Relativistic continuum Hartree Bogoliubov theory for ground-state properties of exotic nuclei

Shape coexistence in nuclei appears to be unique in the realm of finite many-body quantum systems It differs from the various geometrical arrangements that sometimes occur in a molecule in that in a molecule the various arrangements are of the widely separated atomic nuclei In nuclei the various ''arrangements'' of nucleons involve (sets of) energy eigenstates with different electric quadrupole properties such as moments and transition rates, and different distributions of proton pairs and neutron pairs with respect to their Fermi energies Sometimes two such structures will ''invert'' as a function of the nucleon number, resulting in a sudden and dramatic change in ground-state properties in neighboring isotopes and isotones In the first part of this review the theoretical status of coexistence in nuclei is summarized Two approaches, namely, microscopic shell-model descriptions and mean-field descriptions, are emphasized The second part of this review presents systematic data, for both even- and odd-mass nuclei, selected to illustrate the various ways in which coexistence is observed in nuclei The last part of this review looks to future developments and the issue of the universality of coexistence in nuclei Surprises continue to be discovered With the major advances in reaching to extremes of proton-neutronmore » number, and the anticipated new ''rare isotope beam'' facilities, guidelines for search and discovery are discussed« less

Shape coexistence in atomic nuclei

Relativistic symmetries in nuclei and hadrons

The concept of spontaneous symmetry breaking is applied to the rotating mean field of nuclei. The description is based on the tilted-axis cranking model, which takes into account that the rotational axis can take any orientation with respect to the deformed density distribution. The appearance of rotational bands in nuclei is analyzed, focusing on weakly deformed nuclei at high angular momentum. The quantization of the angular momentum of the valence nucleons leads to new phenomena. Magnetic rotation represents the quantized rotation of the anisotropic current distribution in a near spherical nucleus. The restricted amount of angular momentum of the valence particles causes band termination. The discrete symmetries of the mean-field Hamiltonian provide a classification scheme of rotational bands. New symmetries result from the combination of the spatial symmetries of the density distribution with the vector of the angular momentum. The author discusses in detail which symmetries appear for a reflection-symmetric density distribution and how they show up in the properties of the rotational bands. In particular, the consequences of rotation about a nonprincipal axis and of breaking the chiral symmetry are analyzed. Also discussed are which symmetries and band structures appear for non-reflection-symmetric mean fields. The consequences of breaking the symmetry with respect to gauge and isospin rotations are sketched. Some analogies outside nuclear physics are mentioned. The application of symmetry-restoring methods to states with large angular momentum is reviewed.

Spontaneous symmetry breaking in rotating nuclei

The concept of boson realization (or mapping) of Lie algebras appeared first in nuclear physics in 1962 as the idea of expanding bilinear forms in fermion creation and annihilation operators in Taylor series of boson operators, with the object of converting the study of nuclear vibrational motion into a problem of coupled oscillators. The physical situations of interest are quite diverse, depending, for instance, on whether excitations for fixed- or variable-particle number are being studied, on how total angular momentum is decomposed into orbital and spin parts, and on whether isotopic spin and other intrinsic degrees of freedom enter. As a consequence, all of the semisimple algebras other than the exceptional ones have proved to be of interest at one time or another, and all are studied in this review. Though the salient historical facts are presented in the introduction, in the body of the review the progression is (generally) from the simplest algebras to the more complex ones. With a sufficiently broad view of the physics requirements, the mathematical problem is the realization of an arbitrary representation of a Lie algebra in a subspace of a suitably chosen Hilbert space of bosons (Heisenberg-Weyl algebra). Indeed, if one includes the study of odd nuclei, one is forced to consider the mappings to spaces that are direct-product spaces of bosons and (quasi)fermions. Though all the methods that have been used for these problems are reviewed, emphasis is placed on a relatively new algebraic method that has emerged over the past decade. Many of the classic results are rederived, and some new results are obtained for odd systems. The major application of these ideas is to the derivation, starting from the shell model, of the phenomenological models of nuclear collective motion, in particular, the geometric model of Bohr and Mottelson and the more recently developed interacting boson model of Arima and Iachello. A critical discussion of those applications is interwoven with the theoretical developments on which they are based; many other applications are included, some of practical interest, some simply to illustrate the concepts, and some to suggest new lines of inquiry.

Boson realizations of Lie algebras with applications to nuclear physics

Relating the pseudospin symmetry back to the Dirac equation through the framework of relativistic Hartree-Bogoliubov (RHB) theory, the pseudospin approximation in real nuclei is discussed. From the Dirac equation, the mechanism behind the pseudospin symmetry was studied and the pseudospin symmetry is shown to be connected with the competition between the centrifugal barrier (CB) and the pseudospin orbital potential (PSOP), which is mainly decided by the derivative of the difference between the scalar and vector potentials. With the scalar and vector potentials derived from a self-consistent relativistic Hartree-Bogoliubov calculation, the pseudospin symmetry and its energy dependence in real nuclei is discussed.

Pseudospin symmetry in relativistic mean field theory

Based on the Relativistic continuum Hartree-Bogoliubov (RCHB) theory, the pseudo-spin approximation in exotic nuclei is investigated in Zr and Sn isotopes from the proton drip line to the neutron drip line. The quality of the pseudo-spin approximation is shown to be connected with the competition between the centrifugal barrier (CB) and the pseudo-spin orbital potential ( PSOP ). The PSOP depends on the derivative of the difference between the scalar and vector potentials dV/dr. If dV/dr = 0, the pseudo-spin symmetry is exact. The pseudo-spin symmetry is found to be a good approximation for normal nuclei and to become much better for exotic nuclei with highly diffuse potential, which have dV/dr ∼ 0. The energy splitting of the pseudo-spin partners is smaller for orbitals near the Fermi surface ( even in the continuum ) than the deeply bound orbitals. The lower components of the Dirac wave functions for the pseudo-spin partners are very similar and almost equal in

Pseudospin symmetry in Zr and Sn isotopes from the proton drip line to the neutron drip line

We found the hidden symmetry of the pseudospin approximation which is inherent in the Dirac equation both for the spherical potential and for the deformed potential with axially symmetric shape. For both cases, we found two kinds of conditions, i.e., an exact condition and an approximate condition for the pseudospin symmetry. The former condition is not always satisfied in real nuclei, but the latter condition is realistic and applicable to real nuclei.

Hidden pseudospin symmetry in the Dirac equation

We apply alternative representations of Holstein-Primakoff boson expansion to the particle-rotor model as useful probes to test the stability and the physical contents of the exact solution. The diagonal representations with the total and single-particle angular momenta along the same axis (longitudinal case) and along two perpendicular axes (transverse case) are employed according to the system with rigid or hydrodynamical moments of inertia (MoI). The longitudinal case gives the normal wobbling mode for rigid MoI, but does not give a stable solution for hydrodynamical MoI. The transverse case applied for hydrodynamical MoI describes a few low spin states with reduced alignment of the total spin in the parallel direction to the single-particle spin, but these states differ from the wobbling mode which involves a quantized unit of rotational angular momentum. Employing the angular-momentum-dependent rigid MoI which is derived from the self-consistent Hartree-Fock-Bogoliubov equation together with angular momentum and particle-number constraints, we obtain good fitting not only to the energy-level scheme, but also to the electromagnetic transition rates and the mixing ratio for $^{135}\mathrm{Pr}$.

Stability of the wobbling motion in an odd-mass nucleus and the analysis of Pr 135

An extensive application of the Holstein-Primakoff boson expansion to both single-particle angular momentum and total angular momentum provides an algebraic solution, which gives a good approximation to the exact results for the particle-rotor model with one high-j nucleon coupled to a triaxially deformed core. Two kinds of quantum numbers classify the rotational bands characteristic of the particle-rotor model and lead the selection rules for the interband and intraband transitions. The algebraic solution is compared with the experimental data for the odd mass isotopes $^{163,165,167}\mathrm{Lu}$ as testing grounds.

Kazuko Sugawara-Tanabe

Papers

Pseudospin symmetry in relativistic mean field theory

Pseudospin symmetry in Zr and Sn isotopes from the proton drip line to the neutron drip line

Hidden pseudospin symmetry in the Dirac equation

Stability of the wobbling motion in an odd-mass nucleus and the analysis of Pr 135

Algebraic description of triaxially deformed rotational bands in odd mass nuclei