Computational and Group-Theoretical Methods in Nuclear Physics

Conference Photograph

Conference Organization

Preface

Introduction

Contents

No abstract received.

Science is not done in a vacuum – occasions like this give one an opportunity to step back and take an inventory of those individuals who have been part of one’s scientific journey: ‘Masters’ from whom one has learned, ‘Students’ of whom one must always be one; ‘Nexgens’ to whom one looks to carry major campaigns forward. Though circuitous my path at times did seem, I would claim to have enjoyed the best of all – Masters who led by example, Students who grew beyond their teacher, and Nexgens who have stayed the course!

The pseudo-SU(3) model has been extensively used to study normal parity bands in even-even and odd-mass heavy deformed nuclei. The use of a realistic Hamiltonian that mixes many SU(3) irreps has allowed for a successful description of energy spectra and electromagnetic transition strengths. While this model is powerful, there are situations in which the intruder states must be taken into account explicitly. The quasi-SU(3) symmetry is expected to complement the model, allowing for a description of nucleons occupying normal and intruder parity orbitals using a unified formalism.

Partial dynamical symmetry (PDS) extends and complements the concepts of exact and dynamical symmetry. It allows one to remove undesired constraints from an algebraic theory, while preserving some of the useful aspects of a dynamical symmetry, and to study the effects of symmetry breaking in a controlled manner. An example of a PDS in an interacting fermion system is presented. The associated PDS Hamiltonians are closely related with a realistic quadrupole-quadrupole interaction and provide new insights into this important interaction.

We make a systematic study of correlations in the chart of calculated masses of Moller and Nix and we find that it is possible to reduce the rms by 20%. The correlations can have important consequences in the errors as signaling the presence of chaos, as was recently proposed.

The phenomenom of emerging regular spectral features from random interactions is addressed in the context of the interacting boson model. A mean-field analysis links different regions of the parameter space with definite geometric shapes. The results provide a clear and transparent interpretation of the high degree of order that has been observed before in numerical studies.

The evidence that pseudospin symmetry is a relativistic symmetry is reviewed. Search for pseudospin symmetry beyond the mean field approximation is motivated.

The nuclear shell model allows several analytical solutions which broadly can be divided in two classes: pairing models and rotational models. The latter are based on Elliott’s SU(3) symmetry which presupposes LS coupling. The search for solvable rotational models that can accommodate a departure from LS coupling has been an important theme of nuclear structure in which pseudo-spin symmetry has played a pivotal role. In this contribution the arguments that justify a departure from SU(4) symmetry and a move towards a pseudo-LS or pseudo-SU(4) scheme are reviewed.

The development of collectivity and shape transitions in nuclei is discussed, from the perspective of structural evolution, both with nucleon number and spin.

Richardson’s exact solution of the pairing model can be generalized to three families of exactly solvable models for interacting bosons and fermions. We focus on the rational family and show how to map these models onto classical two-dimensional electrostatic problems. In the case of fermions, we use the electrostatic mapping of the pairing model to provide a new perspective on the superconducting phase transition in finite nuclei. In the case of bosons, we show that this class of models displays a second-order phase transition to a fragmented state in which only the two lowest boson states are macroscopically occupied and suggest that this provides a new mechanism for sd dominance in interacting boson models of nuclei.

Exact solutions for low-lying J = 0 states of 2k nucleons interacting with one another through an isovector charge-independent pairing interaction are derived by using the Bethe ansatz method. The results show that a set of highly nonlinear equations must be solved for k ≥ 3.

We describe collective vibrations of nuclei within the framework of relativistic quasi-particle random phase approximation with non-linear meson couplings. Pairing correlations are described by a finite range effective particle-particle interaction of Gogny type. The quasi-particle random phase equations are solved in the canonical basis and collective strength distributions are discussed for the recently discovered low-lying collective E1-modes and for isoscalar dipole excitations.

Precise measurements of the intensities for superallowed Fermi 0⁺ → 0⁺ β^- decays have provided a powerful test of the CVC hypothesis at the level of 3 × 10^-4 and have led to a disagreement with unitarity for the CKM matrix at the 98% confidence level. It is essential to address possible trivial explanations for the apparent non-unitarity such as uncertainties in the calculated isospin symmetry-breaking corrections. We have carefully studied the ⁷⁴Rb to ⁷⁴Kr β^- decay in order to measure the total non-analog β-decay branching and especially the β decay branching to the 0⁺ state at 509 keV in ⁷⁴Kr. We observed experimentally a non-analog branch of 336(20) × 10^-5 and deduce with the aid of a recent shell model calculation an analog branch of 99.5(1)%. The branching to the 509 keV level is < 5 × 10^-4, which confirms a recent theoretical estimate of the isospin mixing in this level and its analog in ⁷⁴Rb. We also show that high-precision, complete spectroscopy must be performed to obtain meaningful β decay branching ratios.

Two important pieces of nuclear structure are many-body collective deformations and single-particle spin-orbit splitting. The former can be well-described microscopically by simple SU(3) irreps, but the latter mixes SU(3) irreps, which presents a challenge for large-scale, ab initio calculations on fast modern computers. Nonetheless, SU(3)-like phenomenology remains even in the face of strong mixing. The robustness of band structure is reminiscent of robust, pairing collectivity that arises from random two-body interactions.

We report on large-scale applications of the ab initio, no-core shell model with the primary goal of achieving an accurate description of nuclear structure from the fundamental inter-nucleon interactions. In particular, we show that realistic two-nucleon interactions are inadequate to describe the low-lying structure of ¹⁰B, and that realistic three-nucleon interactions are essential.

Nuclear structure requires solutions to the complicated quantum many-body problem. I discuss an initial implementation of the coupled-cluster method for nuclear structure calculations and apply our method to a preliminary study of ⁴He.

This presentation explains why models with a dynamical symmetry often work extraordinarily well even in the presence of large symmetry breaking interactions. A model may be a caricature of a more realistic system with a “quasi-dynamical” symmetry. The existence of quasi-dynamical symmetry in physical systems and its significance for understanding collective dynamics in complex nuclei is explained in terms of the precise mathematical concept of an “embedded representation”. Examples are given which exhibit quasi-dynamical symmetry to a remarkably high degree. Understanding this unusual symmetry and why it occurs, is important for recognizing why dynamical symmetries appear to be much more prevalent than they would otherwise have any right to be and for interpreting the implications of a model’s successes. We indicate when quasi-dynamical symmetry is expected to apply and present a challenge as to how best to make use of this potentially powerful algebraic structure.

Algebraic approach to the integrability condition called shape invariance is briefly reviewed. Various applications of shape-invariance available in the literature are listed. A class of shape-invariant bound-state problems which represent two-level systems are examined. These generalize the Jaynes-Cummings Hamiltonian. Coherent states associated with shape-invariant systems are discussed. For the case of quantum harmonic oscillator the decomposition of identity for these coherent states is given. This decomposition of identity utilizes Ramanujan’s integral extension of the beta function.

We investigate 1-D inelastic collision and resonance states in a strong localized potential by using nonlinear wave functions (solitons) in order to prevent wave function spreading. For small excitations energy the nonlinear terms can be neglected inside the potential region and we obtained exact solutions for the entire real axis: solitons in empty space region and Schrödinger states in the interaction region.

We discuss the possibility of producing a new kind of nuclear system by putting a few antibaryons inside ordinary nuclei. The structure of such systems is calculated within the relativistic mean-field model assuming that the nucleon and antinucleon potentials are related by the G–parity transformation. The presence of antinucleons leads to decreasing vector potential and increasing scalar potential for the nucleons. As a result, a strongly bound system of high density is formed. Due to the significant reduction of the available phase space the annihilation probability might be strongly suppressed in such systems.

A simple model for QCD is presented. It is based on a Lipkin model, consisting of two levels for the quarks coupled to a boson level, representing pairs of gluons with color and spin zero. The basic ingredients are pairs of quark-antiquark coupled to combinations of flavor and spin, in addition to the gluon pairs. The interaction part of the Hamiltonian is a particle non-conserving interaction and commutes with spin, flavor, parity and charge conjugation. The four parameters of the Hamiltonian are adjusted to the meson spectrum at low energy, corrected for flavor mixing and Gell’man-Okubo terms of the two lowest meson nonets. The states exhibit mixture of quarks, antiquarks and gluons. In the second part of the contribution, the partition function is constructed and several observables calculated, like particle ratios and absolute production rates. The model exhibits a hint for a possible transition at temperature 0.170GeV from the Quark-Gluon Plasma to the hadron gas.

The pick-up reaction has been recently the object of a detailed experimental study¹. Its main interest is to test if the nuclear dynamical supersymmetry (SUSY) is a valid model to describe nuclei in this region. We compare the spectroscopic strengths obtained with different forms of the transfer operator and show that by introducing some parameters to weigh the different orders of their expansion we obtain better agreement with the experimental data.

Using relations between wave functions obtained in the framework of the relativistic mean field theory, we investigate the effects of pseudospin and spin symmetry breaking on the single nucleon wave functions in spherical nuclei. In our analysis, we apply both relativistic and non-relativistic self-consistent models as well as the harmonic oscillator model.

The solution of the E(5) Hamiltonian for finite well depth is described, and the effects of finite depth on observables are discussed.

The observation of neutrinoless double beta decay would provide evidence about neutrino masses and their Dirac or Majorana character. We study the neutrinoless double electron capture accompanied by photon emission for the nucleus of ¹⁵⁶Dy using the pseudo SU(3) framework.

The algebraic mean field theory of the symplectic algebra sp(3,R) is studied for solutions to nuclear collective motion. This theory is a general method that can be applied to any dynamical symmetry model. Predicted energy levels agree most closely with experiment when the body is neither rigid nor irrotational. The theory is applied to nuclei with the su(3) label μ = 0.

The level scheme of well deformed nucleus ¹⁵⁹Gd has been experimentally established by means of radiative neutron capture and single neutron transfer reactions. Previous identification of states is confirmed and expanded. Levels with vibrational components are investigated. The structure of this nucleus is described within the quasiparticle-phonon model.

The spectrum and wave functions of ⁴⁴Ti are studied in oblique-basis calculations using spherical and SU(3) shell-model states. Although the results for ⁴⁴Ti are not as good as those previously reported for ²⁴Mg, due primarily to the strong spinorbit interaction that generates significant splitting of the single-particle energies that breaks the SU(3) symmetry, a more careful quantitative analysis shows that the oblique-basis concept is still effective. In particular, a model space that includes a few SU(3) irreducible representations, namely, the leading irrep (12,0) and next to the leading irrep (10,1) including its spin S = 0 and 1 states, plus spherical shell-model configurations (SSMC) that have at least two valence nucleons confined to the f_7/2 orbit – the SM(2) states, provide results that are compatible with SSMC with at least one valence nucleon confined to the f_7/2 orbit – the SM(3) states.

We compute accurate approximations to the low-lying states of ⁴⁴Ti by ground-state factorization. Energies converge exponentially fast as the number of retained factors is increased, and quantum numbers are reproduced accurately.

The behavior of the and bands in a sequence of deformed even-even rare earth nuclei, organized into F-spin multiplets of the Sp(4,R) scheme, is explored. The complex nature of these states and the collective bands built on them is interpreted in terms of the microscopic proton-neutron pseudo-SU(3) shell model.

Properties of pairing and higher-order interactions are investigated with a Sp_(q) (4) model. The q-deformation introduces an order parameter for a ‘phase transition’.

Normal parity bands in ¹⁵⁷Gd, ¹⁶³Dy and ¹⁶⁹Tm are studied using the pseudo SU(3) shell model. Energies and B(E2) transition strengths between states belonging to low-lying, same-parity rotational bands in each nuclei are considered. The pseudo SU(3) basis includes states with pseudo spin 0 and 1, and and , for even and odd nucleon numbers, respectively. States with pseudo-spin 1 and must be included for a proper description of some excited bands and M1 transition strengths.

We apply a recently proposed matrix-coherent state approach for configuration mixing Hamiltonians in the context of the IBM ¹, to describe the evolving geometry of the neutron deficient Pb isotopes. The potential energy surface of ¹⁸⁶Pb has three well developed minima, which correspond to spherical, oblate and prolate shapes, in close agreement with recent experimental measurements and deformation dependent mean-field calculations ². We find that the mixing between the three configurations ³ is probably overestimated in the fit, since the oblate minimum is blocked in the full calculation. A slight modification of the mixing parameters, however, gives rise again to a remarkably similar shape for the potential surface of this neutron-deficient isoptope, when compared with the mean-field calculations. Moving away from mid-shell, towards the heavier Pb isotopes, the deformed minima tend to disappear immediately. Our analysis suggests that the method may be a reliable tool for the study of geometrical aspects of shape coexistence phenomena in nuclei.

Photos from The Banquet

List of Participants