INS-Rep.-909 INSTITUTE FOR NUCLEAR STUDY Jan. 1992 UNIVERSITY OF TOKYO Tanashi, Tokyo 188 JNS-PO9_ Japan JP9202239 Nuclear Collective Dynamics and Chaos Fumihiko SAKATA and Toshio MARUMORit) Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo 188, Japan t) Institute of Physics, University of Tsukuba, Ibaraki 305, Japan To appear in Direction in Chaos, Vol.4 (Series Editor: Bai-Lin Hao), World Scientific Publication Inc. To appear in Direction in Chaos, Vol.4 (Series Editor: Bai-Lin Hao), World Scientific Publication Inc. Nuclear Collective Dynamics and Chaos Fumihiko SAKATA and Toshio MARUMORlt) Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo 188, Japan t) Institute of Physics, University of Tsukuba, Ibaraki 305, Japan ABSTRACT The present status and future problems in both the classical-level theory and full quantum theory of nuclear collective dynamics are discussed by putting special emphasis on their relation to the classical and quantum order-to-chaos transition dynamics, respectively. The non- linear dynamics between the collective and single-particle excitation modes of motion specific for the finite, self-sustained and self-organizing system as the nucleus is discussed within the time-dependent Hartree- Fock (TDHF) theory, the basic equation of which is shown to be formally equivalent to the Hamilton's canonical equations of motion in the classical nonlinear dynamical system. An importance to relate the structure of the TDHF symplectic manifold with an inexhaustible rich structure of the classical phase space in the nonlinear system is stressed. A full quantum theory of nuclear collective dynamics is proposed under a dictation of what has been developed in the classical-level TDHF theory. It is shown that the proposed quantum theory enables us to explore exceeding complexity of the Hilbert space. It is discussed that a resonant denomi- nator known as a source of the extraordinary rich structure of the phase space trajectories, also plays a decisive role in generating a rich structure of the quantum Hilbert space. I CONTENTS 1. INTRODUCTION 1.1. Ordered and Chaotic Motions in Nuclei 1.2. Single-Particle Dynamics 1.3. Order-to-Chaos Transition Mechanism in Nuclei in TDHF Manifold 1.4. Quantum Order-to-Chaos Transition Mechanism in Nuclei 2. SINGLE-PARTICLE DYNAMICS OF NUCLEAR SYSTEM 2.1. Single-particle Algebra 2.2. Hartree-Fock Theory 2.3. Symplectic Structure of the TDHF-Manifold 2.4. Constrained Hartree-Fock Theory 2.5. Adiabatic Time-Dependent Hartree-Fock Theory 3. CLASSICAL-LEVEL THEORY OF LARGE-AMPLITUDE COLLECTIVE MOTION 3.1. Self-Consistent Collective Coordinate Method 3.2. Self-Consistent Constrained Hartree-Fock Method 3.3. Stability and Separability Conditions of Collective Submanifold 3.4. Resonant Interaction Between Relevant and Irrelevant Degrees of Freedom 4. QUANTUM THEORY OF NUCLEAR COLLECTIVE DYNAMICS 4.1. Concepts of Quantum "Integrability" and Quantum "Chaos" 4.2. Basic Idea of Quantum Theory of Nuclear Collective Dynamics 4.3. Basic Equations for Determining the Dynamical Representation 4.4. Dynamical Representation and Extraction of Collective Subspace 4.5. How to Evaluate Stochasticity of Individual Eigenfunctions 4.5. Resonant Quantum Number 5. APPLICATION OF QUANTUM THEORY 5.1. Simple Model Hamiltonian 5.2. Fractional Parentage Plot 5.3. Stochasticity of Ensemble of Eigenstates 5.4. Structure of Eigenfunctions 5.5. Inexhaustible Rich Structure of the Hilbert Space 1.Introduction 1.1. Ordered and Chaotic Motions in Nuclei The objective of this article is to review the recent development towa- rd the microscopic theory of nuclear collective dynamics, which intends to describe the large-amplitude collective excited states situated far from the "ground state"("vacuum") characterized by a certain local stable mean- field, and to investigate dynamical transition mechanism from one local vacuum to another local vacuum in a finite, quantum many-body system. As is generally recognized, it is one of the main subjects in various fields of contemporary physics to explore states of matter located at large- amplitude (large-fluctuation) regions, e.g. far from a vacuum, a ground state specified by a stable mean-field or a thermo-dynamical state of equi- libration, etc. The individual states of matter in small-amplitude regions have been so far extensively studied and are well described by means of the random phase approximation (RPA) or the linear response theory, etc. On the other hand, various experimental investigations on the states of matter in the large-amplitude region are being achieved by using the recent state-of-art experimental facilities in both accelerators and in complex detector systems. Since the states of matter in the large-ampli- tude region usually show a quite different character from those in the small-amplitude region, it is very promising subject to develop a quantum theory of many-body problem which gives dynamical explanations how some characteristic feature of the states of matter in the small-amplitude region persists, transfigures, dissipates or terminates as they evolve toward the large-amplitude region, and how a new characteristic feature specific for the large-amplitude region comes out. In this sense, we are now in a new era of the physical science, which enable us to explore evo- lution of matter [1,2], in addition to the conventional physics of individual states of matter, on the basis of the recent diverse experimental observa- tions on various states of matter. In exploring the above stated new field of quantum mechanics, it is very instructive to learn enormous progress which has been developed during the last few decades in the general the- ory of dynamical systems such as the theory of nonlinear oscillations for Hamiltonian systems of classical mechanics [3,41. A history of nuclear physics may be summarized by stating that there have been incessant struggles for finding a proper position of contradicto- ry concepts; regularity and complexity. Since the nucleus satisfies satura- tion property, each nucleon inside the nucleus is supposed to interact only with its neighboring nucleons through the short-range strong nu- clear force. Consequently, it was hard to expect any ordered regular motion of the nucleons inside the nucleus. Based on this idea of N. Bohr, there introduced the compound nucleus model [5] as well as the liquid drop model [6], which implicitly imply chaotic complex motions of indi- vidual nucleons inside the nucleus so that only collective behaviors of the nucleus as a whole can be treated as regular motions like the surface vibrations, the giant resonances, the nuclear fission process, etc. On the other hand, Mayer and Jensen (7,8] proposed the independent-particle model (the shell model) which is based on the idea that the nucleons inside the nucleus are moving regularly and independently within an average mean-field. The striking success of the shell model proposed two fundamental problems for the study of nuclear structure at that time. The first problem is to understand why the ordered, regular single-p>article motions are justified for such a many-body system of nucleons with strong interaction. The second problem is how to find a proper place for com- plementary modes of motion referring to the single-particle and collec- tive motions. The Bruckner theory [9] struggled against the first funda- mental problem and succeeded in deriving a self-consistent average mean-field characterizing the regular single-particle mode, by properly taking into account both the strong two-body scattering correlations and the Pauli principle among nucleons inside the nucleus. As for the second fundamental problem, the collective model of A. Bohr and Mottelson [10] was proposed by allowing the average mean-field to be time dependent and deformable. The basic idea underlying the collective model is the following [2J. In the shell model, the average mean-field is assumed to be of a spherical "equilibrium" shape from the outset. In the finite quantum system as the nucleus, however, one may expect large fluctuations of the mean-field around the equilibrium (stable) shape, because of the strong self-consis- tency between the single-particle motions and the mean-field. Such fluc- tuations of the mean-field are simply the collective modes of motion, which has been treated by the liquid drop model as the ordered collective motions. One of the outstanding features of the collective model of Bohr and Mottelson is in its explanation of the rotational spectra. When the fluctuations become large, there may occur an equilibrium (stable) defor- - i - mation of the mean-field due to the strong nonlinear self-consistency between the single-particle motions and the mean-field. This stable deformation introduces a spatial orientation of the system and breaks the rotational invariance, so that the ordered collective rotation manifests itself in order to restore the broken symmetry. A further progress to accommodate the two contradictory concepts. regularity and complexity, was made by Strutinski [11] just after the dis- covery of the fission isomer. He considered the nuclear binding energy to be divided into two parts; one given by an "average" level density and the other due to the shell-structure correction around the former, and pro- posed to evaluate the former by means of the liquid drop model and the latter by means of the shell model. It was surprising that the inclusion of shell structure into the liquid drop model (shell correction method) brought about a successful result in reproducing the second minimum (another stable mean-field located far from the first local minimum) of the collective potential. This method turned out to give a reasonable explanation of the fission isomers, asymmetric fission, etc. Namely, the regular nucleonic motion plays a decisive role even in the large-amplitude region where the liquid drop model has been regarded to give a success- ful description. 1.2.