Halo phenomena in deformed nuclei are investigated within a deformed relativistic Hartree Bogoliubov (DRHB) theory. These weakly bound quantum systems present interesting examples for the study of the interdependence between the deformation of the core and the particles in the halo. Contributions of the halo, deformation effects, and large spatial extensions of these systems are described in a fully self-consistent way by the DRHB equations in a spherical Woods-Saxon basis with the proper asymptotic behavior at large distance from the nuclear center. Magnesium and neon isotopes are studied and detailed results are presented for the deformed neutron-rich and weakly bound nucleus $^{44}$Mg. The core of this nucleus is prolate, but the halo has a slightly oblate shape. This indicates a decoupling of the halo orbitals from the deformation of the core. The generic conditions for the occurence of this decoupling effects are discussed.
We explore the rotational feature of deformed halos by performing the angular momentum projection (AMP) on the ground state wave functions obtained from the deformed relativistic Hartree-Bogoliubov theory (DRHBc) in continuum. The DRHBc+AMP approach self-consistently describes the coupling between single particle bound states and the continuum not only in the ground state but also in rotational states. The rotational modes of deformed halos in $^{42}$Mg and $^{44}$Mg are investigated by studying properties of rotational states such as excitation energy, configuration, and density distribution.…
The structure of Ne, Na, Mg, Al, Si and S nuclei near the neutron drip-line region is investigated in the frame-work of relativistic meannfield (RMF) and non-relativistic Skyrme Hartree-Fock formalisms. The drip-line of these nuclei are pointed out. We analysed the large deformation structures and many of these neutron rich nuclei are quite deformed. New magic number are seen for these deformed nuclei.
In this contribution we present some recent results about neutron halos in deformed nuclei. A deformed relativistic Hartree-Bogoliubov theory in continuum has been developed and the halo phenomenon in deformed weakly bound nuclei is investigated. These weakly bound quantum systems present interesting examples for the study of the interdependence between the deformation of the core and the particles in the halo. Magnesium and neon isotopes are studied and detailed results are presented for the deformed neutron-rich and weakly bound nuclei 42Mg. The core of this nucleus is prolate, but the halo …
Covariant density functional theory is used to analyze the evolution of low-lying M1 strength in superfluid deformed nuclei in the framework of the self-consistent Relativistic Quasiparticle Random Phase Approximation (RQRPA). In nuclei with a pronounced neutron excess two scissor modes are found. Besides the conventional scissor mode, where the deformed proton and neutron distributions oscillate against each other, a new soft M1 mode is found, where the deformed neutron skin oscillates in a scissor like motion against a deformed proton-neutron core.
An extension of the previously proposed Variable Moment of Inertia with Softness model VMIS has been applied to describe successfully the effect of backbending in some deformed e-e nuclei. The model shows good results in the rare earth and actinide regions.
The 62Sm, 64Gd, 64Dy, 70Y b, 72Hf and 74W nuclei are classified as deformed nuclei. Low-lying bands are one of the most fundamental excitation modes in the energy spectra of deformed nuclei. In this paper a theoretical analysis of the experimental data within the phenomenological model is presented. The energy spectra of ground states are calculated. It is found the low-lying spectra of ground band states are in good agreement with the experimental data.
We calculate the binding energy and deformation parameter for light nuclei with and without pairing using a deformed relativistic mean field model. The role of BCS-type pairing effect is analyzed for Ne and F isotopes. The calculated odd-even staggering and the deformation parameters argue strongly against the role of pairing in the light nuclei.
We discuss the notion of partial dynamical symmetry in relation to nuclear spectroscopy. Explicit forms of Hamiltonians with partial $SU(3)$ symmetry are presented in the framework of the interacting boson model of nuclei. An analysis of the resulting spectrum and electromagnetic transitions demonstrates the relevance of such partial symmetry to the spectroscopy of axially deformed nuclei.
We present analytic expressions of the Coulomb interaction between a spherical and a deformed nuclei which are valid for all separation distance. We demonstrate their significant deviations from commonly used formulae in the region inside the Coulomb radius, and show that they remove various shortcomings of the conventional formulae.
An article by Covello, Gargano and Itaco [Phys.Rev. C 56, 3092 (1997)] tries to find evidence for the important role of the residual tensor force between the valence proton and neutron in doubly odd deformed nuclei. It is shown that observable effects discussed by these authors do not fully justify their rather strong conclusions.
Weakly-bound deformed nuclei have been studied by the Skyrme Hartree-Fock-Bogoliubov (HFB) approach in large coordinate-space boxes. In particular, the box-size dependence of the HFB calculations of weakly-bound deformed nuclei are investigated, including the particle density and pairing density distributions at nuclear surfaces, the near-threshold resonant and continuum quasiparticle spectra, and energetic properties. The box size may have larger influences in pairing properties than in other bulk properties. We demonstrate that large-box calculations of weakly-bound nuclei are important to p…
Octupole deformed shapes in neutron-rich actinides and superheavy nuclei as well as extremely deformed shapes of the N~Z light nuclei have been investigated within the framework of covariant density functional theory. We confirmed the presence of new region of octupole deformation in neutron-rich actinides with the center around Z~96, N~196 but our calculations do not predict octupole deformation in the ground states of superheavy Z~108 nuclei. As exemplified by the study of 36Ar, the nodal structure of the wavefunction of occupied single-particle orbitals in extremely deformed structures allo…
A study of the effect of deformation and pairing on the development of halo nuclei is presented. Exploratory three-body $core+n+n$ calculations show that both the NN interaction and the deformation/excitation of the core hinder the formation of the halo. Preliminary self-consistent mean-field calculations are used to search for regions in the nuclear chart where halos could potentially develop. These are also briefly discussed.
Magnetic form factors corresponding to elastic electron scattering from odd-A nuclei are presented. The calculations are carried out in plane-wave Born approximation. The one-body properties are obtained in a deformed self-consistent mean-field calculation based on a Skyrme HF+BCS method. Collective effects are also included in the cranking approximation. Results on several stable nuclei are compared with the available experimental information. It is shown that a deformed formalism improves the agreement with experiment in deformed nuclei, while reproducing equally well spherical nuclei by tak…
The consequences of the spontaneous breaking of rotational symmetry are investigated in a field theory model for deformed nuclei, based on simple separable interactions. The crucial role of the Ward-Takahashi identities to describe the rotational states is emphasized. We show explicitly how the rotor picture emerges from the isoscalar Goldstone modes, and how the two-rotor model emerges from the isovector scissors modes. As an application of the formalism, we discuss the M1 sum rules in deformed nuclei, and make connection to empirical information.
Potential energies, moments of inertia, quadrupole and octupole moments of dinuclear systems are compared with the corresponding quantities of strongly deformed nuclei. As dinuclear system we denote two touching nuclei (clusters). It is found that the hyperdeformed states of nuclei are close to those of nearly symmetric dinuclear systems, whereas the superdeformed states are considered as states of asymmetric dinuclear systems. The superdeformed and hyperdeformed states constructed from two touching clusters have large octupole deformations. The experimental measurement of octupole deformation…
We present a study of deformed nuclei in the framework of the sdg interacting boson model utilizing both numerical diagonalization and analytical $1/N$ expansion techniques. The focus is on description of high-spin states which have recently become computationally accessible through the use of computer algebra in the $1/N$ expansion formalism. A systematic study is made of high-spin states in rare-earth and actinide nuclei.
The single-particle spectrum of deformed shell-model states in nuclei, is shown to exhibit a supersymmetric pattern. The latter involves deformed pseudospin doublets and intruder levels. The underlying supersymmetry is associated with the relativistic pseudospin symmetry of the nuclear mean-field Dirac Hamiltonian with scalar and vector potentials.
The thermal evolution of the shell correction energy is investigated for deformed nuclei using Strutinsky prescription in a self-consistent relativistic mean-field framework. For temperature independent single-particle states corresponding to either spherical or deformed nuclear shapes, the shell correction energy $Δ_{sc}$ steadily washes out with temperature. However, for states pertaining to the self-consistent thermally evolving shapes of deformed nuclei, the dual role played by the single-particle occupancies in diluting the fluctuation effects from the single-particle spectra and in drivi…
Non-trivial geometrical effects in relativistic central collisions of deformed nuclei are studied using a simple version of optical Glauber model. For very small impact parameters large centrality and eccentricity fluctuations are observed. In very high multiplicity collisions of oblate nuclei (e.g. Au and Cu) significant fraction of events with elliptic flow strength v_2 dependent on oblateness beta_2 is predicted.
Background: Collective excitations of nuclei and their theoretical descriptions provide an insight into the structure of nuclei. Replacing traditional phenomenological interactions with unitarily transformed realistic nucleon-nucleon interactions increases the predictive power of the theoretical calculations for exotic or deformed nuclei. Purpose: Extend the application of realistic interactions to deformed nuclei and compare the performance of different interactions, including phenomenological interactions, for collective excitations in the sd-shell. Method: Ground-state energies and charge r…
From the semi-empirical formalisms of Bohr-Mottelson, a new model, based on the effect of beta- and gamma- head energies and the variable moment of inertia, was developed to calculate the ground state rotational band of almost all deformed e-e nuclei. The model can be tuned for nuclei with experimentally available beta- and gamma-vibrational energies as well as for those with unmeasured beta- and gamma-vibrations.
Reaction cross-sections are calculated using the Coulomb modified Glauber model for deformed target nuclei. The deformed nuclear matter density of the target is expanded into multipoles of order k = 0,2,4.The reaction cross-sections between some spherical and deformed nuclei are studied at energy range (10-1000 MeV/nucleon). The most significant effects in the intermediate energy range are the Coulomb field and in-medium effect that modified the trajectory of the incident beams. Introducing the deformation effect beside the Coulomb field and in-medium effect improves the agreement with the exp…
We investigate the deformation properties of atomic nuclei in a hadronic chiral SU_f(3) model approach. The parameters are fitted to hadron mass properties and adjustments for spherical finite nuclei have been performed. Using these parameters the deformation of a series of light and heavy nuclei are obtained in a two-dimensional self-consistent calculation. In addition we discuss a case of superdeformation in a heavy nucleus.
This work presents an older, but still interesting description of a superfluid condensate with two components (protons and neutrons), confined by time-dependent deformed harmonic oscillator potentials either in collective uniform rotation, or opposite angular oscillation.
We formulate a new scheme of the Hartree-Fock-Bogoliubov mean-field theory applicable to weakly bound and pair correlated deformed nuclei using the coordinate-space Green's function technique. On the basis of a coupled-channel representation of the quasiparticle wave function expanded in terms of the partial waves, we impose the correct boundary condition of the asymptotically out-going waves on the continuum quasiparticle states. We perform numerical analysis for $^{38}$Mg to illustrate properties of the continuum quasiparticle states and the pair correlation in deformed nuclei near the neutr…
We consider deformed nuclei within an effective theory that exploits the small ratio between rotational and vibrational excitations. For even-even nuclei, the effective theory predicts small changes in the rotational constants of bands built on multi-phonon excitations that are linear in the number of excited phonons. In 166Er and 168Er, this explains the main variations of the rotational constants of the two-phonon gamma vibrational bands. In 232Th, the effective theory correctly explains the trend that the rotational constants decrease with increasing spin of the band head. We also study the…
We discuss characteristic features of SU(3) partial dynamical symmetry in relation to nuclear spectroscopy and compare with previous broken-SU(3) calculations for ^{168}Er.
The effect of deformation on the excitation of multiple giant dipole resonances is studied. Analytical expressions are derived in the framework of the interacting boson model for the energies and E1 properties of giant dipole resonances in spherical and deformed nuclei, and a numerical treatment of transitional nuclei is proposed. Coulomb-excitation cross sections are calculated in $^{238}$U and in the samarium isotopes.
In the frame work of the hydrodynamical model, a new model of the ground state rotational bands of deformed e-e nuclei is developed by introducing the variable moment of inertia, and the effect of beta- and gamma- vibrational bands. The model is applied to calculate the energies of the ground state band of 158Dy. The results of our calculations are in close agreement with data compared with other existing models.
The properties of kaonic nuclei are studied using a two-dimensional Skyrme-Hartree-Fock model with a $KN$ Skyrme force. We focus in particular on the instability of the solutions for a too strong $KN$ interaction, which determines a maximum value of the kaon binding in this approach. We then analyze the change of the deformation properties of several core-deformed nuclei caused by the added kaon, and find a shrinking of the core and in some cases a complete loss of deformation.
Deformation properties of weakly bound nuclei are discussed in the deformed single-particle model. It is demonstrated that in the limit of a very small binding energy the valence particles in specific orbitals, characterized by a very small projection of single-particle angular momentum onto the symmetry axis of a nucleus, can give rise to the halo structure which is completely decoupled from the rest of the system. The quadrupole deformation of the resulting halo is completely determined by the intrinsic structure of a weakly bound orbital, irrespective of the shape of the core.
Pseudospin symmetry is a relativistic symmetry of the Dirac Hamiltonian with scalar and vector mean fields equal and opposite in sign. This symmetry imposes constraints on the Dirac eigenfunctions. We examine extensively the Dirac eigenfunctions of realistic relativistic mean field calculations of deformed nuclei to determine if these eigenfunctions satisfy these pseudospin symmetry constraints.
Initial eccentricity of the interacting volume of QCD matter created in collisions of prolate and oblate nuclei is studied using optical Glauber simulation. Collisions of prolate 165-Ho + 165-Ho nuclei and Au+Au nuclei are investigated (assuming oblate ground-state deformation for 197-Au nucleus). Qualitative predictions are obtained for RHIC energies.
A formula to evaluate the effects of a general deformation on the Coulomb direct contribution to the energy of the Isobaric Analog State (IAS) is presented and studied via a simple yet physical model. The toy model gives a reasonable account of microscopic deformed Hartree-Fock-Bogolyubov (HFB) calculations in a test case, and provides a guidance when predicting unknown IAS energies. Thus, deformed HFB calculations, to predict the IAS energies, are performed for several neutron-deficient medium-mass and heavy nuclei which are now planned to be studied experimentally.
The relationship between the deformed harmonic oscillator and the formation of molecular cluster structures, whereby valence neutrons are exchanged between cluster cores, is examined. It is found that there is a strong connection between the properties of the valence orbitals associated with deformed structures in the deformed harmonic oscillator and the molecular orbitals created by linear combinations of single-centre orbitals around nuclear clusters. The conclusion is that in addition to the appearance of clustering in the deformed harmonic oscillator that \emph{every} prolate deformed clus…
Examples of the change of neutron shell-structure in both weakly-bound and resonant neutron one-particle levels in nuclei towards the neutron drip line are exhibited. It is shown that the shell-structure change due to the weak binding may lead to the deformation of those nuclei with the neutron numbers $N \approx$ 8, 20, 28 and 40, which are known to be magic numbers in stable nuclei. Nuclei in the "island of inversion" are most easily and in a simple manner understood in terms of deformation. As an example of spectroscopic properties other than single-particle energies, magnetic moments of so…
A collective vector-boson model with broken SU(3) symmetry is applied to several deformed even-even nuclei. The model description of ground and $γ$ bands together with the corresponding B(E2) transition probabilities is investigated within a broad range of SU(3) irreducible representations (irreps) $(λ,μ)$. The SU(3)-symmetry characteristics of rotational nuclei are analyzed in terms of the bandmixing interactions.
The giant dipole resonance (GDR) in deformed nuclei is analyzed using the self-consistent separable random-phase-approximation (SRPA) with Skyrme forces SkT6, SkM$^*$, SLy6 and SkI3. The deformed nuclei $^{150}$Nd and $^{238}$U are used as representative rare-earth and actinide samples. Dependencies of the dipole strength distributions on some basic characteristics of the Skyrme functional and nuclear matter properties (isoscalar and isovector effective masses, time-odd contributions) are discussed. Particular attention is paid to the fragmentation structure of the GDR strength which are shown…
Using an angular momentum projected single particle basis, a pnQRPA approach is used to study the $2νββ$ properties of ten isotopes, exhibiting various quadrupole deformations. The mother and daughter nuclei exhibit different quadrupole deformations. Since the projected basis enables a unified description of deformed and spherical nuclei, situations where the nuclei involved in the double beta decay process are both spherical, both deformed or one spherical and another deformed, can be treated through a sole formalism. Dependence of single $β^-$ and $β^+$ strength distribution on atomic mass n…
We have investigated the effects of the neutron-proton interaction in several doubly odd deformed nuclei within the framework of the particle-rotor model. In this paper, we show some selected results of our study which evidence the importance of the tensor-force effects.
Spontaneous symmetry breaking in non-relativistic quantum systems has previously been addressed in the framework of effective field theory. Low-lying excitations are constructed from Nambu-Goldstone modes using symmetry arguments only. We extend that approach to finite systems. The approach is very general. To be specific, however, we consider atomic nuclei with intrinsically deformed ground states. The emergent symmetry breaking in such systems requires the introduction of additional degrees of freedom on top of the Nambu-Goldstone modes. Symmetry arguments suffice to construct the low-lying …
Techniques from effective field theory are applied to nuclear rotation. This approach exploits the spontaneous breaking of rotational symmetry and the separation of scale between low-energy Nambu-Goldstone rotational modes and high-energy vibrational and nucleonic degrees of freedom. A power counting is established and the Hamiltonian is constructed at next-to-leading order.
We describe and apply a version of the finite amplitude method for obtaining the charge-changing nuclear response in the quasiparticle random phase approximation. The method is suitable for calculating strength functions and beta-decay rates, both allowed and forbidden, in axially-deformed open-shell nuclei. We demonstrate the speed and versatility of the code through a preliminary examination of the effects of tensor terms in Skyrme functionals on beta decay in a set of spherical and deformed open-shell nuclei. Like the isoscalar pairing interaction, the tensor terms systematically increase a…
The Relativistic Hartree-Fock-Bogoliubov model for axially deformed nuclei (RHFBz) is introduced. The model is based on an effective Lagrangian with density-dependent meson-nucleon couplings in the particle-hole channel, and the pairing part of the Gogny force is used in the pairing channel. The RHFBz quasiparticle equations are solved by expansion in the basis of a deformed harmonic oscillator. Illustrative RHFBz calculations are performed for Carbon, Neon and Magnesium isotopes. The effect of the explicitly including the pion field is investigated for binding energies, deformation parameters…
Deformed Hartree-Fock-Bogoliubov calculations for finite nuclei are carried out. As residual interaction, a Brueckner G-matrix derived from a meson-exchange potential is taken. Phenomenological medium modifications of the meson masses are introduced. The binding energies, radii, and deformation parameters of the Carbon, Oxygen, Neon, and Magnesium isotope chains are found to be in good agreement with the experimental data.
A semi-microscopic model to study the neutron and proton induced backbending phenomena in some deformed even-even nuclei from the rare earth region, is proposed. The space of particle-core states is defined by the angular momentum projection of a quadrupole deformed product state. The backbending phenomena are described by mixing four rotational bands, defined by a set of angular momentum projected states, and a model Hamiltonian describing a set of paired particles moving in a deformed mean field and interacting with a phenomenological deformed core. The ground band corresponds to the configu…
We discuss three practical methods for performing Eulerian rotations of Slater determinants in a three-dimensional Cartesian geometry. In addition to the straightforward application of the active form of the quantum mechanical rotation operator, we introduce two methods using a passive position-space rotation followed by an active spin-space rotation, one after variation and the other before variation. These methods can be used to initialize reactions involving deformed nuclei where a particular alignment of the deformed nuclei with respect to the collision axis is desired. We show that doing …
Experimental data on Coulomb breakup and neutron removal indicate that 31Ne is one of the heaviest halo nuclei discovered so far. The possible ground state of 31Ne is either 3/2- coming from p-wave halo or 1/2+ from s-wave halo. In this work, we develop a treatable model to include deformed wave functions and a dynamical knockout formalism which includes the dependence on the nuclear orientation to study the neutron removal from 31Ne projectiles at energies around E=200 MeV/nucleon. A detailed account of the effects of deformation on cross sections and longitudinal momentum distributions is ma…
Deformed neutron halo nuclei and soft dipole excitations in the 40<A<90 mass region
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Department of Physics, Anhui Normal University（安徽师范大学物理系）；RIKEN Nishina Center for Accelerator-Based Science（理化学研究所仁科加速器科学中心）；Center for Mathematics and Physics, University of Aizu（会津大学数学与物理中心）；School of Nuclear Science and Technology, University of Chinese Academy of Sciences（中国科学院大学核科学与技术学院）；Institute for Advanced Simulation (IAS-4), Forschungszentrum Jülich（于利希研究中心高级模拟研究所（IAS-4））；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
We formulate a new Hartree-Fock-Bogoliubov method applicable to weakly bound deformed nuclei using the coordinate-space Green's function technique. An emphasis is put on treatment of quasiparticle states in the continuum, on which we impose the correct boundary condition of the asymptotic out-going wave. We illustrate this method with numerical examples.
A method to implement realistic nucleon-nucleon residual interaction (Bonn CD force) based on the Brueckner G-matrix into the Quasiparticle Random Phase Approximation (QRPA) for deformed nuclei is formulated. The two-neutrino double decay for ground state to ground state transitions $^{76}$Ge$ \to ^{76}$Se and $^{150}$Nd $\to ^{150}$Sm is calculated along with the Gamow-Teller strength distributions. The effect of deformation on the observables is studied.
A deformed relativistic Hartree-Bogoliubov (DRHB) model is developed aiming at a proper description of exotic nuclei, particularly deformed ones with large spatial extension. In order to give an adequate description of both the contribution of the continuum and the large spatial distribution in exotic nuclei, the DRHB equations are solved in a Woods-Saxon basis in which the radial wave functions have proper asymptotic behaviors at large distance from the nuclear center which is crucial for the formation of halo. The formalism and the numerical procedure of the DRHB model in a Woods-Saxon basis…
UU collisions can provide about 30% larger densities compared to central PbPb ones. New aspect is generation of rather deformed initial states. We show that those can be effectively used to resolve a number of outstanding issues, from corrections to hard processes, elliptic flow (the QGP push issue), and the mechanism of $J/ψ$ suppression. UU collisions are studied by a simple Monte-Carlo model, and it is shown how selecting two control parameters - the number of participant nuclei and deformation - one can select particular geometry of the collision.
Band mixing calculations in rapidly rotating well-deformed nuclei are presented, investigating the properties of energy levels and rotational transitions as a function of excitation energy. Substantial fragmentation of E2 transitions is found for $E_x \gsim$ 800 keV above yrast, which represents the onset of rotational damping. Above $E_x \approx $ 2 MeV, energy levels and E2 strengths display fluctuations typical of quantum chaotic systems, which are determined by the high multipole components of the two-body residual interaction.
We develop a new framework of the deformed quasiparticle-random-phase approximation (QRPA) where the Skyrme density functional and the density-dependent pairing functional are consistently treated. Numerical applications are carried out for the isovector dipole and the isoscalar quadrupole modes in the spherical $^{20}$O and in the deformed $^{26}$Ne nuclei, and the effect of the momentum dependent terms of the Skyrme effective interaction for the energy-weighted sum rule is discussed. As a further application, we present for the first time the moments of inertia of $^{34}$Mg and $^{36}$Mg usi…
We developed new parameterizations of local regularized finite-range pseudopotentials up to next-to-next-to-next-to-leading order (N3LO), used as generators of nuclear density functionals. When supplemented with zero-range spin-orbit and density-dependent terms, they provide a correct single-reference description of binding energies and radii of spherical and deformed nuclei. We compared the obtained results to experimental data and discussed benchmarks against the standard well-established Gogny D1S functional.
Using a new approximate analytic parameter-free proxy-SU(3) scheme, we make predictions of shape observables for deformed nuclei, namely beta and gamma deformation variables, and compare these with empirical data and with predictions by relativistic and non-relativistic mean-field theories.
In two previous papers, the Kerman-Klein-Donau-Frauendorf (KKDF) model was used to study rotational bands of odd deformed nuclei. Here we describe backbending for odd nuclei using the same model. The backbending in the neighboring even nuclei is described by a phenomenological two band model, and this core is then coupled to a large single-particle space, as in our previous work. The results obtained for energies and M1 transition rates are compared with experimental data for 165Lu and for energies alone to the experimental data for 179W. For the case of 165Lu comparison is also made with prev…
