Electron and proton scattering off nuclei with neutron and proton excess. Matteo Vorabbi

Transcription

1 UNIVERSITÀ 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

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3 Contents 1 Introduction 1 2 Relativistic Mean-Field Models Introduction The Density-Dependent Model The Lagrangian Density Stationary Solutions of the Equations of Motion Rotationally Invariant Systems Covariant Density Functional Theory The Density-Dependence of the Couplings Pairing Correlations Non-Relativistic Theory Covariant Density Functional Theory with Pairing Results for Neutron and Proton Densities Electron Scattering Historical Introduction The Nuclear Response Elastic and Quasi-Elastic Scattering on Exotic Nuclei Elastic Scattering Theoretical Framework Results Parity-Violating Elastic Scattering Theoretical Framework Results for Oxygen and lcium Isotopic Chains Neutron Density of 28 Pb Inclusive Quasi-Elastic Scattering Relativistic Green s Function Model Results Summary and Conclusions iii

4 CONTENTS 4 Theoretical Foundations for Proton Scattering Introduction The First-Order Optical Potential The Spectator Expansion The Impulse Approximation The KMT Multiple Scattering Theory Fixed Beam Energy Approximation The Optimum Factorization Approximation Kinematical Variables The Nucleon-Nucleon Potential The CD-Bonn Potential The Chiral Potential The Nucleon-Nucleon Transition Matrix Theoretical Results for Nucleon-Nucleon Amplitudes Relativistic Kinematics and the Scattering Observables Kinematics Relativistic Kinematics The Møller Factor The Transition Amplitude in Partial Wave Representation The Scattering Observables Treatment of the Coulomb Potential Results for Scattering Observables Conclusions and Future Perspectives Summary and Conclusions 139 A The Singlet-Triplet Scattering Amplitudes 143 B Numerical Solution of the Lippmann-Schwinger Equation 145 B.1 The Matrix-Inversion Method B.1.1 The Nucleon-Nucleon Scattering B.2 The relativistic Lippmann-Schwinger equation C Angular Momentum Projections 153 D Abbreviations 157 iv

5 Chapter1 Introduction The study of atomic nuclei has a long history and represents one of the most exciting 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 Chromodynamics (QCD), neutrons and protons are taken as the relevant degrees of freedom for present nuclear theories. The properties of nuclei are thus characterized by the number of nucleons and by the ratio between neutron and proton numbers. In nature we have found only 263 stable nuclei surrounded by radioactive ones. Some of these unstable nuclei can be found on Earth, some are manufactured and several thousand nuclei are the yet unexplored exotic species. A large number of these radioactive nuclei are characterised by beta decay, while, for heavier nuclei, processes like emission of alpha particles or spontaneous fission dominate due to the important role played by the electromagnetic interaction. In the nuclear chart, moving away from stable nuclei by adding either protons or neutrons, we reach the boundaries for nuclear particle stability, called drip lines. The nuclei beyond the drip lines are unbound to nucleon emission, that is, for these systems the strong interaction is unable to bind all the A nucleons in 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 neutron numbers 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. As a result, the nuclear landscape separating the neutron drip line and the valley of stability is large and difficult to probe experimentally. In this region of the nuclear chart we can find new unexpected phenomena which are important not only for nuclear physics but also for astrophysics and cosmology. The structure of the nuclei is expected to change significantly as the limit of nuclear stability is approached in neutron excess. Due to the systematic variation in the spatial distribution of nucleonic densities and the increased importance of the pairing 1

6 1. Introduction field, the average nucleonic potential is modified when approaching the neutron drip line. With the increase of the neutron number, the single-particle neutron potential becomes very shallow and for drip-line nuclei this results in a new shell structure. On the proton-rich side of the valley of stability, physics is different than in nuclei with a large neutron excess. The Coulomb barrier tends to localize the proton density in the nuclear interior and thus nuclei beyond the proton drip line are quasi-bound with respect to proton emission. In spite of this, the effects associated with the weak binding are also present in proton drip-line nuclei. Moreover, N = Z proton-rich nuclei are very important for the study of nuclear pairing. Due to the large spatial overlaps between neutron and proton single-particle wave functions, these systems are expected to exhibit unique manifestations of proton-neutron pairing. Nuclei beyond the proton drip line are ground-state proton emitters. Proton radioactivity is an excellent example of the elementary three-dimensional quantum mechanical tunneling and its investigation will open up a wealth of exciting new physics. Compared with most physical systems, nuclei are difficult to study and the reason lies in the strength of the nuclear interaction, which results in a very tightly bound system. Accordingly, we cannot proceed as easily as in atomic physics, where the wavefunctions of individual electrons are studied by bombarding the atom with photons or electrons or X-rays and then measuring the energies, momenta and angular momenta of the ejected electrons. In the case of nuclei, an equivalent investigation would require photons or particles of high enough energy to overcome the nuclear binding, which is tipically 8 MeV per nucleon. There are basically two ways to investigate nuclear properties, namely radioactive decay and nuclear reactions. The former is an important source of information: it often provides the first information on any newly produced nucleus and it is the basis for precision tests of the Standard Model and other things. Thus it is extremely useful but unfortunately we have no control over radioactive 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 investigating the structure of hadron systems such as atomic nuclei and their constituents. Electron- and nucleon-scattering reactions are the most important ones historically employed for such a purpose. Electron-scattering reactions have provided the most complete and detailed information on nuclear and nucleon structure. Electrons predominantly interact with nuclei through the electromagnetic interaction, which is well known from Quantum Electrodynamics (QED) and is weak compared with the strength of the interaction between hadrons. The scattering process is therefore adequately treated assuming the validity of the Born approximation, i.e., the one-photon exchange mechanism between the electron and the target. Thevirtualphoton, liketherealone, hasameanfreepathmuchlargerthanthe 2

7 target dimensions, thus exploring the whole target volume. This contrasts with hadron probes, which are generally absorbed at the target surface. Moreover, the ability to vary independently the energy and momentum of the exchanged virtual photon transferred to the nucleus makes it possible to map the nuclear response as a function of its excitation energy with a spatial resolution that can be adjusted to the scale of processes that need to be studied. On the other hand, the scattering of neutrons and protons from nuclei has an even longer history and represents a very important tool to investigate the interaction of a nucleon with a many-nucleon system. In the case of elastic scattering the reaction may be cast in the form of an optical potential which is used as input for the Schrödinger or Lippmann-Schwinger (LS) equation depending on the coordinate or momentum space where we work. The solution of these equations permits then to compute the elastic differential cross section. The comparison of the calculated elastic scattering observables with available data is the first crucial test of an optical potential. After its validity has been checked in comparison with elastic nucleon-nucleus (NA) scattering data, the optical potential can then be used for calculations on a wide variety of inelastic processes and nuclear reactions, where it is a crucial and critical input. In particular, in electron scattering the optical potential is the basic ingredient to describe final-state interactions (FSI) in quasi-elastic (QE) exclusive and inclusive electron-nucleus scattering. This makes nucleon scattering very important for many purposes and, in particular, a microscopic theoretical approach to this computational tool allows us to get more insight about nuclear structure. Several decades of experimental and theoretical work on these reactions have provided a wealth of information on the properties of stable nuclei. The use of electron and nucleon probes can be extended to exotic nuclei. The detailed study of the properties of nuclei far from the stability line and the evolution of nuclear properties with respect to the asymmetry between the number ofneutronsandprotonsisoneofthemajortopicsofinterest inmodern nuclear physics. In the next years the advent of radioactive ion beam facilities will provide a large amount of data on unstable nuclei. These facilities will offer unprecedented opportunities to study the structure of exotic unstable nuclei through electron and nucleon scattering. In particular, for electron scattering, kinematically complete experiments will become feasible for the first time, allowing a clean separation of different reaction channels as well as a reduction of the unavoidable radiative background seen in conventional experiments. Therefore, even applications using stable isotope beams will be of interest. In this work we present and discuss theoretical results obtained within the frameworks of electron and proton scattering off nuclei with neutron and proton excess. In particular, our aim is to study the evolution of nuclear properties with the increase of neutron and proton number. For this reason, we performed our calculations along isotopic and isotonic chains. The dissertation 3

8 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 we show the results obtained for neutron and proton densities of the nuclei considered in the next chapters. The study of the evolution of nuclear properties requires a good knowledge of nuclear-matter distributions for neutrons and protons separately. The ground-state densities reflect the basic properties of effective nuclear forces providing fundamental nuclear structure information and represent the basic ingredient to compute electron- and proton-scattering observables. In Chapter 3 we present and discuss the theoretical results obtained for electron scattering. In particular, we consider elastic and QE reactions that allow us to investigate bulk and single-particle properties of nuclei. For these calculations we consider both isotopic and isotonic chains. The chapter is divided in three sections devoted to describe the theory and the results for elastic scattering, parity-violating elastic scattering, and inclusive QE scattering. Elastic electron scattering makes it possible to measure with excellent precision only charge densities and therefore proton distributions. It is much more difficult to measure neutron distributions. Our present knowledge of neutron densities comes primarily from hadron-scattering experiments, the analysis of which requires always model-dependent assumptions about strong nuclear forces at low energies. A model-independent probe of neutron densities is provided by parity-violating elastic electron scattering, where direct information on the neutron density can be obtained from the measurement of the parity-violating asymmetry A pv parameter, which is defined as the difference between the cross sections for the scattering of right- and left-handed longitudinally polarized electrons. This quantity is related to the radius of the neutron distribution and provides a robust model-independent measurement of it. In QE scattering the response of the nucleus to the electron probe is dominated by the singleparticle dynamics and by the process of one-nucleon knockout, where the probe interacts with only one nucleon which is then ejected from the nucleus with a direct knockout mechanism. The emitted nucleon can be detected in coincidence with the scattered electron and suitable kinematics conditions can be envisaged where we can assume that the residual nucleus is left in a discrete eigenstate. This is the case of the exclusive scattering, where the final nuclear state is completely determined. In the inclusive scattering only the scattered electron is detected, the final nuclear state is not determined and the cross section includes all the available final nuclear state, but the main contribution in the region of the QE peak still comes from the interaction on single nucleons. The inclusive scattering corresponds to an integral over all available nuclear states and it is directly related to the dynamics of the initial nuclear ground state. In Chapter 4 we introduce the general NA scattering problem stated in momentum space. In this space the description of the interaction between 4

9 the incoming nucleon and the target nucleus is described by the transition matrix, obtained solving the general many-body LS equation. This problem can be cast in a simple one-body equation for the general transition matrix and a complicated many-body one for the optical potential that enters the first equation. The optical potential operator can be written as a sum of A many-body terms, where the first one is a two-body term given by the sum over all nucleons of the interaction between the projectile nucleon and the i- th target nucleon inside the nucleus. The first term of this expansion defines the first-order optical potential and leads to a manageable equation that is further simplified using the optimum factorization approximation, in which, the form of the optical potential is factorized into the product of the NN transition matrix and the neutron and proton densities. The rest of the chapter is devoted to the computation of the NN transition matrix that represents, with the nucleon densities, the second basic ingredient for the construction of a microscopic theoretical optical potential. Particular attention is given to the different NN potentials necessary to the construction of the transition matrix. In fact, a novelty of our calculations is represented by the use of Chiral potentials as input in the NN LS equation to obtain the transition matrix. At the end of the chapter we present and discuss the results obtained with different NN potentials for the Wolfenstein amplitudes, which are proportional to the central and spin-orbit part of the NN transition matrix. In Chapter 5 we describe the calculational framework needed to employ the optical potential developed in Chapter 4 in calculations of the elastic scattering observables: the differential cross section, the analyzing power, and the spin rotation. The calculation is performed in the NA center-of-mass frame and the optical potential and the transition matrix are decomposed in the partial wave repesentation. A section is also dedicated to the algorithm employed to include in the model the Coulomb interaction between the projectile nucleon and the target nucleus. Finally, we present and discuss the theoretical results of the scattering observables obtained with different NN potentials. First, calculations are performed for stable nuclei and compared with available experimental data, then they are extended to nuclei with neutron excess. To be consistent with the work of Chapter 3 we take under consideration the same oxygen and calcium isotopic chains. Finally, in Chapter 6 we draw our conclusions and give some perspectives for possible improvements and future developments in the construction of a microscopic optical potential. 5

10 6 1. Introduction

11 Chapter2 Relativistic 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 at the forefront of nuclear science. Recent experiments have been probing nuclei under extreme conditions of high spin, large deformations, high excitation energies and neutron-to-proton ratios at their limit of stability. A wealth of experimental data on such nuclei is expected in the near future from new radioactive-beam facilities. In neutron-rich nuclei the weak binding energy of the outermost neutrons and the effects of the coupling between bound states and particle continuum produce different exotic phenomena allowing the growth of regions in nuclei with very diffuse neutron density in which we can find the formation of a neutron skin for heavy nuclei or halo structures for light ones. With the increase of the number of neutrons the effective nuclear potential is modified and this produces a suppression of shell effects with the resulting disappearance of magic numbers. Moreover, very neutron-rich nuclei are important for nuclear astrophysics because they offer the opportunity to study pairing phenomena in systems with strong density variations that determine the astrophysical conditions for the formation of neutron-rich stable isotopes. On the proton-rich side of the valley of stability physics is different than in nuclei with a large neutron excess. Extremely proton-rich nuclei are important both for nuclear structure studies and in astrophysical applications. Because of the Coulomb barrier, which tends to localize the proton density in the nuclear interior, nuclei beyond the proton drip line are quasi-bound with respect to proton emission. These systems are characterized by exotic ground-state decay modes, such as direct emission of charged particles and β-decays with large Q- values. The phenomenon of proton emission from the ground-state has been extensively investigated in medium-heavy and heavy spherical and deformed nuclei. The properties of many proton-rich nuclei play an important role in 7

12 2. Relativistic Mean-Field Models the process of nucleosynthesis by rapid-proton capture. Nuclei with an equal number of protons and neutrons are very important in studies of exotic nuclear structure. They permit to study the details of the effective off-shell neutron-proton interaction because of the high symmetry between the degrees of freedom of protons and neutrons that occupy the same shell-model orbitals. Light N = Z nuclei are β-stable, while heavier nuclei are found close to the proton drip line and they display a variety of structure phenomena: shape coexistence, super-deformation, alignment of proton-neutron pairs, and proton radioactivity from highly excited states. During the last decades nuclear structure theory has evolved from the macroscopic and microscopic descriptions of structure phenomena in stable nuclei, towards more exotic ones far from the valley of β-stability. The development of mean-field theories has reported significant progress, in particular, in describing the structure phenomena of medium-heavy and heavy nuclear systems for which an accurate description is obtained by mean-field calculations based on nuclear energy density functionals. These models are based on concepts of non-renormalizable effective relativistic field theories and density functional theory and provide an interesting theoretical framework for studies of nuclear structure phenomena at and far from the valley of β-stability. A well known example of an effective theory of nuclear structure is Quantum Hadrodynamics (QHD): a field theoretical framework of Lorentz-covariant, meson-nucleon or point-coupling models of nuclear dynamics. A variety of nuclear phenomena have been described with QHD based models such as nuclear matter or properties of finite spherical and deformed nuclei. The effective Lagrangians of QHD is compound by known long-range interactions constrained by symmetries and a set of generic short-range interactions. All QHD based models are consistent with the symmetries of QCD: Lorentz invariance, parity invariance, electromagnetic gauge invariance, isospin and chiral symmetry. However, pions are not explicitly included in the RMF calculation and the effects of correlated two-pion exchange are implicitly taken into account through the phenomenological scalar, isoscalar σ mean-field. RMF models of nuclear structure are phenomenological, with parameters adjusted to reproduce the nuclear matter equation of state and a set of global properties of spherical closed-shell nuclei. Lorentz scalar and four-vector nucleon self-energies are at the basis of all QHD models. Although it is not possible to experimentally verify the presence of the large scalar and vector potentials, from nuclear matter saturation and spin-orbit splittings in finite nuclei the magnitude of these potentials can be estimated around several hundred MeV in the nuclear interior. In application to finite nuclei, the scalar and the vector nucleon self-energies are treated phenomenologically and in some models they arise from the exchange of σ and ω mesons. The strength parameters are determined by nuclear matter saturation and nuclear structure data. However, in order to reproduce the empirical bulk 8

13 2.2. The Density-Dependent Model properties of finite nuclei it is necessary to consider density-dependent coupling constants adjusted to Dirac-Brueckner self-energies in nuclear matter. In applications to open-shell nuclei pairing correlations are very important and they have to be included in the RMF framework. In the BCS 1 model, the pairing force is taken into account phenomenologically from experimental odd-even mass differences. For exotic nuclei far from the valley of stability this is, however, only a poor approximation. For nuclei close to the drip lines a unified and self-consistent treatment of mean-field and pairing correlations is required. This led to the development of the relativistic Hartree-Bogoliubov (RHB) model that represents the relativistic extension of the conventional Hartree-Fock-Bogoliubov (HFB) [2] framework. Such a model has been successfully employed in analyses of structure phenomena in exotic nuclei far from the valley of stability and it is crucial for the description of ground-state properties in weakly bound systems. In Section 2.2 we present the characteristics of the density-dependent mesonexchange (DDME) RMF model: we start from the Lagrangian density and we derive the Dirac equation for the nucleon and the Klein-Gordon equations for all mesonic fields. In order to obtain a description of ground-state properties of nuclei, the equations of motion are then derived for the static case and assuming spherical symmetry for all nuclei. In Section 2.3 we present the RHB model based on a covariant density functional approach that gives a unified description of mean-field and pairing correlations. In this model the pairing force is treated non-relativistically, but at this stage this approximation is fully justified. Finally, in Section 2.4 we present our theoretical results for neutron and proton densities of oxygen and calcium isotopic chains and N = 14, 2, and 28 isotonic ones, considered in Refs. [3,4] for a study of nuclear properties with electron scattering calculations and presented in Chapter The Density-Dependent Model The Lagrangian Density In the standard representation of QHD the nucleus is described as a system of Dirac nucleons coupled to the exchange mesons and the electromagnetic field through an effective Lagrangian. The isoscalar scalar σ-meson, the isoscalar vector ω-meson, and the isovector vector ρ-meson build the minimal set of meson fields that together with the electromagnetic field is necessary for a quantitative description of bulk and single-particle nuclear properties [5 8]. The model is defined by the Lagrangian density L = L N +L m +L int. (2.1) 1 The name derives from the Bardeen-Cooper-Schrieffer theory of superconductivity applied in nuclear physics to describe the pairing interaction between nucleons in the atomic nucleus. 9

14 2. Relativistic Mean-Field Models The first term L N denotes the Lagrangian of the free nucleon L N = ψ(iγ µ µ m)ψ, (2.2) where m is the bare nucleon mass and ψ denotes the Dirac spinor. The term L m is the Lagrangian of the free meson fields and the electromagnetic field L m = 1 2 µσ µ σ 1 2 m2 σ σ2 1 4 Ω µνω µν m2 ω ω µω µ 1 4 R µν R µν m2 ρ ρ µ ρ µ 1 4 F µνf µν, (2.3) with the corresponding masses m σ, m ω, m ρ, and Ω µν, R µν, F µν are field tensors Ω µν = µ ω ν ν ω µ, R µν = µ ρ ν ν ρ µ, F µν = µ A ν ν A µ, (2.4) where arrows denote isovectors. Boldface symbols will be used for vectors in ordinary space. In the previous equations, σ, ω µ, and ρ µ are the scalar, vector isoscalar, and vector isovector meson fields, while A µ is the electromagnetic field. In L int is contained the minimal set of interaction terms L int = g σ ψσψ gω ψγ µ ω µ ψ g ρ ψ τγ µ ρ µ ψ e ψ 1 τ 3 γ µ A µ ψ, (2.5) 2 where τ 3 is the third isospin matrix and g σ, g ω, g ρ, and e are the coupling constants. Already in the earliest applications of the RMF framework it was realized, however, that this simple model, with interaction terms only linear in the meson fields, does not provide a quantitative description of complex nuclear systems. An effective density dependence was introduced [9] by replacing the quadratic σ-potential 1 2 m2 σ σ2 with a quartic potential U(σ) = 1 2 m2 σσ 2 + g 2 3 σ3 + g 3 4 σ4. (2.6) This potential includes the non-linear σ self-interactions with two additional parameters g 2 and g 3. This particular form of the non-linear potential has become standard in applications of RMF models, although additional nonlinear interaction terms, both in the isoscalar and isovector channels, have also been considered [1 13]. The problem with including additional interaction terms, however, is that the empirical data set of bulk and single-particle properties of finite nuclei can only determine six or seven parameters in the general expansion of the effective Lagrangian in powers of the fields and their derivatives [14]. Moreover, implementation of the covariant density functional (see section 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] 1

15 2.2. The Density-Dependent Model and use density-dependent couplings: g σ, g ω, and g ρ are assumed to be vertex functions of Lorentz-scalar bilinear forms of the nucleon operators. In most applications of the density-dependent hadron field theory the meson-nucleon couplings are functions of the vector density ρ v = j µ j µ with j µ = ψγ µ ψ. (2.7) In a relativistic framework the couplings could also depend on the scalar density ψψ. Nevertheless, expanding in ψ ψ is the natural choice, for several reasons. The vector density is connected to the conserved baryon number, unlike the scalar density for which no conservation law exists. The scalar density is a dynamical quantity, to be determined self-consistently by the equations of motion, and expandable in powers of the Fermi momentum. For the meson-exchange models it has been shown that the dependence on vector density alone provides a more direct relation between the self-energies of the density-dependent hadron field theory and the Dirac-Brueckner microscopic self-energies [16]. In the following the vector density dependence of the meson-nucleon couplings will be assumed. The single-nucleon Dirac equation is derived by variation of Lagrangian (2.1) with respect to ψ [ γ µ ( i µ Σ µ ) ( m+σ )] ψ =, (2.8) with the nucleon self-energies defined by the following relations: Σ = g σ σ, Σ µ = g ω ω µ +g ρ τ ρ µ +e (1 τ 3) (2.9) A µ +Σ R µ. 2 The density dependence of the vertex functions g σ, g ω, and g ρ produces the rearrangement contribution Σ R µ to the vector self-energy Σ R µ = j ( µ gω ψγ ν ψω ν + g ρ ψγ ν τψ ρ ν + g ) σ ψψσ, (2.1) ρ v ρ v ρ v ρ v wherej µ = ρ v u µ andu µ isthefour-velocity. Theinclusionoftherearrangement self-energies is essential for the energy-momentum T µν conservation: T µν = g µν L+ i L ( µ φ i ) ν φ i, φ i = ψ,ψ,σ,ω µ,ρ µ,a µ. (2.11) If the rearrangement terms are omitted, the relation µ T µν = 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 the Poisson equation for the electromagnetic potential ( +m 2 σ ) σ = gσ ψψ, ( +m 2 ω ) ω µ = g ω ψγ µ ψ, ( +m 2 ρ ) ρ µ = g ρ ψγ µ τψ, (2.12) A µ = e ψγ µ1 τ 3 ψ. 2 11

16 2. Relativistic Mean-Field Models 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 nuclear ground state. The A nucleons, described by a Slater determinant Φ of singleparticle spinorsψ i, (i = 1,2,...,A), move independently intheclassical meson fields. The densities can be expressed as simple sums of bilinear products of baryon amplitudes. In applications to nuclear matter and finite nuclei, the densities are calculated in the no-sea approximation: the Dirac sea of states with negative energies does not contribute to the densities and currents. For a nucleus with A nucleons A ψγ m ψ = ψ i (r,t)γ m ψ i (r,t), (2.13) i=1 where the summation is performed only over occupied orbits in the Fermi sea of positive energy states and Γ m represents the model vertices. The set of coupled equations (2.8) and (2.12) define the RMF model. RMF models can also be formulated without explicitly including mesonic degrees of freedom. Meson-exchange interactions can be replaced by local four-point interactions between nucleons. It has been shown that the relativistic point-coupling models [18 21] are completely equivalent to the standard meson-exchange approach. In order to describe properties of finite nuclei at a quantitative level, the point-coupling models include also some higher order interaction terms. For instance, six-nucleon vertices ( ψψ) 3, and eight-nucleon vertices ( ψψ) 4 and [( ψγ µ ψ)( ψγ µ ψ)] Stationary Solutions of the Equations of Motion In this dissertation we are interested in the description of ground state properties of nuclei and in order to do it we look for static solutions of the equations of motion 2.8 and In this case the nucleon spinors are eigenvectors of the stationary Dirac equation which gives the single-particle energies ǫ i : [ α( i V (r))+βm (r)+v(r)+σ R (r) ] ψ i (r) = ǫ i ψ i (r). (2.14) The effective mass m is determined by the scalar field m (r) = m+g σ σ(r). (2.15) The potentials V(r) and V (r) denote the time-like and space-like components of the vector self-energy Σ µ, respectively. The term Σ R is the rearrangement self-energy of Eq. (2.1). Due to the charge conservation only the 3rd component of the isovector ρ-meson contributes. The equations of motion can be further simplified if we restrict ourselves to systems with time-reversal invariance, for example even-even nuclei in the ground state. In this case there are no net currents and the spatial components of the vector self-energy vanish. The Dirac equation (2.14) reduces to [ iα +βm (r)+v(r)+σ R (r) ] ψ i (r) = ǫ i ψ i (r). (2.16) 12

17 2.2. The Density-Dependent Model The Klein-Gordon equations and the Poisson equation reduce to ( +m 2 σ ) σ(r) = gσ ρ s (r), ( +m 2 ω ) ω (r) = g ω ρ v (r), ( +m 2 ρ ) ρ 3 (r) = g ρ ρ tv (r), (2.17) A (r) = eρ c (r). The Eqs. (2.17) can be solved analitically using the Green s function method: σ(r) = g σ (ρ v (r))d σ (r,r )ρ s (r )d 3 r, ω (r) = g ω (ρ v (r))d ω (r,r )ρ v (r )d 3 r, (2.18) ρ 3 (r) = g ρ (ρ v (r))d ρ (r,r )ρ tv (r )d 3 r, A (r) = e D c (r,r )ρ c (r )d 3 r. The propagator for the massive meson field reads and for the photon D φ (r,r ) = e m φ r r 4π r r, (2.19) D c (r,r ) = The densities which enter Eqs. (2.17) read 1 4π r r. (2.2) ρ s (r) = ρ v (r) = ρ tv (r) = ρ c (r) = A ψ i (r)βψ i(r), i=1 A ψ i (r)ψ i(r), i=1 A ψ i (r)τ 3ψ i (r), i=1 A i=1 ψ i (r) 1 τ 3 2 ψ i (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 the Dirac equation (2.16) for the spinors ψ i. The results are used to calculate the densities of Eqs. (2.21) which form the sources of Eqs. (2.17). From this set of equations we calculate the meson fields and construct a new set of scalar and vector potentials. We repeat this cycle until the convergence is achieved. 13

18 2. Relativistic Mean-Field Models The total energy of the system is calculated by integrating the T component of the energy-momentum tensor (2.11) over the entire nucleus: E RMF = A i=1 d 3 rψ i( iα +βm ) ψi d 3 r ( g σ ρ s σ +g ω ρ v ω +g ρ ρ tv ρ 3 ). (2.22) When the solution of the self-consistent equations has been obtained, the total binding energy is corrected with the microscopic estimate for the energy of center-of-mass motion E cm = P2 cm 2Am, (2.23) where P cm is the total momentum of a nucleus with A nucleons [22] Rotationally Invariant Systems A nucleus with a proton or neutron closed major shell always exhibits spherical symmetry. The corresponding density and fields depend only on the radial coordinate r. The Dirac spinor is characterized by the single-particle angular momentum quantum numbers j and m, the parity π, and the isospin projection m t = ±1/2 for neutron and proton, respectively: ( ) f(r)φl,j,m (θ,φ,s) ψ(r,s,t) = χ ig(r)φ l,j,m (θ,φ,s) t,mt, (2.24) where χ t,mt is the isospin wave function. The orbital angular momentum l for the large, and l for the small component of the Dirac spinor are determined by the angular momentum j and the parity π: l = j + 1 2, 1 l = j 2 l = j 1 2, 1 l = j + 2 for π = ( 1) j+1 2, for π = ( 1) j 1 2. (2.25) The spin-angular function Φ l,j,m is a two-dimensional spinor with the angular momentum quantum numbers ljm, Φ l,j,m (θ,φ,s) = ( ) 1 lml jm Y m l l (θ, φ)χ1,ms, (2.26) m l,m s where Y m l l (θ,φ) arethe spherical harmonics andχ1 the spinwave functions.,ms 2 The Dirac equation reduces to a set of coupled ordinary differential equations for the radial functions f(r) and g(r) ( [m (r)+v(r)]f(r)+ r κ 1 ) g(r) = ǫf(r), r ( r + κ+1 r 2 m s ) f(r) [m (r) V(r)]g(r) = ǫg(r), 14 2 (2.27)

19 2.2. The Density-Dependent Model where κ = ±(j + 1/2) for j = l 1/2. In spherical coordinates the densities are expressed in terms of the radial functions ρ s (r) = ρ v (r) = ρ tv (r) = ρ c (r) = A (2j i +1) ( f i (r) 2 g i (r) 2), i=1 A (2j i +1) ( f i (r) 2 + g i (r) 2), i=1 A t i (2j i +1) ( f i (r) 2 + g i (r) 2), i=1 A (1 t i )(2j i +1) ( f i (r) 2 + g i (r) 2), i=1 (2.28) where t i = 1 for neutrons, and t i = 1 for protons. The densities define the source terms for the spherical Klein-Gordon equations ( 2 r 2 ) 2 r r +m2 φ φ(r) = s φ (r), (2.29) where m φ are the meson masses for φ = σ,ω,ρ and zero for the photon. The source term reads g σ (ρ v (r))ρ s (r) for the σ-field, g ω (ρ v (r))ρ v (r) for the ω-field, s φ (r) = (2.3) g ρ (ρ v (r))ρ tv (r) for the ρ-field, eρ c (r) for the Coulomb field Covariant Density Functional Theory The modern interpretation of QHD is based on concepts of the Effective Field Theory (EFT) and the Density Functional Theory (DFT). The DFT enables a description of the nuclear many-body problem in terms of a universal energy density functional and the mean field approach to nuclear structure represents an approximate implementation of the Kohn-Sham DFT [23 25], which is successfully used in the treatment of different quantum many-body problems. Energy functionals of the ground-state density are the basis of the DFT approach and for RMF models they are functionals of the ground-state scalar density and baryon currents. In the description of nuclear ground-state properties we employ the self-consistent mean-field implementation of QHD: the RHB model. By employing global effective interactions, adjusted to reproduce empirical properties of symmetric and asymmetric nuclear matter, and bulk properties of simple, spherical and stable nuclei, the current generation of self-consistent mean-field methods has achieved a high level of accuracy in the description of the properties of ground and excited states in arbitrarily heavy 15

20 2. Relativistic Mean-Field Models nuclei, exotic nuclei far from β-stability, and in nuclear systems at the nucleon drip lines. The equations of motion of the RMF model can be derived starting from a density functional. The energy-momentum tensor determines the total energy of the nuclear system and for the static case we have: E RMF [ψ, ψ,σ,ω µ, ρ µ,a µ ] = A i=1 d 3 rψ i( α p+βm ) ψi + 1 d 3 r [ ( σ) 2 +m 2 2 σσ 2] 1 d 3 r [ ( ω) 2 +m 2 ω 2 ω2] 1 d 3 r [ ( ρ) 2 +m 2 ρ 2 ρ2] 1 d 3 r( A) d 3 r [ g σ ρ σ σ +g ω j µ ω µ +g ρ j µ ρ µ +ej cµ A µ]. By using the definition of the relativistic single-nucleon density matrix (2.31) ˆρ(r,r,t) = A ψ i (r,t) ψ i (r,t), (2.32) i=1 the total energy can also be rewritten as a functional of the density matrix ˆρ and of the meson fields E RMF [ˆρ,φ m ] = Tr [( α p+βm )ˆρ ] ± 1 d 3 r [ ( φ m ) 2 [( ] +m 2 φ 2 m] φ2 +Tr Γm φ m )ˆρ. (2.33) The trace operation involves a sum over the Dirac indices and an integral in coordinate space. The index m is used as generic notation for all mesons and the photon. The equations of motion are obtained from the classical variational principle δ t2 t 1 dt{ Φ i t Φ E RMF [ˆρ,φ m ]} =. (2.34) The equation of motion for the density matrix reads i tˆρ = [ ĥ(ˆρ,φ m ), ˆρ ] (2.35) andthesingle-particlehamiltonianĥisobtainedfromthefunctionalderivative of the energy with respect to the single-particle density matrix ˆρ ĥ = δe δˆρ. (2.36) 16

21 2.3. Pairing Correlations The Density-Dependence of the Couplings The density dependence of the meson-nucleon couplings can be parameterized in a phenomenological way by the functionals of Refs. [26, 27]. The coupling of the σ-meson and ω-meson to the nucleon field reads where g i (ρ) = g i (ρ sat )f i (x) for i = σ,ω, (2.37) f i (x) = a i 1+b i (x+d i ) 2 1+c i (x+d i ) 2 (2.38) is a function of x = ρ/ρ sat, and ρ sat denotes the baryon density at saturation in symmetric nuclear matter. The eight real parameters in Eq. (2.38) are not independent. The five constraints f i (1) = 1, f σ (1) = f ω (1), and f i () =, reduce the number of independent parameters to three. Three additional parameters in the isoscalar channel are: g σ (ρ sat ), g ω (ρ sat ), and m σ, the mass of the phenomenological sigma-meson. For the ρ-meson coupling the functional form of the density dependence is suggested by Dirac-Brueckner calculations of asymmetric nuclear matter [28] g ρ (ρ) = g ρ (ρ sat )exp[ a ρ (x 1)]. (2.39) The isovector channel is parameterized by g ρ (ρ sat ) and a ρ. For the masses of the ω and ρ mesons the free values are used: m ω = 783MeV and m ρ = 763 MeV. The eight independent parameters, seven coupling parameters and the mass of the σ-meson, are adjusted to reproduce properties of symmetric and asymmetric nuclear matter, binding energies, charge radii and neutron radii of 12 spherical nuclei. 2.3 Pairing Correlations Non-Relativistic Theory Pairing correlations are essential for a correct description of structure phenomena in spherical open-shell nuclei and in deformed nuclei and they have to be included in the model. For nuclei close to the β-stability line, pairing has been included in the RMF model in the form of the simple BCS approximation [29]. In this approximation for open-shell nuclei pairing is treated phenomenologically using a schematic ansatz in which the constant gap approximation is made and the empirical value of the gap parameter ˆ is given by the five-point formula [3] ˆ (5) (N) = 1 8[ M(N +2) 4M(N +1) +6M(N) 4M(N 1)+M(N 2) ], (2.4) 17

22 2. Relativistic Mean-Field Models where M(N) is the mass for a nucleus with N neutrons. We used this approximation in Ref. [3] where we considered oxygen and calcium isotopic chains and consequently only the neutron gap was needed. However, for nuclei far from stability the BCS model presents only a poor approximation. In particular, in drip-line nuclei the Fermi level is found close to the particle continuum and the BCS model is unable to correctly describe the scattering of nucleonic pairs from bound states to the positive energy continuum allowing the partial occupation of the high levels in the continuum. The HFB theory[2] provides a unified description of ph- and pp-correlations at a mean-field level by using two average potentials: the self-consistent Hartree- Fock field ˆΓ which encloses all the long-range ph-correlations, and a pairing field ˆ which sums up the pp-correlations. The ground state of a nucleus is described by a generalized Slater determinant Φ which represents the vacuum with respect to independent quasiparticles. The quasiparticle operators are defined by the unitary Bogoliubov transformation of the single-nucleon creation and annihilation operators: α + k = l U lk c + l +V lk c l, (2.41) where U lk, V lk are the HFB wave functions. The index l denotes an arbitrary basis, for instance the harmonic oscillator states. In the coordinate space representation l = (r,σ,τ), where σ and τ are the spin and isospin indices, respectively. The HFB wave functions determine the Hermitian single-particle density matrix ˆρ ll = Φ c + l c l Φ = ( V V T) ll (2.42) and the antisymmetric pairing tensor ˆκ ll = Φ c l c l Φ = ( V U T) ll. (2.43) According to Valatin [31] these two densities can be combined into the generalized density matrix ( ) ρ κ R = κ 1 ρ. (2.44) In the case of a nuclear Hamiltonian of the form Ĥ = ǫ l c + l c l + 1 ṽ lm 4 l mc + l c+ m c m c l (2.45) l ll mm with a general NN interaction v, the expectation value Φ Ĥ Φ can be expressed as a function of the Hermitian density matrix ˆρ and the antisymmetric pairing tensor ˆκ. From the variation of this energy functional with respect to ˆρ and ˆκ, the HFB equations are derived (ĥ λ )( ) ( ) ˆ Uk Uk = E ˆ ĥ +λ V k. (2.46) k V k 18

23 2.3. Pairing Correlations The single-nucleon Hamiltonian reads ĥ = ˆǫ+ ˆΓ. (2.47) The two self-consistent potentials ˆΓ and ˆ are obtained as ˆΓ ll = ṽ lm l m ˆρ mm (2.48) mm and ˆ ll = m<m ṽ ll mm ˆκ mm. (2.49) The chemical potential λ is determined by the particle number subsidiary condition in such a way that the expectation value of the particle number operator in the ground state equals the number of nucleons. The column vectors denote the quasiparticle wave functions, and E k are the quasiparticle energies. If the nuclear many-body problem is formulated in the framework of the DFT,onedoesnotstartfromatwo-bodyinteractionasgiveninEq. (2.45),but rather from an energy functional E[R] = E[ˆρ, ˆκ] that depends on the densities ˆρ and ˆκ. The generalized Hamiltonian H is obtained as a functional derivative of the energy with respect to the generalized density H = δe δr = ( ) ĥ ˆ. (2.5) ˆ ĥ The single-particle Hamiltonian ĥ results from the variation of the energy functional with respect to the Hermitian density matrix ˆρ ĥ = δe δˆρ (2.51) and the pairing field is obtained from the variation of the energy functional with respect to the pairing tensor ˆ = δe δˆκ. (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 fully relativistic theory of the pairing force and it can be shown that the RHB equations can be rewritten in the same form as Eqs. (2.46). However, it must be emphasized that pairing in nuclei is a fully non-relativistic phenomenon in which pairing effects are restricted to an energy window of a few MeV around the Fermi level. Moreover, the energy scale of pairing effects is well separated from 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 no experimental evidence for any relativistic effect in the nuclear pairing field ˆ. From these considerations it is therefore justified to use a hybrid model with a non-relativistic pairing interaction in a phenomenological approach based on a DFT, such as the RHB framework used in this dissertation. 19

24 2. Relativistic Mean-Field Models Covariant Density Functional Theory with Pairing The RHB model can be derived within the framework of covariant DFT. When pairing correlations are included, the energy functional depends not only on the density matrix ˆρ and the meson fields φ m, but in addition also on the pairing tensor: E RHB [ˆρ,ˆκ,φ m ] = E RMF [ˆρ,φ m ]+E pair [ˆκ], (2.53) where E RMF [ˆρ,φ m ] isthe RMFfunctional defined ineq. (2.33), andthe pairing energy E pair [ˆκ] is given by E pair [ˆκ] = 1 4 Tr[ˆκ V ppˆκ ]. (2.54) The term V pp denotes a general two-body pairing interaction. The equation of motion for the generalized density matrix reads i t R = [H(R),R] (2.55) and the generalized Hamiltonian H is obtained as a functional derivative of the energy with respect to the generalized density H RHB = δe RHB (ĥd δr = m λ ˆ ) ˆ ĥ D +m+λ. (2.56) The self-consistent mean field ĥd is the Dirac Hamiltonian. In the static case with time-reversal symmetry ĥ D = iα +V (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 pp abcd (r,r )κ cd (r,r ), (2.58) where a,b,c,d denote quantum numbers that specify the Dirac indices of the spinors, 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 openshell nucleus [33]. It is determined by the solutions of the Hartree-Bogoliubov equations )( ) ( ) (ĥd m λ ˆ Uk (r) Uk (r) ˆ ĥ D +m+λ = E V k (r) k. (2.59) V k (r) The chemical potential λ is determined by the particle number subsidiary condition in order that the expectation value of the particle number operator in 2

25 2.3. Pairing Correlations the ground state equals the number of nucleons. The column vectors denote the quasiparticle wave functions, and E k are the quasiparticle energies. The dimension of the RHB matrix equation is two times the dimension of the corresponding Dirac equation. For each eigenvector (U k,v k ) with positive quasiparticle energy E k >, there exists an eigenvector (Vk,U k ) with quasiparticle energy E k. Since the baryon quasiparticle operators satisfy fermion commutationrelations, thelevels E k and E k cannotbeoccupiedsimultaneously. For the solution that corresponds to a ground state of a nucleus with even particle number, one usually chooses the eigenvectors with positive eigenvalues E k. The RHB equations are solved self-consistently, with potentials determined in the mean-field approximation from solutions of static Klein-Gordon equations (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 of baryon amplitudes ρ s (r) = k> V k (r)γ V k (r), ρ v (r) = k> ρ tv (r) = k> V k (r)v k(r), V k (r)τ 3V k (r), (2.6) ρ c (r) = k> V k (r)1 τ 3 V k (r), 2 where the sum over positive-energy states corresponds to the no-sea approximation. The self-consistent solution of the Dirac-Hartree-Bogoliubov integrodifferential equations and Klein-Gordon equations for the meson fields determines the ground state of a nucleus. The eigensolutions of Eq.(2.59) form a set of orthogonal(normalized) single quasiparticle states. The corresponding eigenvalues are the single quasiparticle energies E k. Theself-consistent iterationprocedureisperformedinthebasis of quasiparticle states. In order to obtain a better understanding of the structure of the Hartree-Bogoliubov wave function Φ, the self-consistent quasiparticle eigenspectrum is then transformed into the canonical basis of single-nucleon states. By definition the canonical basis { ϕ µ (r) } diagonalizes the singlenucleon density matrix ˆρ in Eq. (2.42) ˆρ ϕ µ = v 2 µ ϕ µ. (2.61) The transformation to the canonical basis determines the energies and pairing matrix elements ǫ µ = ϕ µ ĥd m ϕ µ and µ = ϕ µ ˆ ϕ µ (2.62) 21