International Journal of Modern Physics E

We first review the development of the Richardson-Gaudin exactly-solvable pairing models and then discuss several new models based on rank-two algebras and their applications to problems in nuclear structure.

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.

Performing shell model calculations for heavy nuclei is a long-standing problem in nuclear physics. The shell model truncation in the configuration space is an unavoidable step. The Projected Shell Model (PSM) truncates the space under the guidance of the deformed mean-field solutions. This implies that the PSM uses a novel and efficient way to bridge the two conventional methods: the deformed mean-field approximations, which are widely applied to heavy nuclei but able to describe the physics only in the intrinsic frame, and the spherical shell model diagonalization method, which is most fundamental but feasible only for small systems. We discuss the basic philosophy in construction of the PSM (or generally PSM-like) approach. Several examples from the PSM calculations are presented. Astrophysical applications are emphasized.

With the intrinsic coherent state formalism and the angular momentum projection, we study the shape phase structure of the yrast states in the dynamical symmetries of the IBM. We found that the states in the U(5) symmetry can undergo a rotation driven vibrational to axially rotational shape phase transition if the interaction parameters take negative values smaller than the critical ones. It shows that the U(5) symmetry of the IBM1 is an appropriate approach to describe the rotation driven shape phase transition along the yrast line of individual nucleus as the interaction parameters are taken in a special region. The O(6) symmetric yrast states may involve a phase transition from γ-soft rotation to triaxial rotation as the angular momentum increases, if the interaction parameters are specially chosen. And the yrast states in SU(3) symmetry always appear in the axially prolate shape phase.

Quantum phase transitional patterns in the whole parameter space of the consistent-Q Hamiltonian in the Interacting Boson Model are studied based on an implemented Fortran code for numerical computation of the matrix elements in the SU(3) Draayer-Akiyama basis. Results with respect to both ground and some excited states of the model Hamiltonian are discussed. Quantum phase transitional behavior under a variety of parameter situations is shown. It is found that transitional behavior of excited states is more complicated. Pt isotopes are taken as examples in illustrating the prolate-oblate shape phase transition.

Lifetimes of excited states in ¹²⁸Ce were measured using the recoil distance Doppler-shift (RDDS) and the Doppler-shift attenuation (DSAM) methods. The experiments were performed at the Wright Nuclear Structure Laboratory of Yale University. Excited states of ¹²⁸Ce were populated in the ¹⁰⁰Mo(³²Si,4n) reaction at 120 MeV and the nuclear γ decay was measured with an array of eight Clover detectors positioned at forward and backward angles. The deduced yrast transition strengths together with the energies of the levels within the ground-state (gs) band of ¹²⁸Ce are in agreement with the predicted values for the X(5) critical point symmetry. Thus, we suggest ¹²⁸Ce as a benchmark X(5) nucleus in the mass A ≈ 130 region.

The existence of quasi-dynamical symmetry (QDS) in physical systems and its significance for understanding the persistence of rotational structure in nuclei is explained in terms of the mathematical concept of an embedded representation. We consider the spectra obtained by coupling two SU(3) irreps by means of a quadrupole-quadrupole interaction. For a particular large value of this interaction, the two irreps combine to form a single (strongly-coupled) irrep while for zero interaction the weakly-coupled results are mixtures of many irreps. A notable result is the persistence of the rotor character of the low-energy states for a wide range of the interaction strength which can be explained by coherent mixing of SU(3) irreps. Such a coherent mixing of representations is an indication that the model has a QDS. Also notable is the fact that, for very weak interaction strengths, the rotational states of the yrast band approach those of a vibrational sequence while the B(E2) transition strengths remain close to those of an axially symmetric rotor.

Based on a nucleon-pair shell model, a projected SD-pair shell model is proposed. It is found that within this model, the classical limiting cases in interacting boson model and the experimental spectrum can all be reproduced approximately.

To investigate the pairing phase transition in finite systems with the particle-number conservation, we formulate a new BCS-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. This theory enables us to distinguish the symmetry-restoring fluctuation (SRF) from additional quantum fluctuations. By numerical calculations, it is found that the phase transition is compatible with the conservation law in this theory, and that the SRF shifts up the critical temperature. Having correct zero-temperature limit, this theory also gives new interpretation on the Pauli blocking effect in systems with odd particle number.

Using the realistic nucleon-nucleon interaction, we studied the pairing interaction potential for ¹S₀ and ³S₁ channel in neutron and nuclear matter. The explicit formulas are given for the calculation. It is found that the screening using realistic force is much smaller than that using effective interaction, for instance, Gogny force or G-matrix.

In the frame of deformed Skyrme Hartree-Fock (HF) model with pairing correlations, the strong mass number dependence of quadrupole deformations in sd and pf shell nuclei with mass A=(16 ~ 56) is studied as a clear manifestation of the evolution of nuclear deformation in nuclear many-body systems. The competition between the deformation driving particle-vibration coupling and the shell structure is shown by a systematic study on the ratios of the protons to neutrons quadrupole moments in nuclei with T=|T_z|=1. The mass number dependence of deformations obtained by deformed HF calculations is compared with the results of shell model and experimental data.

The decay by proton emission of deformed nuclei is discussed. The Coriolis coupling term is included in the non-adiabatic approach and calculations with non-spherical spin-orbit interactions are performed. Within this model, the half-lives of deformed drip-line odd-even nuclei are shown to be in good agreement with experimental data.

Yrast states in neutron rich nuclei ¹⁸⁸Os and ¹⁹⁰Os have been populated using the deep-inelastic reaction ⁸²Se + ¹⁹²Os at 460 MeV beam bombarding energy. A thick ¹⁹²Os target (>50 mg/cm²) with 0.2 mm Ta backing was used to stop all of the recoils in the target, minimising the broadening of the lines due to Doppler shift. High fold γ-γ coincidences were aquired with the 4π spectrometer GASP detector array. Three and two dimensional gamma-ray matrices were used to construct level schemes of the nuclei of interest up to an yrast state of 22.

In the framework of a modified statistical abrasion-ablation (SAA) model, the isospin effect and isoscaling behavior in projectile fragmentation have been systematically investigated. The normalized peak differences and reduced isoscaling parameters are found to decrease with (Z_proj-Z)/Z_proj and have no significant dependence on the size of reaction systems. The linear dependence of α and |β| on or for projectile-like fragments is demonstrated, which is obtained for light fragments from multifragmentation in the grand-canonical limit. Assuming a Fermi-gas behavior, the excitation energy dependence of the symmetry energy coefficients are tentatively extracted from α and β which looks consistent with the experimental data. It is pointed out that the reduced isoscaling parameters can be used as an observable to study the excitation extent of system and asymmetric nuclear equation of state in heavy ion collisions.

Probabilities of angular momentum I projected states from six particles Cranking Shell Model (CSM) as a function of cranking frequency ω are calculated and its validity is studied carefully. The generalized eigenvalue equations of I projected states are solved. All the results show that the I projected states of the CSM eigenfunctions have totally different properties from the rigid rotor approximated by a phenomenological model.

The structure of parity doublet rotational bands in odd-A actinides is investigated in the di-nuclear system model, in which the cluster-type shape is described by collective oscillations in the mass asymmetry degree of freedom. The odd nucleon is assumed to move in the mean field produced by the even-even di-nuclear core, which is approximated by a Woods-Saxon potential with quadrupole and octupole deformation. The calculated spin dependence of the parity splitting is in good agreement with the experimental data.

The phenomena of giant halo and halo of neutron-rich even-Ca isotopes are investigated and compared in the framework of the relativistic continuum Hartree-Bogoliubov (RCHB) and non-relativistic Skyrme Hartree-Fock-Bogoliubov (HFB) calculations. With two parameter sets for each of the RCHB and the Skyrme HFB calculations, it is found that although halo phenomena exist for Ca isotopes near neutron drip line in both calculations, the halo of the Skyrme HFB calculations starts at a more neutron-rich nucleus than that of the RCHB calculations, and the RCHB calculations have larger neutron root-mean-square (rms) radii systematically in N ≥ 40 than those of the Skyrme HFB calculations. The former difference comes from difference in shell structure. The reasons for the latter can be partly explained by the neutron 3s_1/2 orbit, which causes more than 50 % of the difference among the four calculations for neutron rms radii at ⁶⁶Ca.

We review some of the empirical and theoretical evidence supporting pseudospin symmetry in nuclei as a relativistic symmetry. We show that the eigenfunctions of realistic relativistic nuclear mean fields approximately conserve pseudospin symmetry. We discuss the implications of pseudospin symmetry for magnetic dipole transitions and Gamow-Teller transitions between states in pseudospin doublets. We explore a more fundamental rationale for pseudospin symmetry in terms of quantum chromodynamics (QCD), the basic theory of the strong interactions. We show that pseudospin symmetry in nuclei implies spin symmetry for an anti-nucleon in a nuclear environment.

The differences between measured masses and Liquid Drop Model (LDM) predictions have well known regularities, which can be analyzed as a two-dimensional texture on the N-Z plane. The remaining microscopic effects, obtained after removing the smooth LDM mass contributions, have proved difficult to model. They contain all the information related to shell closures, nuclear deformation and the residual nuclear interactions, and display a well defined pattern. In the present work the more than 2000 known nuclear masses are studied as an array in the N-Z plane viewed through a mask, behind which the approximately 7000 unknown unstable nuclei that can exist between the proton and neutron drip lines are hidden. Employing a Fourier transform deconvolution method these masses can be predicted. Measured masses are reconstructed with and r.m.s. error of less than 100 keV. Potential applications of the present approach are outlined.

The simplest of the two-body random matrix ensembles (TBRE) is the embedded Gaussian orthogonal ensemble of two-body interactions [EGOE(2)] for spinless fermion systems. With m fermions in N single particle states, EGOE(2) and similarly EGUE(2) [the embedded Gaussian unitary ensemble] are generated by the SU(N) algebra. For these ensembles results, obtained using SU(N) Wigner-Racah algebra, for lower order cross correlations between spectra with different particle numbers are given. For fermions with spin degree of freedom one has EGOE(2)-s and similarly EGUE(2)-s, both generated by U(2Ω) ⊃ U(Ω) ⊗ SU(2) algebra with SU(2) generating m particle spins S and 2Ω = N. For these ensembles numerical and first analytical results for cross correlations between spectra with different particle numbers and spins are given. As further extensions, it is possible to construct EGOE(2)-(s,p) generated by U(2Ω) ⊃ [U(Ω) ⊃ SO(Ω)] ⊗ SU(2) where SO(Ω) corresponds to pairing, for nuclear shell model the EGOE(2)-JT generated by U(N) ⊃ SO_J(3) ⊗ SU_T(2) algebra etc. On the other hand EGOE's with extended group symmetries of the shell model and the interacting boson models of nuclei, in particular via trace propagation for energy centroids give new insights into regular structures seen in the ground state region of nuclei. Several new examples for energy centroids generated by random 2 and 3-body interactions are given.

The nuclear shell model exhibits dynamical and geometrical aspects. The former are determined by the two-body matrix elements of the residual interaction while the latter are due to the two-body operators that couple pairs of fermions to good quantum numbers of spin, isospin, and parity. We analyze the geometric aspects of the shell model and understand some of its generic properties regarding the generation of chaos in nuclear spectra and correlations between spectra with different quantum numbers. In particular, we address the preponderance of spin-0 ground states.

The experimental information on the observed nearly degenerate bands in the N = 75 isotones, in particular ¹³⁴Pr, which is considered as the best candidate for chiral bands, if critically analyzed, shows that most properties of the bands, in particular the recently measured branching ratios and lifetimes, are in clear disagreement with the interpretation of the two bands as chiral bands. For I = 14 - 18 in ¹³⁴Pr, where the observed energies are almost degenerate, a value of 2.0(4) for the ratio of the transition quadrupole moments of the two bands was obtained, which implies a considerable difference in the nuclear shape associated with the two bands. The insufficiency of the near-degeneracy criterion to trace nuclear chirality is clearly emphasized.

This paper described the nuclear astrophysical studies using the unstable ion beam facility GIRAFFE, by indirect measurements. We measured the angular distributions for some single proton or neutron transfer reactions, such as ⁷Be(d,n)⁸B, ¹¹C(d,n)¹²N, ⁸Li(d,n)⁹Be, ⁸Li(d,p)⁹Li and ¹³N(d,n)¹⁴O in inverse kinematics, and derived the astrophysical S-factors or reaction rates of ⁷Be(p,γ)⁸B, ¹¹C(p,γ)¹²N, ⁸Li(n,γ)⁹Li, ¹³N(p,γ)¹⁴O by asymptotic normalization coefficient, spectroscopic factor, and R-matrix approach at astrophysically relevant energies.

Bulk properties of nuclear matter associated with the equation of state, the nuclear matter incompressibility coefficient, K_nm, in particular, are very important physical quantities required in the study of properties of nuclei, supernova collapse, neutron stars and heavy-ion collisions. Here we review the current status of K_nm and the experimental and theoretical methods used to determine the value of K_nm from the excitation cross-sections and the transition strength distributions of compression modes in nuclei, mainly the the isoscalar giant monopole resonance and the isoscalar giant dipole resonance.

The roles of anti-nucleon degrees of freedom in nuclear relativistic models are discussed. They are necessary for sum rules. It is also pointed out that the no-sea approximation, where the divergent terms due to the anti-nucleon degrees of freedom are neglected, has no physical justification and is a serious problem. These facts are shown in an analytic way by using the sum rule and RPA response function for the giant Gamow-Teller states.

We study the β decay from an even-even nucleus ¹²⁴Ba to an odd-odd nucleus ¹²⁴Cs in the interacting boson-fermion-fermion model (IBFFM). The even-even nucleus is described by the proton-neutron interacting boson model (IBM2) while the odd-odd nucleus is by the proton-neutron interacting boson-fermion-fermion model (IBFFM2). The result shows that the model is applicable to β-decay as well as the energy levels and the electromagnetic properties.

HIRFL has been upgraded for basic research on nuclear physics, atomic physics, irradiative material and biology from beginning of this decade. So far, the main performances of HIRFL have improved in the beam species from light ion to uranium and the maximum beam intensities reaching ~10μA from SFC, 1.5 μA from SSC. Therefore, some experiments have been performed during this period, especially, on new isotope synthesis and unstable nuclear physics. The new upgrading project Cooling Storage Ring (CSR) is under commissioning by ~2p μA carbon beam stripping injection. About 10⁹ C ion have stored inside CSRm, and part of them have been cooling down by the electron cooler. The acceleration of CSRm also has been test successful. Some future experiment are under development.

Gamma-ray spectroscopy associated to the clear identification of the emitting nuclei is a key to understand in a coherent way the nuclear structure of the elements located far from stability. The coupling of very efficient gamma-ray arrays and spectrometers has pushed away the limits and opened up new possibilities in nuclear physics studies. These combinations give access to a comprehensive picture of both the reaction mechanism and the nuclear structure of a given nucleus. In this talk, the various possibilities offered at GANIL with such a coupling will be given.

The FAIR facility with its high-energy and high-intensity secondary radioactive ion beams will be unique worldwide. The HIgh-resolution in-flight SPECtroscopy (HISPEC) project will use these rare beams at energies of E/A = 3–150 MeV for studies in nuclear structure, reactions and astrophysics. A brief review of the physics questions addressed and the proposed experimental setup are presented.

This contribution describes DESPEC (DEcay SPECtroscopy) which together with HISPEC (see the corresponding contribution to this conference) constitutes one of the proposed and accepted joint experiments of NuSTAR (Nuclear STucture, Astrophysics and Reactions), the part of the new FAIR facility that focuses on the production and study of beams of radioactive nuclei created by the fragmentation method.

I have enjoyed very much this international conference on nuclear structure. It is my great pleasure to summarize what we have learnt in this conference.