Collective Motion and Phase Transitions in Nuclear Systems

PREFACE.

Organising Committee and List of Participants.

CONTENTS.

The dynamics of nuclear collective motion is investigated in the case of reflection-asymmetric deformation, with the purpose of describing the critical point of phase transitions between different shapes. The model is based on the Bohr hydrodynamical approach and employs a new parametrization of the octupole and quadrupole degrees of freedom, valid for nuclei close to the axial symmetry. Three particular cases are discussed in some detail: octupole critical point in nuclei which already possess a permanent quadrupole deformation (²²⁶Th); octupole vibrations in nuclei at the X(5) critical point of quadrupole mode (¹⁵⁰Nd, ¹⁵²Sm); and critical point in both quadrupole and octupole modes (²²⁴Ra, ²²⁴Th). Results are compared with experimental data.

A unified description of four positive and four negative parity rotational bands for nuclei with and without octupole deformation is presented within an extended version of the coherent state model. Signatures for octupole deformation in the ground as well as in the excited bands are pointed out. Specific features of octupole deformed nuclei related with the electric and magnetic transition probabilities are presented. Comments concerning the project of enlarging the quadrupole and octupole boson system by coupling a set of particles leading to a final system with a chiral symmetry are added. To save the space, only part of the applications performed in the last decade were reviewed.

A method for finding reflection asymmetric or symmetric saddle-point nuclear shapes with axial symmetry is presented. The shape is a solution of an Euler-Lagrange equation, derived by solving the variational problem of minimization of the deformation energy. By introducing phenomenological shell corrections one obtains minima of deformation energy at the saddle-point for binary fission of ^230,234,238U nuclei at a non-zero mass asymmetry. Ternary, quaternary, and multicluster fission is also discussed.

The symmetric three center shell model has been constructed in order to account for the transition of the level scheme of one parent nucleus towards three equal spherical fragments. The model calculate the energy levels using a three center oscillator potential for partially overlapped nuclei. The oscillator levels and the spin-orbit and l² terms are dependent on the distance between the centers of the side fragments, thus on the elongation of the ternary configuration. At the end the independent level schemes of three separated spherical nuclei are obtained. Calculations are applyied to the splitting of ¹⁴⁴Nd into three ⁴⁸Ca spherical systems.

Ion traps have become an essential tools for nuclear studies. They have been applied widely for measurements of atomic masses with unprecedented precision, but recently they have also been developed for spectroscopic studies, where they can provide clear benefits compared to conventional techniques. In these lecture notes, recent results from Penning trap projects for radioactive ions are discussed with special emphasis on JYFLTRAP project in the Department of Physics, University of Jyväskylä.

The parameter independent (up to overall scale factors) predictions of the X(5)-β², X(5)-β⁴, and X(3) models, which are variants of the X(5) critical point symmetry developed within the framework of the geometric collective model, are compared to two-parameter calculations in the framework of the interacting boson approximation (IBA) model. The results show that these geometric models coincide with IBA parameters consistent with the phase/shape transition region of the IBA for boson numbers of physical interest (close to 10). ¹⁸⁶Pt and ¹⁷²Os are identified as good examples of X(3), while ¹⁴⁶Ce, ¹⁷⁴Os and ¹⁵⁸Er, ¹⁷⁶Os are identified as good examples of X(5)-β² and X(5)-β⁴ behavior respectively.

A γ-rigid veraion (with γ = 0) of the X(5) critical point symmetry is constructed. The model, to be called X(3) since it is proved to contain three degrees of freedom, utilizes an infinite well potential, is based on exact separation of variables, and leads to parameter free (up to overall scale factors) predictions for spectra and B(E2) transition rates, which are in good agreement with existing experimental data for ¹⁷²Os and ¹⁸⁶Pt. An unexpected similarity of the β₁-bands of the X(5) nuclei ¹⁵⁰Nd, ¹⁵²Sm, ¹⁵⁴Gd, and ¹⁵⁶Dy to the X(3) predictions is observed.

A survey of chaotic dynamics in atomic nuclei is presented, using on the one hand standard statistics of quantum chaos studies, and on the other a new approach based on time series analysis methods. First we emphasize the energy and isospin dependence of nuclear chaoticity, based on shell-model energy spectra fluctuations in Ca, Sc and Ti isotopes, which are analyzed using standard statistics such as the nearest level spacing distribution and the Dyson-Mehta Δ₃ statistic. For all the Ca isotopes, in the ground state region the energy level fluctuations show strong deviations from GOE predictions. When one or two neutrons are replaced by protons, Sc is closer to GOE and Ti is even more chaotic. Thus we find a clear isospin dependence in the degree of nuclear chaoticity. Afterward, we discuss chaos in nuclei using a new approach based on the analogy between the sequence of energy levels and a discrete time series. Considering the energy spectrum fluctuations of quantum systems as a discrete time series, we suggest the following conjecture: The energy spectra of chaotic quantum systems are characterized by 1/f noise. Moreover, we show that the spectra of integrable quantum systems exhibit 1/f² noise. Although an exact proof of this conjecture is not available yet, we use random matrix theory to derive theoretical expressions that reproduce to a good approximation the power laws of type 1/f and 1/f² characteristic of chaotic and integrable systems, respectively. We note that 1/f noise is a very ubiquitous property, since many complex systems in nature and in human society exhibit the same kind of fluctuations as chaotic quantum systems.

The lecture describes a number of indirect methods in nuclear astrophysics using radioactive beams: Coulomb dissociation, transfer reactions (the ANC method), breakup of loosely bound nuclei at intermediate energies, other spectroscopic measurements. The examples chosen are be drawn from the experiments the author was involved together with his group at Texas A&M University. One example discussed in particular is that of the reactions used to determine the S₁₇ astrophysical factor for the ⁷Be(p, γ)⁸B reaction, crucial for the understanding the solar neutrino problem. We discuss also the case of the proton drip line nucleus ²³A1. From its study we extract data to determine the stellar reaction rates for ²²Mg(p,γ)²³Al and ²²Na(p,γ)²³Mg. Both breakup and beta-decay methods were used in this study.

We outline two methods for facing the nuclear eigenvalue problem, one suitable for low-lying spectroscopy and the other for describing collective modes. The first is an an important sampling algorithm which generates a subset of exact eigensolutions of the nuclear shell model Hamiltonian within a truncated space. The other is an equation of motion method which generates iteratively a microscopic multiphonon basis and is therefore especially suitable for the investigation of collective modes. Numerical tests of both methods are presented.

First in a single-j-shell calculation (j = f_7/2) we discuss various symmetries, e.g., two to one in ⁴⁴Ti vs. ⁴³Ti. We note that the wave function amplitudes for T(higher) states are coefficients of fractional parentage, and that orthogonality of T(higher) and T(lower) states leads to useful results. Then we consider what happens if T = 0 two-body matrix elements are set equal to zero. We find a partial dynamical symmetry with several interesting degeneracies. It is noted that some formulae developed for identical particles also apply to different (companion) problems involving mixed systems of protons and neutrons. In the g_9/2 shell, where one can have for the first time seniority violation for identical particles, we find some interesting yet unproven results. Finally, we discuss shell-model calculations for the magnetic moments of different nuclei, such as ⁵²Ti, even-even Ca isotopes, and N = Z nuclei.

The present paper is comprised of two parts. First, we give a brief survey of the theoretical framework for microscopic shell-model calculations starting from the free nucleon-nucleon potential. In this context, we discuss the use of the low-momentum nucleon-nucleon (NN) interaction V_{low – k} in the derivation of the shell-model effective interaction and emphasize its practical value as an alternative to the Brueckner G-matrix method. Then, we present some results of recent studies of nuclei near doubly magic ¹³²Sn, which have been obtained starting from the CD-Bonn potential renormalized by use of the V_{low – k} approach. The comparison with experiment shows how shell-model effective interactions derived from modern NN potentials are able to provide an accurate description of nuclear structure properties.

Nuclear superfludity in exotic nuclei close to the drip lines and in the inner crust matter of neutron stars have common features which can be treated with the same theoretical tools. In the first part of my lecture I discuss how two such tools, namely the HFB approach and the linear response theory can be used to describe the pairing correlations in weakly bound nuclei, in which the unbound part of the energy spectrum becomes important. Then, using the same models, I shall discuss how the nuclear superfluidity can affect the thermal properties of the inner crust of neutron stars.

We present the microscopic description of some exotic nuclear structure phenomena and decays identified experimentally at low, intermediate and high spins in the A≃ 70 mass region within the complex VAMPIR approaches. Special emphasis is put on the very large many-nucleon model spaces required in order to determine the structure of the wave functions.

Thermal fluctuations of quasiparticle number are included making use of the secondary Bogolyubov's transformation, which turns quasiparticles operators into modified-quasiparticle ones. This restores the unitarity relation for the generalized single-particle density operator, which is violated within the Hartree-Fock-Bogolyubov (HFB) theory at finite temperature. The resulting theory is called the modified HFB (MHFB) theory, whose limit of a constant pairing interaction yields the modified BCS (MBCS) theory. Within the MBCS theory, the pairing gap never collapses at finite temperature T as it does within the BCS theory, but decreases monotonously with increasing T. It is demonstrated that this non-vanishing thermal pairing is the reason why the width of the giant dipole resonance (GDR) does not increase with T up to T ~ 1 MeV. At higher T, when the thermal pairing is small, the GDR width starts to increase with T. The calculations within the phonon-damping model yield the results in good agreement with the most recent experimental systematic for the GDR width as a function of T. A similar effect, which causes a small GDR width at low T, is also seen after thermal pairing is included in the thermal fluctuation model.

Anharmonic features of the low-lying collective states in cadmium, ruthenium and molybdenum isotopes have been investigated systematically by using the Microscopic Anharmonic Vibrator Approach (MAVA). MAVA is based on a large single-particle valence space and a realistic nuclear Hamiltonian which is used to generate the one-phonon states by the use of the Quasiparticle Random-Phase Approximation (QRPA). The same Hamiltonian is also used to introduce anharmonicities into the description of the low-lying excited states leading to dynamical splitting of the energies of the two-phonon vibrational states. Comparison of the calculated energies and B(E2) values with the available data points to mixing between anharmonic vibrations and deformed intruder degrees of freedom in the case of cadmium isotopes, a shape transition in the case of ruthenium isotopes and the discussed molybdenum isotopes are suggested to be closer to anharmonic vibrators than deformed rotors.

Relativistic Hartree-Bogoliubov (RHB) theory is a powerful tool for the description of the properties of exotic systems. It is described in terms of a covariant density functional and with the use of only a limited number of phenomenological parameters, the theory is able to provide a unified description of nuclear structure properties throughout the periodic table. Here, the covariant density functional theory in nuclei, its various extensions and several applications for nuclei away from stability line, are presented.

The conceptual framework of self-consistent mean-field models is presented. After discussing the parameter adjustment of these models, their overall predictive power is investigated, and, as a special application, predictions for fission barriers of superheavy nuclei are studied.

Heavy Ion Collisions, HIC, represent a unique tool to probe the in-medium nuclear interaction in regions away from saturation and at high nucleon momenta. In this report we present a selection of reaction observables particularly sensitive to the isovector part of the interaction, i.e. to the symmetry term of the nuclear Equation of State, EoS. At low energies the behavior of the symmetry energy around saturation influences dissipation and fragment production mechanisms. A very good tracer appears to be the isospin transport during the reaction dynamics. Predictions are shown for deep-inelastic and fragmentation collisions induced by neutron rich projectiles. Differential flow measurements will also shed lights on the controversial neutron-proton effective mass splitting in asymmetric matter. The high density symmetry term can be derived from isospin effects on heavy ion reactions at higher energies, a few AGeV range, that can even allow a direct study of the covariant structure of the isovector interaction in the hadron medium. We work within a relativistic transport frame, beyond a cascade picture, consistently derived from effective Lagrangians, where isospin effects are accounted for in the mean field and collision terms. Rather sensitive observables are proposed from collective flows and from pion-kaon production (π⁻/π⁺, K⁰/K⁺ yields). For the latter point relevant non-equilibrium effects are stressed. The possibility of the transition to a mixed hadron-quark phase, at high baryon and isospin density, is finally suggested. Some signatures could come from an expected neutron trapping effect.

Various aspects of the nuclear multifragmentation phenomenon are discussed from the point of view of the Microcanonical Multifragmentation Model (MMM) model. This model provides results in very good agreement with experimental data and predicts a first order phase transition in nuclear matter. An analysis performed with MMM aiming to identify a statistically equilibrated stage in the dynamical path provided by a transport code (Stochastic Mean Field) is described. As a result, a distinct statistically equilibrated stage corresponding to the time of 140 fm/c was identified.

The role of spinodal instabilities in nuclear fragmentation is investigated. A thermodynamical and dynamical analysis based on Landau theory of Fermi liquids is employed. It is shown that in the low density region of the phase diagram asymmetric nuclear matter can be characterized by a unique spinodal region, defined by the instability against isoscalarlike fluctuation, as in symmetric nuclear matter. Everywhere in this density region the system is stable against isovectorlike fluctuations related to the species separation tendency. Nevertheless, this instability in asymmetric nuclear matter induce isospin distillation leading to a more symmetric liquid phase and a more neutron rich gas phase.

Thermal and phase properties of nuclear systems are briefly reviewed within an information theory approach. Such theory allows treating on the same ground extended systems at the thermodynamic limit, as nuclear matter in the inner crust of neutron stars, and finite size, short-lived systems, as excited nuclei produced in heavy ion collisions. Different related issues including the pertinence of equilibrium in systems finite in size and time, ensemble inequivalence, and the effect of Coulomb interactions are discussed.

The nuclear equation-of-state is fundamental to both understanding systems as diverse as nuclei and neutron stars. Light-ion-induced reactions have served well to elucidate the behavior of excited nuclear material near the valley-of-stability. Heavy-ion reactions with systems of varying neutron-to-proton ratio (N/Z) are currently being used to gain a greater understanding of the equation-of-state away from the valley-of-stability.

In this lecture we review the theoretical investigation of heavy ion collisions in order to obtain information on the nuclear equation-of-state (EOS). We discuss the present knowledge of the EOS, and stress, in particular, the large uncertainty about the density dependence of the symmetry energy. We develop the treatment of heavy ion collisions with transport theory and non-equilibrium effects. We then discuss investigations both of the high density EOS with intermediate energy collisions and of the low density EOS in the Fermi energy regime. At the high density we make connections with neutron stars. At low density we discuss the fragmentation process and, in particular, the role and treatment of fluctuations and the dynamical fragment formation.

Ab inito calculations for the nuclear many-body problem make predictions for the density and isospin dependence of the nuclear equation-of-state (EOS) far away from the saturation point of nuclear matter. I compare predictions from microscopic and phenomenological approaches. Constraints on the EOS derived from heavy ion reactions, in particular from subthreshold kaon production, as well as constraints from neutron stars are discussed.

The concept of a nuclear molecule or a dinuclear system assumes two touching nuclei which carry out motion in the internuclear distance and exchange nucleons by transfer. The dinuclear model can be applied to nuclear structure, to fusion reactions leading to superheavy nuclei and to multi-nucleon transfer reactions.

We investigate the spectroscopic information which can be extracted from low-lying rotational yields in the cold fission of ²⁵²Cf. A fissioning state is considered as a resonance in the potential well between the emitted fragments. As fissioning states we select those resonances which are oriented close to the pole-to-pole configuration in the overlapping region. We predict a strong dependence of decay yields upon the quadrupole and hexadecapole deformation parameters. Predictions of rotational yields for ten possible cold splittings of ²⁵²Cf are given.

The steep falloff in the fusion cross section at energies far below the Coulomb barrier is discussed in the frame of the Coupled Channels metod for various symmetric and asymmetric projectile-target combinations where this phenomenon was very recently discovered. We incorporate a repulsive core in the nuclear potential which accounts for the saturation of nuclear matter and therefore provides the correct value of the incompressibility. The result of the incorporation of the nuclear Equation of State is that the internal part of the sudden heavy-ion potential becomes more shallow and consequently the fusion cross sections decreases at bombarding energies far below the barrier. The importance of the inclusion of the low-lying 2⁺ and 3⁻ states in both target and projectile as well as mutual and two-phonon excitations of these states is highlighted. In overall we obtain a very good fit to the data for the cases ⁵⁸Ni+⁵⁸Ni, ⁶⁴Ni+⁶⁴Ni and ⁶⁴Ni+⁷⁴Ge and a satisfactory fit for the more asymmetric case ⁶⁴Ni+¹⁰⁰Mo. We also present the analysis of the data using diagnostic tools well suited for deep sub-barrier energies, such as the astrophysical factor and the logarithmic derivative ans spin distribution. We predict, in particular, a distinct double peaking in the S-factor for the far subbarrier fusion of ⁵⁸Ni+⁵⁸Ni which should be tested experimentally. The relation between anti-resonances and the maximum in the average angular momentum is discussed.

The main questions which are still open concerning the optical model potential for α-particles at low energies are discussed with respect to the α-particle elastic scattering as well as α-induced reactions and α-particle emission. In order to understand the differences between the optical potential parameters used to describe the α-particle emission from excited compound residual nuclei, as compared to those determined by analysis of α-particle elastic scattering, the double-folding model is involved within semi-microscopic analysis of the α-particle elastic scattering on medium-mass nuclei, at energies below 30 MeV, and next involved within calculations of (n, α) reaction cross sections. The corresponding phenomenological optical potentials which match each other in the outer limit of the nuclear surface are shown to be described by a temperature-dependent shape of the nuclear density, which produces decreased central values and a larger diffuseness of the DF real potential.

The final nuclear states in pre-equilibrium (PE) reactions, which link the extreme mechanisms of compound nucleus and direct reactions, lie usually in the continuum of the nuclear excitation spectrum. Since both semi-classical models and quantum-statistical theories describe the PE processes as passing through a series of particle-hole excitations caused by two-body interactions, the initial target-projectile interactions within the diffuse nuclear surface limit the energy of the possible hole excitation due to the shallower nuclear potential in this region. The radial dependences of the nucleon's mean free path and the probability for the first PE nucleon-nucleon collision are pointing out the surface character of this interaction even at low energies, while improved cross-section calculations are made possible by use of partial level densities including surface effects as well as energy-dependent single-particle level densities, in the framework of the Geometry-Dependent Hybrid PE model.

We present a coupled channel method for calculating the emission rates based on selfconsistent models for nuclear structure and low-energy dynamics. Phenomenological adjustment of model parameters is discussed in detail. The α-decay properties of some new superheavy elements under current experimental research are estimated using the shell model preformation amplitude and resonant reaction amplitudes for open decay channels. Some extensions and applications of the method to resonant particle spectroscopy technique in studies of α-clustering and fine structure with position-sensitive charge particle detectors are discussed.

We review various issues related to the direct detection of constituents of dark matter, which are assumed to be Weakly Interacting Massive Particles (WIMPs). We specifically consider heavy WIMPs such as: 1) The lightest supersymmetric particle LSP or neutralino. 2) The lightest Kaluza-Klein particles in theories of extra dimensions and 3) other extensions of the standard model. In order to get the event rates one needs information about the structure of the nucleon as well as as the structure of the nucleus and the WIMP velocity distribution. These are also examined Since the expected event rates for detecting the recoiling nucleus are extremely low and the signal does not have a characteristic signature to discriminate against background we consider some additional aspects of the WIMP nucleus interaction, such as the periodic behavior of the rates due to the motion of Earth (modulation effect). Since, unfortunately, this is characterized by a small amplitude we consider other options such as directional experiments, which measure not only the energy of the recoiling nuclei but their direction as well. In these, albeit hard, experiments one can exploit two very characteristic signatures: a)large asymmetries and b) interesting modulation patterns. Furthermore we extended our study to include evaluation of the rates for other than recoil searches such as: i) Transitions to excited states, ii) Detection of recoiling electrons produced during the neutralino-nucleus interaction and iii) Observation of hard X-rays following the de-excitation of the ionized atom.

In the first part of this review I concentrate on the nuclear-structure problems related to nuclear double beta decay. The present status of the nuclear matrix element calculations of the neutrinoless double beta decay is presented. Independent probes of the involved virtual transitions are discussed. In the second part I concentrate on nuclear-structure calculations related to detection of the cold dark matter of the Universe. In particular, I discuss the scattering of the SUSY predicted stable neutralino off nuclei. The results can be contrasted with the claimed WIMP detection of the DAMA experiment.

An introduction to physics of in-medium hadrons with special emphasis towards modification of vector meson properties in dense nuclear matter is given. We start from remarkable analogy between the in-medium behavior of atoms in gases and hadrons in nuclear matter. Modifications of vector meson widths and masses can be registered experimentally in heavy-ion collisions by detecting dilepton spectra from decays of nucleon resonances and light unflavored mesons including ρ- and ω-mesons. Theoretical schemes for description of the in-medium hadrons are reviewed and recent experimental results of the NA60 and HADES collaborations on the dilepton production are discussed.

One presents an exact solution of the Frampton-Pisano-Pleitez three-generation 331 gauge model equipped with a special Higgs mechanism and a new kind of Yukawa couplings in unitary gauge. It is shown that the resulting boson mass spectrum depends on a real parameter. There exists a critical value of this parameter for which the different gauge bosons carrying the same charge get the same mass.

In this paper an introduction to the physics of deep inelastic scattering is given together with an account of recent results obtained in electron– or positron–proton collisions at the HERA collider.