Relativistic Density Functional for Nuclear Structure

The following sections are included:

• Preface
• Contents

The concept of density functional theory is introduced. The history of nuclear density functional theory is briefly reviewed. The complexity and specialty of the density functional theories in nuclear system in connection with the degrees of freedom, covariance and pairing correlations are discussed in contrast to Coulombic systems. The existing problems and limits of the nuclear density functional theory are discussed. Perspective on the way to derive ab initio functionals is given.

In this chapter, the covariant energy density functional is constructed with both the meson-exchange and the point-coupling pictures. Several widely used functionals with either nonlinear or density-dependent effective interactions are introduced. The applications of covariant density functional theory are demonstrated for infinite nuclear matter and finite nuclei with spherical symmetry, axially symmetric quadrupole deformation, and triaxial quadrupole shapes. Finally, a relativistic description of the nuclear landscape has been discussed, which is not only important for nuclear structure, but also important for nuclear astrophysics, where we are facing the problem of a reliable extrapolation to the very neutron-rich nuclei.

In this chapter, we will present relativistic mean field (RMF) models with pairing treated by the Bardeen–Cooper–Schrieffer (BCS) and the relativistic Hartree–Bogoliubov (RHB) approaches and applications for exotic nuclear phenomena including nuclear halos, the position of the proton drip line and proton radioactivity, the surface diffuseness and its relation to nuclear exotic phenomena, and the effects of pairing correlations on the nuclear size.

The covariant density functional (CDF) theory with the Fock diagrams, the indivisible part of the effective nuclear interaction, is introduced, including both the relativistic Hartree–Fock and its extension — the relativistic Hartree–Fock–Bogoliubov methods. The specific roles played by Fock diagrams, particularly for the new degrees of freedom associated with the π and ρ-tensor fields and the non-local mean fields, are discussed in determining the nuclear energy functional, the shell structure and the evolution, and nuclear isospin excitations. The existing problems and limits of the CDF theory with Fock terms are also discussed, and the perspective on a new algorithm of dealing with the non-local Fock terms is given.

In this chapter, we will present relativistic mean field (RMF) description of heavy and superheavy nuclei (SHN). We will discuss the shell structure and magic numbers in the mass region of SHN, binding energies and α decay Q values, shapes of ground states and potential energy surfaces and fission barriers. We particularly focus on the multidimensionally-constrained covariant density functional theories (CDFT) and the applications of CDFT to the study of exotic nuclear shapes and fission barriers.

Symmetry is a fundamental concept in quantum physics. The quasi-degeneracy between single-particle orbitals (n, l, j = l + 1/2) and (n −1, l + 2, j = l + 3/2) indicates a hidden symmetry in atomic nuclei, the so-called pseudospin symmetry. Since the pseudospin symmetry was recognized as a relativistic symmetry in 1990s, many special features, including the spin symmetry for anti-nucleons, and many new concepts have been introduced. In this Chapter, we will illustrate the schematic picture of spin and pseudospin symmetries, derive the basic formalism, highlight the recent progress from several different aspects, and discuss selected open issues in this topic.

We review the relativistic mean-field approaches to hypernuclear physics. This includes Lambda hypernuclei, anti-Lambda hypernuclei, and multistrangeness hypernuclei. We particularly focus on the properties of both ground state and collective excitations, hyperon binding energies, spinorbit splittings, magnetic moments, a stabilization of drip-line nuclei, and the hyperon impurity effect on nuclear collectivity. We also discuss briefly the influence of hyperons on neutron stars. We conclude that the relativistic mean-field approaches have achieved a great success in the studies of hypernuclear physics.

The rotating nuclei represent one of most interesting subjects for theoretical and experimental studies. They open a new dimension of nuclear landscape, namely, spin direction. Contrary to the majority of nuclear systems, their properties sensitively depend on time-odd mean fields and currents in density functional theories. Moreover, they show a considerable interplay of collective and single-particle degrees of freedom. In this chapter, I discuss the basic features of the description of rotating nuclei in one-dimensional cranking approximation of covariant density functional theory. The successes of this approach to the description of rotating nuclei at low spin in pairing regime and at high spin in unpaired regime in wide range of deformations (from normal to hyperdeformation) are illustrated. I also discuss the recent progress and open questions in our understanding of the role of proton-neutron pairing in rotating nuclei at N ≈ Z, the physics of band termination and other phenomena in rotating nuclei.

Novel rotational excitations like magnetic and antimagnetic rotations observed in the near spherical nuclei and chiral rotation in the triaxial nuclei have attracted a lot of attention. Unlike the conventional rotation, here the rotational axis does not necessarily coincide with any principle axis of nuclear density distribution, and thus a tilted axis cranking is mandatory to describe them self-consistently in the framework of covariant density functional theory. In this chapter, we introduce the related formalism with the point-coupling covariant density functional and discuss its applications for the magnetic and antimagnetic rotation phenomena. The configuration-fixed covariant density functional theory developed to search for nuclear chiral configurations and favorable triaxial deformation parameters is included as well, which leads to the discoveries of the multiple chiral doublets (MχD) in ¹³³Ce and ¹⁰³Rh.

In this chapter, we introduce the collective small amplitude motion in atomic nucleus, i.e., the vibrational modes which are short time responses of the system to external perturbations. Extensive studies over past several decades have provided valuable information on the structure of the nucleus and the forces of cohesion that are responsible for nuclear binding. This chapter introduces in detail the microscopic theory for the description of the small amplitude motion, the random phase approximation model and its realization within covariant density functional theory. Recent and typical applications of such a model in giant resonances, pygmy strengths and spin-isospin resonances are illustrated, among which, the exotic modes in nuclei far from β-stability line and the inclusion of temperature degree of freedom for astrophysical applications are emphasized.

Extensions of the covariant density functional theory by quasiparticle-vibration coupling (QVC) are discussed. The formalism for one-body and two-body propagators in the nuclear medium allows calculations of single-particle energies and spectroscopic factors as well as the response to various types of excitations. In both cases QVC leads to a fragmentation of states, in agreement with experimental observations. Peculiarities of various 2p2h coupling schemes in the nuclear response function are discussed. The theory of the spin-isospin response includes both QVC and pion exchange and provides a framework for calculations of beta-decay, electron capture and charge-exchange reaction characteristics. The presented approaches are illustrated by realistic calculations for medium-mass and heavy nuclei.

Semi-empirical relativistic energy density functionals (EDFs) or effective interactions implicitly comprise short-range correlations related to the repulsive core of the inter-nucleon interaction, and long-range correlations mediated by nuclear resonance modes. To model spectroscopic properties of finite nuclei, the self-consistent mean-field method must be extended to include collective correlations that arise from restoration of broken symmetries and fluctuations in collective coordinates. These correlations are sensitive to shell effects, vary with particle number, and cannot be included in a universal EDF. We review and compare recent advances in “beyond mean-field“ methods based on relativistic EDFs: the angular-momentum and particle-number projected triaxial generator coordinate method, the five-dimensional quadrupole collective Hamiltonian and the axial quadrupole-octupole collective Hamiltonian models. Illustrative applications include low-energy collective excitation spectra and electromagnetic transition rates of nuclei characterised by quadrupole and/or octupole deformations: ²⁴Mg, ⁷⁶Kr, ²⁴⁰Pu and ²²⁴Ra, in comparison with available data.

With the many successes of covariant density functional theory (CDFT) as seen in the previous chapters, there has been growing interest over the last years to examine directly their applicability in astrophysical nucleosynthesis simulations. This chapter thus concentrates on the very recent applications of CDFT in astrophysics nucleosynthesis, ranging from the calculations of nuclear physics inputs — masses and beta-decay half-lives — for rapid-neutron (r-) and rapid-proton (rp-) capture processes, to the nucleosynthesis studies that employed these inputs and to nuclear cosmochronology. The concepts of nucleosynthesis process and formulas on beta-decays are sketched briefly.

In 1939 Oppenheimer and Volkoff demonstrated using Einstein's theory of general relativity that a neutron star supported exclusively by neutron degeneracy pressure will collapse into a black hole if its mass exceeds seven tenths of a solar mass. Seventy five years after such a pioneering prediction the existence of neutron stars with masses as large as two solar masses has been firmly established. This fact alone highlights the critical role that nuclear interactions play in explaining the structure of neutron stars. Indeed, a neutron star is a gold mine for the study of nuclear phenomena that span an enormous range of densities and neutron-proton asymmetries. Physical phenomena over such diverse scales are best described by a formalism based on Relativistic Density Functional Theory. In this contribution I focus on the synergy between theory, experiment, and observation that is needed to elucidate the myriad of exotic states of matter that are believed to exist in a neutron star.

Three variants of the relativistic mean-field model (RMF) and the nonrelativistic Skyrme–Hartree–Fock model (SHF) are compared. Overall quality, predictive power, and correlations between observables are addressed using statistical analysis on the basis of least squares fits. Appropriate density dependence is a crucial ingredient for good performance of RMF. However, SHF shows still more flexibility particularly in the isovector channel.

The following sections is included:

• Author Index