International Journal of Modern Physics E

Effective interactions and transition operators have a long history in nuclear physics. From the early works of Brueckner and Bethe to the present time, microscopic approaches based on realistic nucleon–nucleon interactions, fit to phase shifts, have produced many useful results and insights. Recently, an ab initio approach has been developed that successfully weds a no-core finite basis state treatment, conserving center of mass motion, with an Hermitian effective Hamiltonian derived from realistic two-nucleon and tri-nucleon interactions. We present this approach with a sample of recent results.

I discuss effective field theories for bound states and narrow resonances near two-body thresholds. I illustrate the method in the case of nucleon-alpha scattering.

A general definition of the model-space effective interaction is given. The energy-independent effective Hamiltonians derived in a time-independent way are classified systematically.

The low-lying dipole and quadrupole states in neutron rich nuclei are studied within the fully self-consistent relativistic quasiparticle random-phase approximation (RQRPA), formulated in the canonical basis of the Relativistic Hartree–Bogoliubov model (RHB), which is extended to include the density dependent interactions. In heavier nuclei, the low-lying E1 excited state is identified as a pygmy dipole resonance (PDR), i.e. as a collective mode of excess neutrons oscillating against a proton–neutron core. Isotopic dependence of the PDR is characterized by a crossing between the PDR and one-neutron separation energies. Already at moderate proton–neutron asymmetry the PDR peak is calculated above the neutron emission threshold, indicating important implications for the observation of the PDR in (γ, γ′) scattering, and on the theoretical predictions of the radiative neutron capture rates in neutron-rich nuclei. In addition, a novel method is suggested for determining the neutron skin of nuclei, based on measurement of excitation energies of the Gamow–Teller resonance relative to the isobaric analog state.

We consider the counter-term describing isoscalar axial two-body currents in the nucleon–nucleon interaction, L_1A, in the effective field theory approach. We determine this quantity using solar neutrino data. We investigate the variation of L_1A when different sets of data are used.

We introduce a new class of exactly solvable boson pairing models using the technique of Richardson and Gaudin. Analytical expressions for all energy eigenvalues and the first few energy eigenstates are given. In addition, another solution to Gaudin's equation is also mentioned. A relation with the Calogero–Sutherland model is suggested.

While the no-core shell model is a state-of-the-art microscopic approach to low-energy nuclear structure, its intense computational requirements lead us to consider time-honored approximations such as the Hartree–Fock (HF) approximation and the random phase approximation (RPA). We review RPA and point out some common misunderstandings, then apply HF + RPA to the no-core shell model. Here the main issue is appropriate treatment of contamination by spurious center-of-mass motion.

Properties of shell model Hamiltonians, and in particular the tri-diagonal matrices obtained with the Lanczos procedure are exploited to determine the density of states and to estimate the ground-state energies. It will be shown that with a few parameters that are adjusted to the moments up to fourth order of the Hamiltonian, one can model the resultant tri-diagonal matrices. An estimate of the ground state energy can quickly be obtained from the modeled matrices, and the density of states for the full matrix can quickly be obtained using the WKB approximation. The accuracy of the ground state is dependent on high moments, six and greater.

A mean-field Hamiltonian with an extended pairing interaction that includes multi-pair excitations is proposed for describing strongly-deformed nuclei. Solutions are easily obtained. The results show that even though the theory includes many-body interaction terms, the one- and two-body parts continue to dominate the dynamics for small values of the pairing strength. As the strength of the pairing interaction grows, however, the higher many-body interaction terms grow in importance. A numerical study of binding energies for the ^154-181Yb isotopes demonstrates the applicability of the theory.

There has been significant progress in the ab initio approaches to the structure of light nuclei. One such method is the ab initio no-core shell model (NCSM). Starting from the realistic two- and three-nucleon interactions, this method can predict the low-lying levels in p-shell nuclei. It is a challenging task to extend the ab initio methods to describe nuclear reactions. In this contribution, we present a brief overview of the NCSM with examples of recent applications as well as the first steps taken toward nuclear reaction applications.

In the no-core shell model formalism we compute effective one- and two-body operators, using the Lee–Suzuki procedure within the two-body cluster approximation. We evaluate the validity of the latter through calculations in reduced model spaces. In particular, we test the results for the two-body system and find that indeed the effective operators in the reduced space reproduce the expectation values or transition strengths computed in the full space. On the other hand, the renormalization for operators in the case of ⁶Li is very weak, suggesting the need for higher-body clusters in computing the effective interaction.

Pseudospin symmetry is an approximate relativistic symmetry of the nucleus as demonstrated by experimental data. This symmetry follows from the fact that the vector and scalar potentials of nucleons moving in a relativistic mean field are approximately equal in magnitude and opposite in sign. QCD sum rules in nuclear matter support this conclusion. Such an observation suggests a fundamental reason for pseudospin symmetry. We review the status of pseudospin symmetry conservation in the nucleon–nucleon interaction.

Relativistic symmetries of the Dirac Hamiltonian with a mixture of spherically symmetric Lorentz scalar and vector potentials are examined from the point of view of supersymmetric quantum mechanics. The cases considered include the Coulomb, pseudospin and spin limits relevant, respectively, to atoms, nuclei and hadrons.

The role of a nearly decoupled pseudospin is investigated. A model of supersymmetry with three kinds of bosons is discussed and compared to data for ¹⁹⁴Pt and ¹⁹⁵Pt. Supersymmetry applied to identical bands is discussed.

Motivated by the success of kinetic theory in the description of observables in intermediate and high energy heavy-ion collisions, we apply kinetic theory to the physics of supernova explosions. The algorithmic implementation for the high-density phase of the iron core collapse is discussed.

Applications of the concept of symmetry in nuclei are reviewed.

A systematic study of nuclei with Z, N~64, 88 and 88, 134 in terms of the spdf-Interacting Boson Approximation model is presented. Various properties are reproduced with a very simple Hamiltonian which predicts an onset of strong octupole correlations at medium spin.

The history and importance of superdeformation in nuclei is briefly discussed. A simple two-level model is then employed to obtain an elegant expression for the branching ratio for the decay via the E1 process in the normal-deformed band of superdeformed nuclei. From this expression, the spreading width Γ^↓ for superdeformed decay is found to be determined completely by experimentally known quantities. The accuracy of the two-level approximation is verified by considering the effects of other normal-deformed states. Furthermore, by using a statistical model of the energy levels in the normal-deformed well, we can obtain a probabilistic expression for the tunneling matrix element V.