power. For the latter one, the potentials reproduce the experimentaldata not only before but also after the minimum region up to 45 degrees.Beyond this value the potentials give different behavior for both Ay and Q.

Similar results are found in Figs. 5.7 and 5.8 where we show the scatteringobservables for 40Ca at 100 and 200 MeV, respectively. At these energies thecross sections are well reproduced while the polarization observables exhibitthe same trends found for 16O. At 100 MeV the experimental data for Ay

are satisfactorily reproduced up the maximum region, while, at 200 MeV, theglobal shape of Ay data is reproduced, but the minima are not deep enough.

On the basis of all these results for 16O and 40Ca we can conclude that theCh-N3LO-600 MeV potential is better than the Ch-N3LO one. In particular,this can be seen for Ay, where the results obtained with Ch-N3LO-600 MeVare closer to the CD-Bonn ones than those obtained with Ch-N3LO. However,for energies above 200 MeV, this agreement is no longer valid and the Chiralpotentials begin to fail with the increasing of the energy. As an example, inFig. 5.9 we display the results for the scattering observables on 16O computedat 318 MeV. We clearly see that at this energy the Ch-N3LO-600 MeV poten-tial is completely unable to describe the data. The Ch-N3LO potential gives asomewhat better description of data because it is able to describe the globalshape of the experimental results and the position of the minima, but the gen-eral agreement is poor. The CD-Bonn potential is the only one that producesan optical potential able to give a reasonable description of data also at theseenergies. Concerning the differences between the two Chiral potentials, weguess that they reside in the regularization scheme employed in the 2PE term.Entem and Machleidt [238] use a dimensional regularization while Epelbaumet al. [239] adopt a regularization scheme based on the spectral function repre-sentation. The first choice permits to obtain a reliable potential over a largerenergy range, up to 290 MeV, but the potential computed at lower energies isless precise than the one regularized with the spectral function. This secondmethod permits to compute a precise potential at low energies but it beginsto fail beyond an energy threshold of 250 MeV. However, we stress that ChPTis a low-momentum expansion and its goal is to perform calculations at lowenergies.

118

5.6. Conclusions and Future Perspectives

After checking the reliability of the developed optical potentials in compar-ison with experimental on 16O and 40Ca at different energies we give now ourresults on oxygen and calcium isotopic chains. Calculations are performed atan energy of 200 MeV and using the CD-Bonn and the Ch-N3LO potentials.

In Fig. 5.10 we show the results of the cross section for the oxygen isotopicchain. The trend of the curves is quite similar to the ones obtained in Chapter 3for elastic electron scattering. With the increasing of the neutron number thepositions of the diffraction minima shift towards smaller scattering angles, i.e.,towards smaller values of the momentum transfer. The shift of the minimatowards smaller q is in general accompanied by a simultaneous increase in theheight of the maxima. The behavior is similar for both results computed withthe two NN potentials.

In Fig. 5.11 we display the results for the analyzing power. With theincreasing of the neutron number the position of the minima is shifted towardssmaller angles and at the same time the value of the minimum is increased.In the same way the maxima are also shifted towards smaller angles but theirvalues are constant. The only difference between the results computed with thetwo NN potentials is given by the values of the minima: the results obtainedwith CD-Bonn are deeper than those obtained with Ch-N3LO.

In Fig. 5.12 we show the results for the the spin rotation. Also in this case,with the increasing on the neutron number the positions of the minima and ofthe maxima are shifted towards smaller angles while the values of the minimaare increased. The only exception is given by the third minimum of 26O and28O. The main difference between the calculations performed with the two NNpotentials is given by the values of the third minimum of Q, which are deeperfor the CD-Bonn potential, but only for the first four isotopes.

Similar results are found for the calcium isotopic chain. In Figs. 5.13, 5.14,and 5.15 we show the theoretical results for the cross section, the analyzingpower, and the spin rotation, respectively. The minima of the cross sectionsare shifted towards smaller angles, but the difference is less pronounced. Thesame situation occurs for the other two observables, that display more minimaand maxima, with a shift of their positions towards smaller angles with theincreasing of the neutron number.

5.6 Conclusions and Future Perspectives

In this chapter we have presented and discussed numerical results for the crosssection, the analyzing power, and the spin rotation in elastic proton scatteringon 16O and 40Ca to investigate the reliability of the optical potentials calculatedwith different NN potentials also in comparison with available data. Thennumerical predictions have been given for oxygen and calcium isotopic chainswith the aim to study the evolution of the calculated scattering observableswith increasing neutron number.

The understanding of the properties of exotic nuclei is one of the major

119


topic of interest in modern nuclear physics and in Chapter 3 we performed afirst study of these systems through electron scattering. This study has beenextended to proton scattering through the definition and the construction, inChapter 4, of a microscopic optical potential that can be used in elastic protonscattering. In this chapter such a microscopic optical potential has indeedbeen used to compute the transition amplitudes and scattering observablesfor elastic proton-nucleus scattering. The case of elastic proton scatteringrepresents the first natural and necessary test of the reliability of an opticalpotential. We note, however, that the optical potential is an important andcritical input for calculations on a wide variety of nuclear reactions and cantherefore be employed in many other situations beyond those considered inthis dissertation.

In many calculations for nuclear reactions phenomenological optical poten-tials are generally used, with parameters fitted to available elastic NA scat-tering data. Available data, however, do not completely constrain the opticalpotential and are till now available only on stable nuclei. A microscopic, moretheoretically founded optical potential can therefore represent an interestingand useful alternative to the use of phenomenological optical potentials, inparticular for calculations on exotic nuclei. The construction of a microscopicoptical potential is, however, a very hard task that in principle requires thesolution of the full many-body problem which is beyond present capabilities.In order to make calculations feasible we must necessarily resort to some ap-proximations.

With the development of many NN potentials in momentum space, thegeneral problem of the scattering of a nucleon from a target nucleus has beenmore conveniently stated in momentum space. The optical potential operatorcan be written in the spectator expansion as a sum of A many-body terms,where the first one is a sum over all nucleons of the two-body term given by theinteraction between the projectile nucleon and the i-th nucleon of the target.Neglecting the other many-body terms we have obtained the first-order opticalpotential. However, the interaction matrix τ between the incoming nucleonand the target nucleon is still extremely difficult to compute. In order to makecalculations simpler, we have adopted the impulse approximation that consistsin replacing τ with the free NN t matrix. In this approximation the mediumeffects contained inside the τ matrix are neglected. As a further simplification,we have then used the optimum factorization approximation that leads to afactorized form in which the optical potential is given by the product of thefree NN t matrix and the nuclear density. This form conserves the off-shellnature of the potential and it has been used in this chapter to compute thescattering observables.

As case studies for the present investigation we have maintained oxygenand calcium isotopic chains. In the calculations the neutron and proton nu-clear densities are obtained considering a system of nucleons coupled to theexchange mesons and the electromagnetic field through an effective Lagrangian.

120


The calculated observables are obtained with different NN potentials. In par-ticular, we have used the CD-Bonn NN potential and two different versionsof the Chiral potential at the fourth order. The Coulomb interaction betweenthe proton and the target nucleus has also been included in the calculations.

First, all results obtained with chiral potentials have been compared withthe results computed with the CD-Bonn potential, which has been alreadyused from other authors in similar calculations. The general agreement incomparison with the results of other authors gives us enough confidence aboutthe correctness of our numerical results. From the comparison of the resultsobtained with different NN potentials we have checked the sensitivity of ouroptical potential to the details of the NN potential. From the comparisonbetween the results obtained with CD-Bonn and with the different versions ofthe Chiral potential, that has been used here for the first time in the calculationof a microscopic optical potential, we have tested the reliability of our newresults with the Chiral potential and we have determined the set of cut-offsleading to the closest agreement with the CD-Bonn results. After we havedetermined the best set of cut-offs, the results of the three selected modelshave been compared with available data on 16O and 40Ca in order to checktheir reliability. Finally, the same models have been used to calculate theelastic scattering observables on oxygen and calcium isotopic chains.

Our results show an evolution of the calculated quantities without discon-tinuities. The increase of the neutron number essentially produces an increaseof the nuclear and proton densities that determines a shift of the cross sectionminima towards smaller angles and, at the same time, a corresponding increaseof the values of the minima. A similar behavior has been also found for themaxima and minima of the other observables Ay and Q, but in this case theheight of the maxima has been found to be constant along all the chain.

The microscopic optical potential obtained in this dissertation is the basicingredient for our investigation of elastic proton scattering on nuclei with aneutron excess. It might be interesting to compare our results with the cor-responding results obtained with phenomenological optical potentials. Withseveral parameters adjusted to fit elastic proton-nucleus scattering data, phe-nomenological potentials are, in general, quite successful in the description ofthese data. Experimental data, however, do not uniquely determine a phe-nomenological optical potential. In addition, they are available till now onlyon stable nuclei, and their extrapolation to unstable nuclei is not without ambi-guity. Our microscopic optical potential is the result of some approximations,but it does not contain adjustable parameters. It has been computed with twobasic ingredients, the nucleon-nucleon potential and the nuclear density, it istherefore more theoretically founded than a phenomenological potential, andwe can expect that it has a greater predictive power when applied to situa-tions where experimental data are not yet available, such as, for instance, tothe study of unstable nuclei. The comparison between the results of micro-scopic and phenomenological optical potentials would therefore be useful to

121


investigate and understand the differences between the two methods used toderived the optical potential. The use of the optical potential can be extendedto calculations of inelastic scattering and to generate the distorted waves forthe analysis of the cross sections for a wide variety of nuclear reactions, forinstance, in QE electron scattering. Also in this case, the comparison withthe corresponding results obtained with phenomenological optical potentials,which have widely been applied to the analysis of nuclear reactions, woulddeserve a careful investigation.

Moreover, some improvements can be made. For instance, we might thinkof including medium effects in the optical potential. In this dissertation wehave always considered the NN t matrix instead of the more complicatedτ one. This means that the nucleon inside the target nucleus is consideredas free and not embedded in the nuclear medium. We have seen that thisapproximation does not prevent us from describing the experimental data forthe differential cross section, but it is presumably a reason of the much lessgood agreement with the analyzing power data. The model could be improvedconsidering the situation in which the target nucleon is inserted in the nuclearmedium and the interactions with the other nucleons are taken into accountby a mean-field potential generated by the rest of the nucleus.

The second possible improvement concerns the use of the Similarity Renor-malization Group (SRG) technique to renormalize the optical potential. Thisprocedure allows to evolve a general off-shell potential in such a manner todisentangle the couplings between low and high momenta, and then permitsto use the renormalized potential in low-energy calculations. The decouplingprocedure is carried out without any cuts in the momentum space but inte-grating out the high momenta degrees of freedom. In this way it is possible toobtain a band diagonal form for the potential that still preserves the off-shellcharacter of the original one. Applying this procedure to the optical potentialcould allow us to extract an on-shell potential which still maintains the originaloff-shell properties.

122


0 10 20 30 40 50 60θ [fm]

1×10-3

1×10-2

1×10-1

1×100

1×101

1×102

1×103

1×104

dσ/d

Ω [

mb/

sr]

Ch-N3LO - 700 MeV

Ch-N3LO - 600 MeV

Ch-N3LO - 500 MeV

Ch-N3LO

CD-Bonn

(a)

0 10 20 30 40 50 60θ [fm]

1×10-3

1×10-2

1×10-1

1×100

1×101

1×102

1×103

1×104

dσ/d

Ω [

mb/

sr]

Ch-N3LO - 700 MeV

Ch-N3LO - 600 MeV

Ch-N3LO - 500 MeV

Ch-N3LO

CD-Bonn

(b)

Figure 5.1: Differential cross section for elastic proton scattering on 16O (panela) and 40Ca (panel b) calculated at 200 MeV with different NN potentials.All calculations are performed without Coulomb interaction.

123


0 10 20 30 40 50 60θ [deg]

-0.2

0

0.2

0.4

0.6

0.8

1A

y

Ch-N3LO - 700 MeV

Ch-N3LO - 600 MeV

Ch-N3LO - 500 MeV

Ch-N3LO

CD-Bonn

(a)

0 10 20 30 40 50 60θ [deg]

0

0.2

0.4

0.6

0.8

1

Ay

Ch-N3LO - 700 MeV

Ch-N3LO - 600 MeV

Ch-N3LO - 500 MeV

Ch-N3LO

CD-Bonn

(b)

Figure 5.2: Analyzing power for elastic proton scattering on 16O (panel a)and 40Ca (panel b) calculated at 200 MeV with different NN potentials. Allthe calculations are performed without the Coulomb interaction.

124


0 10 20 30 40 50 60θ [deg]

-1

-0.5

0

0.5

1Q

Ch-N3LO - 700 MeV

Ch-N3LO - 600 MeV

Ch-N3LO - 500 MeV

Ch-N3LO

CD-Bonn

(a)

0 10 20 30 40 50 60θ [deg]

-1

-0.5

0

0.5

1

Q

Ch-N3LO - 700 MeV

Ch-N3LO - 600 MeV

Ch-N3LO - 500 MeV

Ch-N3LO

CD-Bonn

(b)

Figure 5.3: Spin rotation for elastic proton scattering on 16O (panel a) and40Ca (panel b) calculated at 200 MeV with different NN potentials. All thecalculations are performed without the Coulomb interaction.

125


1×100

1×102

1×104

dσ/d

Ω [

mb/

sr]

Ch-N3LO - 600 MeV

Ch-N3LO

CD-Bonn

-1

-0.5

0

0.5

1

Ay

0 10 20 30 40 50 60θ [deg]

-1-0.8-0.6-0.4-0.2

00.2

Q

Figure 5.4: Scattering observables for elastic proton scattering on 16O com-puted at 100 MeV (laboratory energy) with different NN potentials and in-cluding Coulomb distortion. Data are taken from Refs. [255, 256].

126


1×10-2

1×100

1×102

1×104

dσ/d

Ω [

mb/

sr]

Ch-N3LO - 600 MeV

Ch-N3LO

CD-Bonn

-0.2

0

0.2

0.4

0.6

0.8

Ay

0 10 20 30 40 50 60θ [deg]

-0.5

0

0.5

1

Q


127


1×10-2

1×100

1×102

1×104

dσ/d

Ω [

mb/

sr]

Ch-N3LO - 600 MeV

Ch-N3LO

CD-Bonn

-0.5

0

0.5

1

Ay

0 10 20 30 40 50 60θ [deg]

-0.5

0

0.5

1

Q


128


1×100

1×102

1×104

dσ/d

Ω [

mb/

sr]

Ch-N3LO - 600 MeV

Ch-N3LO

CD-Bonn

0

0.2

0.4

0.6

0.8

Ay

0 10 20 30 40 50 60θ [deg]

-1

-0.5

0

0.5

Q

Figure 5.7: Scattering observables for elastic proton scattering on 40Ca com-puted at 100 MeV (laboratory energy) with different NN potentials and in-cluding Coulomb distortion. Data are taken from Refs. [255, 256].

129


1×10-2

1×100

1×102

1×104

dσ/d

Ω [

mb/

sr]

Ch-N3LO - 600 MeV

Ch-N3LO

CD-Bonn

-0.20

0.20.40.60.8

1

Ay

0 10 20 30 40 50 60θ [deg]

-0.5

0

0.5

1

Q

Figure 5.8: Scattering observables for elastic proton scattering on 40Ca com-puted at 200 MeV (laboratory energy) with different NN potentials and in-cluding Coulomb distortion. Data are taken from Refs. [255, 256].

130


1×10-2

1×100

1×102

1×104

dσ/d

Ω [

mb/

sr]

Ch-N3LO - 600 MeV

Ch-N3LO

CD-Bonn

-1

-0.5

0

0.5

1

Ay

0 10 20 30 40 50 60θ [deg]

-1

-0.5

0

0.5

1

Q


131


0 10 20 30 40 50 60θ [deg]

1×10-3

1×10-2

1×10-1

1×100

1×101

1×102

1×103

1×104

dσ/d

Ω [

mb/

sr]

16O

18O

20O

22O

24O

26O

28O

(a)

0 10 20 30 40 50 60θ [deg]

1×10-3

1×10-2

1×10-1

1×100

1×101

1×102

1×103

1×104

dσ/d

Ω [

mb/

sr]

16O

18O

20O

22O

24O

26O

28O

(b)

Figure 5.10: Differential cross sections for elastic proton scattering on theoxygen isotopic chain computed at 200 MeV (laboratory energy) and includingCoulomb distortion. The results are obtained with the CD-Bonn potential(panel a) and the Ch-N3LO potential (panel b).

132


0 10 20 30 40 50 60θ [deg]

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Ay

16O

18O

20O

22O

24O

26O

28O

(a)

0 10 20 30 40 50 60θ [deg]

0

0.2

0.4

0.6

0.8

1

Ay

16O

18O

20O

22O

24O

26O

28O

(b)

Figure 5.11: The same as in Fig. 5.10 but for the analyzing power.

133


0 10 20 30 40 50 60θ [deg]

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Q

16O

18O

20O

22O

24O

26O

28O

(a)

0 10 20 30 40 50 60θ [deg]

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Q

16O

18O

20O

22O

24O

26O

28O

(b)

Figure 5.12: The same as in Fig. 5.10 but for the spin rotation.

134


0 10 20 30 40 50 60θ [deg]

1×10-3

1×10-2

1×10-1

1×100

1×101

1×102

1×103

1×104

dσ/d

Ω [

mb/

sr]

40Ca

42Ca

44Ca

46Ca

48Ca

50Ca

52Ca

54Ca

56Ca

(a)

0 10 20 30 40 50 60θ [deg]

1×10-3

1×10-2

1×10-1

1×100

1×101

1×102

1×103

1×104

dσ/d

Ω [

mb/

sr]

40Ca

42Ca

44Ca

46Ca

48Ca

50Ca

52Ca

54Ca

56Ca

(b)

Figure 5.13: The same as in Fig. 5.10 but for calcium isotopes.

135


0 10 20 30 40 50 60θ [deg]

-0.2

0

0.2

0.4

0.6

0.8

1

Ay

40Ca

42Ca

44Ca

46Ca

48Ca

50Ca

52Ca

54Ca

56Ca

(a)

0 10 20 30 40 50 60θ [deg]

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Ay

40Ca

42Ca

44Ca

46Ca

48Ca

50Ca

52Ca

54Ca

56Ca

(b)


136


0 10 20 30 40 50 60θ [deg]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Q

40Ca

42Ca

44Ca

46Ca

48Ca

50Ca

52Ca

54Ca

56Ca

(a)

0 10 20 30 40 50 60θ [deg]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Q

40Ca

42Ca

44Ca

46Ca

48Ca

50Ca

52Ca

54Ca

56Ca

(b)


137


138

Chapter 6Summary and Conclusions

In this dissertation we have presented and discussed numerical predictionsfor scattering observables on oxygen and calcium isotopic chains and on theN = 14, 20, and 28 isotonic ones with the aim to investigate their evolutionwith increasing neutron and proton numbers. The understanding of nuclearproperties as a function of the neutron-to-proton ratio is one of the majortopic of interest in modern nuclear physics. Large efforts in this directionshave been done over the last years and many experiments are planned for thefuture. The RIB facilities in different laboratories have opened the possibilityto give insight into nuclear structures which are not available in nature, as theyare not stable, but which are important in astrophysics and had a relevant rolein the nucleosynthesis.

Our investigation is based on the use of electron and proton scattering re-actions. The use of electrons as probe provides a powerful tool to investigatenuclear structure. They predominantly interact through the electromagneticforce and are free from most uncertainties of strong interactions, allowing us toperform very precise measurements of experimental quantities. On the otherhand, elastic proton-nucleus scattering can also provide a lot of informationon nuclear properties, on the interactions and the dynamics which occur in-side a nucleus, and has also been extensively studied. From the theoreticalpoint of view the scattering process is described through the use of the opti-cal potential. This computational tool is important not only to describe theelastic scattering observables but it is also a fundamental input for calcula-tions in many other nuclear reactions, inlcuding electron-induced reactions.In many calculations phenomenological optical potentials are usually adoptedwhich have been determined through a fit to elastic proton scattering dataavailable on stable nuclei. The construction of a theoretical optical potentialbecomes essential for the study of exotic nuclei, for which it is reasonable tothink that such phenomenological optical potentials will be unable to give areliable description of nuclear interactions and properties.

The first scope of this work was to carry out a study of nuclei with neu-tron and proton excess through electron scattering calculations. In order to do

139

6. Summary and Conclusions

it, both elastic and inclusive quasi-elastic electron scattering have been con-sidered. The elastic scattering gives information on the global properties ofnuclei and, in particular, on the different behavior of neutrons and protons.The inclusive QE scattering preferably exploits the single particle aspects ofthe nucleus and, when combined with the exclusive scattering, it is able toexplore the evolution of the single particle model with increasing asymmetrybetween the number of neutrons and protons. Many interesting phenomenahave been predicted in this situation, in particular, the modification of theshell model magic numbers.

In this dissertation, models that have been developed and successfully com-pared with elastic and QE electron scattering data on stable nuclei have beenextended to exotic isotopic and isotonic chains. Our scope was to investigatethe evolution of nuclear properties with increasing number of neutrons andprotons with models of proven reliability in stable nuclei. The possible dis-agreement of the theoretical predictions with the experimental data will bean indication of the insurgence of new phenomena related to the proton-to-neutron asymmetry.

Some important results have been obtained. First of all, the increase ofthe neutron number in isotopic chains extends the neutron densities and also(to a minor extent) the proton ones. The latter have a decrease in the nuclearinterior, in accordance with the results from Refs. [81, 88, 89]. The densitiesof the proton-rich isotopes are significantly extended towards larger distanceswith respect to those of the proton-deficient ones. Secondly, the elastic electronscattering on 16O, 40Ca, on isotopes of Cr using DWBA with RMF chargedensities show that when the neutron number increases, the position of thediffraction minima shifts towards smaller scattering angles in agreement withthe results from the mentioned references. The isotonic chains with N = 14, 20,and 28 have been considered as well. The increase of the proton number alongthe chain leads to a shift of the diffraction minima towards smaller scatteringangles and to an increase of the height of the corresponding maxima of the crosssections. Third, of important interest are the results on parity-violating elasticelectron-nuclei scattering in order to investigate the neutron skin for oxygenand calcium isotopic chains, as well as the results of the neutron density of the208Pb nucleus. Fourth, the inclusive QE electron scattering is calculated withthe RGF model that conserves the global particle flux in all the final channels.

The second scope of this work was the construction of a theoretical opticalpotential for elastic NA scattering. The model is based on the impulse ap-proximation and, after suitable approximations, the optical potential has beenreduced to a simple form in which the NN t matrix and the nuclear density arefactorized, but the optical potential maintains its off-shell character. Calcula-tions require two main ingredients: a NN interaction and neutron and protondensities for the target nucleus. Modern NN interaction have been obtainedusing the CD-Bonn potential and two different versions of the Chiral potential.RMF models have been employed to describe the ground-state properties of

140

nuclei. The obtained optical potential also exhibits an energy dependence thathas not been found to play a significant role at projectile energies up to 200MeV and has been treated in the so called fixed beam energy approximation.The optical potential has been employed to compute scattering observables.First the optical potential has been tested in comparison with data on 16Oand 40Ca at different energies. Then numerical predictions have been given foroxygen and calcium isotopic chains.

Also in this case some important results have been obtained. Together withthe CD-Bonn potential, for the first time two different versions of the Chiralpotential have been used as basic ingredients to calculate the microscopic op-tical potential and the scattering observables. The theoretical results for theNN amplitudes obtained with different NN potentials are presented and dis-cussed. Since ChPT is a low-momentum expansion of QCD, as expected thebehavior of the Chiral potential based on it becomes worse with the increaseof the energy. For example, for the energy of 200 MeV, all the potentials arenot able to reproduce the experimental data of the real part of the cpp am-plitude, but the CD-Bonn and Ch-N3LO potentials give satisfactory results inagreement with the data for all the other amplitudes, while the Ch-N3LO-500MeV potential begins to fall. The Coulomb interaction between the incomingproton and the target with spin 0 and charge Ze is accounted for in comput-ing a microscopic optical potential and the scattering observables from a NNpotential. The problem is that the Fourier transform of the coordinate-spaceCoulomb wave function does not exist in a functional sense. We presentedand discussed the numerical predictions for the scattering observables alongoxygen- and calcium- isotopic chains computed with the CD-Bonn and Chi-ral potentials. It is shown that the calculations with the CD-Bonn potentialare in good agreement with previous results obtained with the same potential.The scattering observables calculated with the CD-Bonn and Ch-N3LO opticalpotentials, as well as with Ch-N3LO-600 MeV are in a reasonable agreement.The results show that the increase of the neutron number produces an increaseof the nuclear and proton densities that leads to a shift of the cross sectionminima towards smaller angles and to a corresponding increase of the heightof the maxima. Finally, for the analyzing power the increase of the neutronnumber leads to a shift of the minima towards smaller angles, while the valueof the minimum is increased.

141

6. Summary and Conclusions

142

Appendix AThe Singlet-Triplet ScatteringAmplitudes

In this appendix we include explicit formulas for the NN spin amplitudes MSν′ν

in terms of the partial wave NN transition amplitudes MJSL′L(κ′, κ;ω) for an

angular momentum quantization axis along the incident beam direction κ. Wehave:

M000 =

2

4π2

∞∑

L=0

PL(cosφ) (2L+ 1)ML,S=0LL ,

M100 =

2

4π2

∞∑

L=0

PL(cosφ)

(L+ 1)ML+1,S=1LL + LML−1,S=1

LL

+√

(L+ 1)(L+ 2)ML+1,S=1L,L+2 +

√

(L− 1)LML−1,S=1L,L−2

,

M111 =

1

4π2

∞∑

L=0

PL(cosφ)

(L+ 2)ML+1,S=1LL + (2L + 1)ML,S=1

LL

+ (L− 1)ML−1,S=1LL −

√

(L + 1)(L+ 2)ML+1,S=1L,L+2

−√

(L− 1)LML−1,S=1L,L−2

,

M110 =

√2

4π2

∞∑

L=1

P 1L(cosφ)

ML+1,S=1LL −ML−1,S=1

LL

+

√

L+ 2

L+ 1ML+1,S=1

L,L+2 −√

L− 1

LML−1,S=1

L,L−2

,

M101 =

√2

4π2

∞∑

L=1

P 1L(cosφ)

− L + 2

L + 1ML+1,S=1

LL +2L+ 1

L(L + 1)ML,S=1

LL

+L− 1

LML−1,S=1

LL +

√

L+ 2

L+ 1ML+1,S=1

L,L+2 −√

L− 1

LML−1,S=1

L,L−2

.

(A.1)

143

A. The Singlet-Triplet Scattering Amplitudes

The sums in the previous expressions run over all partial wave componentsfor which J , L and L′ are greater than or equal to zero and that satisfy thegeneralized Pauli principle (L + S + T = odd). The remaining amplitudes

are obtained from the symmetry relation M1−ν′ −ν = (−1)ν−ν′M1

ν′ν . In theseequations the PL(x) are the Legendre polynomials and

PML (x) = (1 − x2)

M2dM

dxMPL(x) . (A.2)

Using above relations we obtain the a amplitude

a =1

8π2

∞∑

L=0

PL(cosφ)[

(2L+ 1)ML,S=0LL + (2L+ 1)ML,S=1

LL

+ (2L+ 3)ML+1,S=1LL + (2L− 1)ML−1,S=1

LL

]

,

(A.3)

and in the same way the c amplitude

c =i

8π2

∞∑

L=1

P 1L(cosφ)

[(

2L+ 3

L + 1

)

ML+1,S=1LL −

(

2L + 1

L(L+ 1)

)

ML,S=1LL

−(

2L− 1

L

)

ML−1,S=1LL

]

.

(A.4)

144

Appendix BNumerical Solution of theLippmann-Schwinger Equation

In this appendix we give the details of the numerical solution of the Lippmann-Schwinger (LS) equation for the NN and NA systems. In Section B.1 we givethe deails of the matrix-inversion method used to solve the LS equation fora two-nucleon system in the partial wave representation. In Section B.2 weapply the same method to the NA LS equation used to compute the elastictransition amplitude in the NA center-of-mass frame. For this equation wewant to give some details because, for a better description of the experimentaldata, it is treated in the relativistic kinematics.

B.1 The Matrix-Inversion Method

In the patial wave basis the Lipmann-Schwinger equation becomes:

Tl(k′, k;E) = Vl(k

′, k) +2

π

∫ ∞

0

dp p2Vl(k

′, p)Tl(p, k;E)

E − Ep + iǫ, (B.1)

where E is the total kinetic energy of the two incident particles in center ofmass frame at which we solve the equation and it is E = k20/2µ, with µ thereduced mass of the two nucleon system. Knowing that

1

E − Ep + iǫ=

PE −Ep

− iπδ(E − Ep) , (B.2)

where P denotes the principal value of the integral, Eq. (B.1) becomes:


′, k) +4µ

πP∫ ∞

0

dp p2Vl(k

′, p)Tl(p, k;E)

k20 − p2

− i2µk0Vl(k′, k0)Tl(k0, k) .

(B.3)

145

B. Numerical Solution of the Lippmann-Schwinger Equation

Using the principal-value prescription

P∫ ∞

0

dkf(k)

k2 − k20=

∫ ∞

0

dkf(k) − f(k0)

k2 − k20, (B.4)

Eq.(B.3) becomes:


′, k)

− 4µ

π

∫ ∞

0

dpp2Vl(k

′, p)Tl(p, k;E) − k20Vl(k′, k0)Tl(k0, k;E)

p2 − k20− i2µk0Vl(k

′, k0)Tl(k0, k;E) .

(B.5)

In order to solve the Lippman-Schwinger equation in momentum space, weneed first to write a function which sets up the mesh points. We need to dothat since we are going to approximate an integral through

∫ b

a

f(x)dx ≈N∑

i=1

wif(xi) , (B.6)

where we have fixed N lattice points through the corresponding weights wi

and points xi. First we fix the number of mesh points N . We use the functionGAULEG from Numerical Recipes to set up the weights wi and the pointsxi. The routine GAULEG uses Legendre polynomials to fix the mesh pointsand weights: this means that the integral is for the interval [−1, 1], whilethe integral in Eq. (B.5) is for the interval [0,∞]. Thus we need to map theweights and the points from GAULEG to our interval. To do this, we call firstGAULEG and we generate our points in the interval [−1, 1] and it returns themesh points xi and weights wi. Then we map these points over to the limitsin our integral using the following mapping:

yi = const× tanπ

4(1 + xi)

, ωi = constπ

4

wi

cos2

π4(1 + xi)

. (B.7)

If we work with fm−1 units for yi, we set const = 1, otherwise if we choose towork with MeV, we set const = ~c = 197.32696 MeV fm. In our calculationswe chose the latter one.

Next step is to obtain the potential in momentum space. There are differ-ent theoretical potentials and we used the Cd-Bonn and Ch-N3LO potentials.Using the mesh points yi, the weights ωi and omitting the subscript l and theenergy E, we rewrite Eq. (B.5) as

T (k′, k) = V (k′, k) − 4µ

π

N∑

j=1

ωj y2j V (k′, yj) T (yj, k)

y2j − k20

+4µ

πk20

N∑

n=1

ωn

y2n − k20V (k′, k0) T (k0, k) − i2µk0 V (k′, k0) T (k0, k) .

(B.8)

146

B.1. The Matrix-Inversion Method

The previous equation can be rewritten as

T (k′, k) = V (k′, k) −N∑

j=1

[

4µ

π

ωj y2j

y2j − k20

]

V (k′, yj) T (yj, k)

−[

− 4µ

πk20

N∑

n=1

ωn

y2n − k20+ i2µk0

]

V (k′, k0) T (k0, k) .

(B.9)

This equation contains the unknowns T (yi, yj) (with dimension N × N) andT (k0, k0). We can turn Eq. (B.9) into an equation with dimension (N + 1) ×(N+1) with a mesh which contains the original mesh points yj, for j = 1, . . . , Nand the point which corresponds to the energy k0. We consider the latter asthe “observable” point. Now we define new mesh points as

kj ≡ (y1, y2, . . . , yN , k0) , j = 1, 2, . . . , N + 1 , (B.10)

and we also define the array uj:

uj =4µ

π

ωj y2j

y2j − k20, j = 1, 2, . . . , N

uj = −4µ

πk20

N∑

n=1

ωn

y2n − k20+ i2µk0 , j = N + 1 .

(B.11)

Using Eqs. (B.10) and (B.11) we can rewrite Eq. (B.9) as

T (k′, k) = V (k′, k) −N+1∑

j=1

uj V (k′, kj) T (kj, k) . (B.12)

We work on kj points and so we rewrite Eq.(B.12) as

T (ki, kj) +

N+1∑

l=1

ul V (ki, kl) T (kl, kj) = V (ki, kj) . (B.13)

Knowing that

T (ki, kj) =

N+1∑

l=1

δilT (kl, kj) , (B.14)

Eq. (B.13) becomes

N+1∑

l=1

[

δil + ul V (ki, kl)]

T (kl, kj) = V (ki, kj) . (B.15)

Defining the matrix A as

A(ki, kl) ≡ δil + ul V (ki, kl) . (B.16)

147


Eq. (B.15) becomes

N+1∑

l=1

A(ki, kl) T (kl, kj) = V (ki, kj) , ⇒ AT = V . (B.17)

The T matrix is thus obtained by matrix inversion:

T = A−1V . (B.18)

To obtain T we will need to set up the matrices A and V and to invert thematrix A. With the inverse A−1 and performing a matrix multiplication withV , we obtain T . With the T matrix it is possible to compute the phase shiftsδl in each partial wave using the following relation:

Tl(k0, k0;Ek0) = −eiδl sin δl2µk0

. (B.19)

Knowing that

Tl(k0, k0;Ek0) = TR + iTI , (B.20)

we obtain:

δl =1

2arctan

( −4µk0TR1 + 4µk0TI

)

. (B.21)

B.1.1 The Nucleon-Nucleon Scattering

Now we want to apply the LS equation to the case of nucleon-nucleon scat-tering. We represent the two-nucleon states in terms of an |LSJM〉 basis,where S denotes the total spin, L the total orbital angular momentum and Jthe total angular momentum with projection M . In this basis we will denotethe T matrix elements by T JS

L′L ≡ 〈L′SJM |T |LSJM〉 and we have three cases:spin singlet state, uncoupled spin triplet state and coupled triplet states. Inthe following we summarize the equations for these cases.

Spin singlet (L′ = L = J)

T J0JJ (k′, k;E) = V J0

JJ (k′, k) +2

π

∫ ∞

0

dp p2V J0JJ (k′, p)T J0

JJ (p, k;E)

E − Ep + iǫ. (B.22)

Uncoupled spin triplet (L′ = L = J)

T J1JJ (k′, k;E) = V J1

JJ (k′, k) +2

π

∫ ∞

0

dp p2V J1JJ (k′, p)T J1

JJ (p, k;E)

E − Ep + iǫ. (B.23)

148

B.1. The Matrix-Inversion Method

Coupled triplet states

T J1++(k′, k;E) = V J1

++(k′, k)

+2

π

∫ ∞

0

dp p2V J1++(k′, p)T J1

++(p, k;E) + V J1+−(k′, p)T J1

−+(p, k;E)

E − Ep + iǫ,

T J1−−(k′, k;E) = V J1

−−(k′, k)

+2

π

∫ ∞

0

dp p2V J1−−(k′, p)T J1

−−(p, k;E) + V J1−+(k′, p)T J1

+−(p, k;E)

E − Ep + iǫ,

T J1+−(k′, k;E) = V J1

+−(k′, k)

+2

π

∫ ∞

0

dp p2V J1++(k′, p)T J1

+−(p, k;E) + V J1+−(k′, p)T J1

−−(p, k;E)

E − Ep + iǫ,

T J1−+(k′, k;E) = V J1

−+(k′, k)

+2

π

∫ ∞

0

dp p2V J1−−(k′, p)T J1

−+(p, k;E) + V J1−+(k′, p)T J1

++(p, k;E)

E − Ep + iǫ,

(B.24)

where we have introduced the abbreviations:

T J1++ ≡ T J1

J+1,J+1 , T J1−− ≡ T J1

J−1,J−1 , T J1+− ≡ T J1

J+1,J−1 , T J1−+ ≡ T J1

J−1,J+1 .(B.25)

Conventionally, the coupled triplet channels in NN scattering are considered inthis form. For the spin singlet, Eq. (B.22), and for the uncoupled spin triplet,Eq. (B.23), the T matrix is solved applying the matrix-inversion method asshown in the last section. For example, we write Eq. (B.22) in discrete formas

N+1∑

l=1

AJ0JJ(ki, kl) T

J0JJ (kl, kj) = V J0

JJ (ki, kj) , (B.26)

where ki and ωi are the Gauss points and weights and

uj =4µ

π

ωj k2j

k2j − k20, j = 1, 2, . . . , N

uj = −4µ

πk20

N∑

n=1

ωn

k2n − k20+ i2µk0 , j = N + 1 ,

(B.27)

andAJ0

JJ(ki, kl) ≡ δil + ul VJ0JJ (ki, kl) . (B.28)

The matrices AJ0JJ , V J0

JJ and T J0JJ are (N + 1)× (N + 1) matrices and Eq. (B.26)

is a matrix equation,AJ0

JJ TJ0JJ = V J0

JJ , (B.29)

that has the familiar formAX = B , (B.30)

149


where X = T J0JJ is the unknown matrix, and it is solved by matrix inversion:

T J0JJ = (AJ0

JJ)−1V J0JJ . (B.31)

The extension to coupled channels is straightforward. In matrix notation, wehave for the coupled case

(

AJ1++ AJ1

+−

AJ1−+ AJ1

−−

)(

T J1++ T J1

+−

T J1−+ T J1

−−

)

=

(

V J1++ V J1

+−

V J1−+ V J1

−−

)

, (B.32)

where the AJ1++, AJ1

−−, AJ1+− and AJ1

−+ are defined similarly to Eq. (B.28) and

AJ1+− ≡ AJ1

+− − 1 , AJ1−+ ≡ AJ1

−+ − 1 , (B.33)

with 1 the unit matrix. In this case the three matrices of Eq. (B.32) havedimensions (2N + 2) × (2N + 2) and we can solve Eq. (B.32) as we did forEq. (B.29) by matrix inversion:

(

T J1++ T J1

+−

T J1−+ T J1

−−

)

=

(

AJ1++ AJ1

+−

AJ1−+ AJ1

−−

)−1(V J1++ V J1

+−

V J1−+ V J1

−−

)

. (B.34)

B.2 The relativistic Lippmann-Schwinger equa-

tion

In this section we apply the matrix-inversion method to solve the relativis-tic Lippmann-Schwinger equation for the nucleon-nucleus system. The onlydifference with the non-relativistic case is given by the relativistic kinematicsthat changes the energy expressions in the denominator. In the partial waverepresentation the equation is given as


′, k) +2

π

∫ ∞

0

dp p2Vl(k

′, p)Tl(p, k;E)

E − E(p) + iǫ, (B.35)

where

E = Eproj + Etarg =√

k20 +m2proj +

√

k20 +m2targ ,

E(p) = Eproj(p) + Etarg(p) =√

p2 +m2proj +

√

p2 +m2targ ,

and mproj and mtarg are the projectile and the target mass, respectively. Know-ing that

p dp = µ(p) dE(p) , µ(p) =Eproj(p)Etarg(p)

Eproj(p) + Etarg(p),

and using the identity

1

E − E ′ + iǫ=

PE − E ′

− iπδ(E − E ′) ,

150

B.2. The relativistic Lippmann-Schwinger equation

we obtain:


′, k) +2

πP∫ ∞

0

dp p2Vl(k

′, p)Tl(p, k;E)

E −E(p)

− i2µ(k0)k0Vl(k′, k0)Tl(k0, k;E) .

Using the principle-value prescription

P∫ ∞

0

dpf(p)

E(p) − E=

∫ ∞

0

dp

[

f(p)

E(p) − E− 2µ(k0) f(k0)

p2 − k20

]

,

we obtain:


′, k) − 2

π

∫ ∞

0

dpp2Vl(k

′, p)Tl(p, k;E)

E(p) − E

+2

π

∫ ∞

0

dp2µ(k0)k

20Vl(k

′, k0)Tl(k0, k;E)

p2 − k20− i2µ(k0)k0Vl(k

′, k0)Tl(k0, k;E) .

(B.36)

As we made for the non-relativistic equation, we work on gauss points yj andthe respective weights ωj ; the previous equation becomes:

T (k′, k) = V (k′, k)

−N∑

j=1

[

2

π

ωj y2j

√

y2j +m2proj +

√

y2j +m2targ − E

]

V (k′, yj) T (yj, k)

−[

− 4

πµ(k0)k

20

N∑

n=1

ωn

y2n − k20+ i2µ(k0)k0

]

V (k′, k0) T (k0, k) .

This equation contains the unknowns T (yi, yj) (with dimension N × N) andT (k0, k0). We can turn it into an equation with dimension (N + 1) × (N + 1)with a mesh which contains the original mesh points yj, for j = 1, . . . , Nand the point which corresponds to the energy k0, that is considered as the“observable” point. We define new mesh points as

kj ≡ (y1, y2, . . . , yN , k0) , j = 1, 2, . . . , N + 1 ,

and the array uj as

uj =2

π

ωj y2j

√

y2j +m2proj +

√

y2j +m2targ −E

, j = 1, 2, . . . , N

uj = −4

πµ(k0)k

20

N∑

n=1

ωn

y2n − k20+ i2µ(k0)k0 , j = N + 1 .

(B.37)

The computation of the matrix Tl(k′, k;E) is now equal to the non-relativistic

case.

151


152

Appendix CAngular Momentum Projections

The details of the derivation of the results given in Eqs. (5.19) for the angularmomentum projections of the nuclear part of the optical potential of Eq. (5.13)are outlined here. We begin with Eq. (5.13) in the form

U(k′,k;ω) = U c(k′,k;ω) +i

2σ · k′ × k U ls(k′,k;ω) , (C.1)

We note that, form the definig Eqs. (4.59) and (4.60), the quantities U c(k′,k;ω)and U ls(k′,k;ω) are invariant under rotations of the coordinate system anddepend on cos θ = k′ · k as the only angular variable. Accordingly, we canmake the angular momentum expansions (where a stands for either c or ls)

Ua(k′,k;ω) =2

π

∑

JLM

YL 1

2

JM(k′) UaL(k′, k;ω)YL 1

2†

JM (k)

=2

π

∑

LΛ

Y ΛL (k′) Ua

L(k′, k;ω) Y Λ ∗L (k) ,

(C.2)

and thus,

UaL(k′, k;ω) =

π

2

∫

dk′ dk YL 1

2†

JM (k′)Ua(k′,k;ω)YL 1

2

JM(k)

=π

2

∫

dk′ dk Y Λ ∗L (k′)Ua(k′,k;ω)Y Λ

L (k)

=π

2

1

2L + 1

∑

Λ

∫

dk′ dk Y Λ ∗L (k′)Ua(k′,k;ω)Y Λ

L (k)

=π

2

1

4π

∫

dk′ dk PL(k′ · k)Ua(k′,k;ω)

=π

4

∫

dk

∫ +1

−1

dx PL(x)Ua(k′,k;ω)

= π2

∫ +1

−1

dx PL(x)Ua(k′,k;ω) ,

(C.3)

153

C. Angular Momentum Projections

where x = cos θ = k′ · k. For the first central term of Eq. (C.1) the requiredangular momentum projection U c

LJ is thus identified to be U cLJ(k′, k;ω) =

U cL(k′, k;ω) for both values of J = L± 1

2. The second complete spin-orbit term

of Eq. (C.1) can be treated by writing

i

2σ · k′ × k U ls(k′,k;ω) =

1

2σ · k′ U ls(k′,k;ω) σ · k − 1

2k′ · k U ls(k′,k;ω)

≡ F (σ,k′,k;ω) −G(k′,k;ω) .

(C.4)

For the term F we introduce the expansion

F (σ,k′,k;ω) =2

π

∑

JL′LM

YL′ 1

2

JM (k′)FJL′L(k′, k;ω)YL 1

2†

JM (k) , (C.5)

which implements conservation of total angular momentum. The componentsare given by

FJL′L(k′, k;ω) =π

2

∫

dk′ dk YL′ 1

2†

JM (k′)

[

1

2σ · k′ U ls(k′,k;ω) σ · k

]

YL 1

2

JM(k) .

(C.6)When Eq. (C.2) is substituted into Eq. (C.6), we obtain

FJL′L(k′, k;ω) =∑

J ′′L′′M ′′

k′k

2〈JL′M |σ · k′|J ′′L′′M ′′〉

× U lsL′′(k′, k;ω) 〈J ′′L′′M ′′|σ · k|JLM〉 .

(C.7)

Now we make use of the fact that σ · k is a Hermitian, unitary, pseudoscalaroperator that commutes with J2 and Jz. It changes the parity but not thenorm. This leads immediately to the result

σ · k YL 1

2

JM(k) = −Y L 1

2

JM(k) , (C.8)

where L = 2J − L. With this, Eq. (C.7) becomes

FJL′L(k′, k;ω) =k′k

2

∑

L′′

δL′L′′ δL′′L UlsL′′(k′, k;ω) = δL′L

k′k

2U lsL (k′, k;ω) .

(C.9)The expected J dependence of this term is implicitly contained in the definitionof L. The second term of Eq. (C.4) is invariant under rotations of coordinatesystem and can be expanded as in Eqs. (C.2,C.3). The angular momentumcomponent is (x = k′ · k),

GL(k′, k;ω) = π2

∫ +1

−1

dxPL(x)G(k′,k;ω)

= π2

∫ +1

−1

dxPL(x)

[

1

2k′ · k U ls(k′,k;ω)

]

=k′k

2π2

∫ +1

−1

dx xPL(x)U ls(k′,k;ω) .

(C.10)

154

With the help of the recurrence relation

xPL(x) =1

2L+ 1

[

(L + 1)PL+1(x) + LPL−1(x)]

,

Eq. (C.10) becomes

GL(k′, k;ω) =k′k

2

1

2L + 1

[

(L+ 1)U lsL+1(k

′, k;ω) + LU lsL−1(k

′, k;ω)]

, (C.11)

where we have used Eq. (C.3) with a = ls. The angular momentum expansionof Eq. (C.4) for the complete spin-orbit term is thus

i

2σ · k′ × k U ls(k′,k;ω) =

2

π

∑

JLM

YL 1

2

JM(k′) W lsLJ(k′, k;ω)YL 1

2†

JM (k) , (C.12)

where

W lsLJ(k′, k;ω) =

k′k

2

1

2L + 1

[

(2L+ 1)U lsL (k′, k;ω)

− (L+ 1)U lsL+1(k

′, k;ω) − LU lsL−1(k

′, k;ω)]

,

(C.13)

or equivalentlyW ls

LJ (k′, k;ω) = CLJ VlsL (k′, k;ω) , (C.14)

where


k′k

2L + 1

[

U lsL+1(k

′, k;ω) − U lsL−1(k

′, k;ω)]

(C.15)

and

CLJ =1

2

[

J(J + 1) − L(L + 1) − 3

4

]

=

L

2J = L +

1

2

−L + 1

2J = L− 1

2

. (C.16)

When this result is combined with the expansion in Eq. (C.2) for the centralterm of the optical potential, we obtain for the total optical potential

U(k′,k;ω) =2

π

∑

JLM

YL 1

2


2†

JM (k) , (C.17)

where the components are


lsL (k′, k;ω) , (C.18)

in agreement with the results stated in Eqs. (5.19).

155

C. Angular Momentum Projections

156

Appendix DAbbreviations

Here we summarize the abbreviations used throughout this dissertation.

1. QCD: Quantum Chromodynamics.

2. QED: Quantum Electrodynamics.

3. BS: Bethe-Salpeter.

4. LS: Lippmann-Schwinger.

5. QE: Quasi-Elastic.

6. RMF: Relativistic Mean Field.

7. BCS: Bardeen Cooper Schrieffer.

8. NN : Nucleon-Nucleon.

9. NA: Nucleon-Nucleus.

10. FSI: Final-State Interactions.

11. QHD: Quantum Hadrodynamics.

12. RHB: Relativistic Hartree Bogoliubov.

13. HFB: Hartree Fock Bogoliubov.

14. DDME: Density-Dependent Meson-Exchange.

15. EFT: Effective Field Theory.

16. DFT: Density Functional Theory.

17. RIB: Radioactive Ion Beam.

18. RGF: Relativistic Green’s Function.

157

D. Abbreviations

19. DWBA: Distorted-Wave Born Approximation.

20. PWBA: Plane-Wave Born Approximation.

21. PVES: Parity-Violating Electron Scattering.

22. EOS: Equation Of State.

23. 2PF: Two-Parameter Fermi.

24. IA: Impulse Approximation.

25. RIA: Relativistic Impulse Approximation.

26. DWIA: Distorted-Wave Impulse Approximation.

27. RDWIA: Relativistic Distorted-Wave Impulse Approximation.

28. RPWIA: Relativistic Plane-Wave Impulse Approximation.

29. rROP: real (part of the) Relativistic Optical Potential.

30. KMT: Kerman, McManus, and Thaler.

31. OBE: One Boson Exchange.

32. ChPT: Chiral Perturbation Theory.

33. CD: Charge Dependence.

34. LO: Leading Order.

35. NLO: Next-to-Leading Order.

36. NNLO: Next-to-Next-to-Leading Order.

37. N3LO: Next-to-Next-to-Next-to-Leading Order.

38. 1PE: One-Pion Exchange.

39. 2PE: Two-Pion Exchange.

40. 3PE: Three-Pion Exchange.

41. 2NF: Two-Nucleon Forces.

42. 3NF: Three-Nucleon Forces.

43. 4NF: Four-Nucleon Forces.

44. LECs: Low-Energy Constants.

158

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