The interplay between quadrupole and octupole degrees of freedom is discussed in a series of neutron-rich actinides and superheavy nuclei with $92 \le$ Z $\le 110$ and $186 \le$ N $\le 202$. In addition to the static Hartree-Fock-Bogoliubov approach, dynamical beyond-mean-field correlations are taken into account via both parity restoration and symmetry-conserving Generator Coordinate Method calculations based on the Gogny-D1M energy density functional. Physical properties such as correlation energies, negative-parity excitation energies as well as reduced transition probabilities $B(E1)$ and $B(E3)$ are discussed in detail. It is shown that, for the studied nuclei, the quadrupole-octupole coupling is weak and to a large extent the properties of negative parity states can be reasonably well described in terms of the octupole degree of freedom alone.
Collective quadrupole and octupole states are described in a series of Sm and Gd isotopes within the framework of the interacting boson model (IBM), whose Hamiltonian parameters are deduced from mean field calculations with the Gogny energy density functional. The link between both frameworks is the ($β_2β_3$) potential energy surface computed within the Hartree-Fock-Bogoliubov framework in the case of the Gogny force. The diagonalization of the IBM Hamiltonian provides excitation energies and transition strengths of an assorted set of states including both positive and negative parity states.…
An analytic, parameter-free (up to overall scale factors) solution of the Bohr Hamiltonian involving axially symmetric quadrupole and octupole deformations, as well as an infinite well potential, is obtained, after separating variables in a way reminiscent of the Variable Moment of Inertia (VMI) concept. Normalized spectra and B(EL) ratios are found to agree with experimental data for 226-Ra and 226-Th, the nuclei known to lie closest to the border between octupole deformation and octupole vibrations in the light actinide region.
The interplay between the collective dynamics of the quadrupole and octupole deformation degree of freedom is discussed in a series of Sm and Gd isotopes both at the mean field level and beyond, including parity symmetry restoration and configuration mixing. Physical properties like negative parity excitation energies, E1 and E3 transition probabilities are discussed and compared to experimental data. Other relevant intrinsic quantities like dipole moments, ground state quadrupole moments or correlation energies associated to symmetry restoration and configuration mixing are discussed. For the…
The interplay between quadrupole and octupole degrees of freedom is discussed in a series of U, Pu, Cm and Cf isotopes both at the mean-field level and beyond. In addition to the static Hartree-Fock-Bogoliubov approach, dynamical beyond-mean-field correlations are taken into account via both parity restoration and symmetry-conserving Generator Coordinate Method calculations based on the parametrization D1M of the Gogny energy density functional. Physical properties such as correlation energies, negative-parity excitation energies as well as reduced transition probabilities $B(E1)$ and $B(E3)$ …
The analytic quadrupole octupole axially symmetric model, which had successfully predicted 226Ra and 226Th as lying at the border between the regions of octupole deformation and octupole vibrations in the light actinides using an infinite well potential (AQOA-IW), is made applicable to a wider region of nuclei exhibiting octupole deformation, through the use of a Davidson potential (AQOA-D). Analytic expressions for energy spectra and B(E1), B(E2), B(E3) transition rates are derived. The spectra of 222-226Ra and 224,226Th are described in terms of the two parameters phi_0 (expressing the relat…
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…
Quadrupole and octupole deformation energy surfaces, low-energy excitation spectra, and electric transition rates in eight neutron-rich isotopic chains -- Ra, Th, U, Pu, Cm, Cf, Fm, and No -- are systematically analyzed using a quadrupole-octupole collective Hamiltonian model, with parameters determined by constrained reflection-asymmetric and axially-symmetric relativistic mean-field calculations based on the PC-PK1 energy density functional. The theoretical results of low-lying negative-parity bands, odd-even staggering, average octupole deformations $\langleβ_3\rangle$, and $B(E3; 3^-_1\to …
We show that the measured intrinsic octupole moments of $^{220}$Rn, $^{224}$Ra, and $^{226}$Ra constrain the intrinsic Schiff moments of $^{225}$Ra$^{221}$Rn, $^{223}$Rn, $^{223}$Fr, $^{225}$Ra, and $^{229}$Pa. The result is a dramatically reduced uncertainty in intrinsic Schiff moments. Direct measurements of octupole moments in odd nuclei will reduce the uncertainty even more. The only significant source of nuclear-physics error in the laboratory Schiff moments will then be the intrinsic matrix elements of the time-reversal non-invariant interaction produced by CP-violating fundamental physi…
An analytic collective model in which the relative presence of the quadrupole and octupole deformations is determined by a parameter (phi_0), while axial symmetry is obeyed, is developed. The model [to be called the analytic quadrupole octupole axially symmetric model (AQOA)] involves an infinite well potential, provides predictions for energy and B(EL) ratios which depend only on phi_0, draws the border between the regions of octupole deformation and octupole vibrations in an essentially parameter-independent way, and describes well 226-Th and 226-Ra, for which experimental energy data are sh…
A systematic search for axial octupole deformation in the actinides and superheavy nuclei with proton numbers $Z=88-126$ and neutron numbers from two-proton drip line up to $N=210$ has been performed in covariant density functional theory (DFT) using four state-of-the-art covariant energy density functionals representing different model classes. The nuclei in the $Z\sim 96, N\sim 196$ region of octupole deformation have been investigated in detail and the systematic uncertainties in the description of their observables have been quantified. Similar region of octupole deformation exists also in…
In this work we examine the possibility to describe yrast positive- and negative-parity excitations of deformed even-even nuclei through a collective rotation model in which the nuclear surface is characterized by triaxial quadrupole and octupole deformations. The nuclear moments of inertia are expressed as sums of quadrupole and octupole parts. By assuming an adiabatic separation of rotation and vibration degrees of freedom we suppose that the structure of the positive- and negative- parity bands may be determined by the triaxial-rigid-rotor motion of the nucleus. By diagonalizing the Hamilto…
An angular momentum dependence of the parity splitting in the alternating parity bands of nuclei with strong octupole correlations is considered basing on the model of the octupole motion in a one-dimensional potential well conserving axial symmetry. A sign reversal of the parity splitting at higher values of the angular momentum is interpreted as a result of the Coriolis coupling to K=1 octupole mode. New experimental data on the spectra of the alternating parity octupole bands are used.
The cranked Skyrme III effective Hamiltonian is applied for the analysis of the rotational dependence of the quadrupole and octupole moments in Ra, Th, and U isotopes. A comparison of the intrinsic electric dipole moments calculated in the model with available experimental and theoretical values is presented. It is found that the non-axial octupole deformation $Y_{32}$ becomes favorable at high spins for the actinide nuclei.
The evolution of quadrupole and octupole collectivity and their coupling is investigated in a series of even-even isotopes of the actinide Ra, Th, U, Pu, Cm, and Cf with neutron number in the interval $130\leqslant N\leqslant 150$. The Hartree-Fock-Bogoliubov approximation, based on the parametrization D1M of the Gogny energy density functional, is employed to generate potential energy surfaces depending upon the axially-symmetric quadrupole and octupole shape degrees of freedom. The mean-field energy surface is then mapped onto the expectation value of the $sdf$ interacting-boson-model Hamilt…
$J^π$= 0$^+$ states have been investigated in the actinide nucleus ${}^{240}$Pu up to an excitation energy of 3 MeV with a high-resolution (p,t) experiment at $E_{p}$= 24 MeV. To test the recently proposed $J^π$= 0$^+_2$ double-octupole structure, the phenomenological approach of the spdf-interacting boson model has been chosen. In addition, the total 0$^+$ strength distribution and the $0^+$ strength fragmentation have been compared to the model predictions as well as to the previously studied (p,t) reactions in the actinides. The results suggest that the structure of the 0$^+_2$ states in th…
The quadrupole-octupole coupling and the related spectroscopic properties have been studied for the even-even light actinides $^{218-238}$Ra and $^{220-240}$Th. The Hartree-Fock-Bogoliubov approximation, based on the Gogny-D1M energy density functional, has been employed as a microscopic input, i.e., to obtain (axially symmetric) mean-field potential energy surfaces as functions of the quadrupole and octupole deformation parameters. The mean-field potential energy surfaces have been mapped onto the corresponding bosonic potential energy surfaces using the expectation value of the $sdf$ Interac…
We discuss the octupole deformation properties of many even-even actinide isotopes. The analysis is carried out in the mean field framework (Hartree-Fock-Bogoliubov approximation) by using the axially symmetric octupole moment as a constraint. A one-dimensional octupole collective Hamiltonian is used to obtain properties like excitation energies or $E1$ and $E3$ transition probabilities of the negative parity band-heads associated to the lowest lying $1^{-}$ and $3^{-}$ states. The evolution of these values with neutron number is discussed and a comparison with available experimental data is m…
Nuclear fission of heavy (actinide) nuclei results predominantly in asymmetric mass-splits. Without quantum shells, which can give extra binding energy to these mass-asymmetric shapes, the nuclei would fission symmetrically. The strongest shell effects are in spherical nuclei, so naturally the spherical "doubly-magic" ${^{132}}$Sn nucleus (${Z=50}$ protons), was expected to play a major role. However, a systematic study of fission has shown that the heavy fragments are distributed around ${Z=52}$ to 56, indicating that ${^{132}}$Sn is not the only driver. Reconciling the strong spherical shell…
A systematic analysis of low-lying quadrupole and octupole collective states is presented, based on the microscopic energy density functional framework. By mapping the deformation constrained self-consistent axially symmetric mean-field energy surfaces onto the equivalent Hamiltonian of the $sdf$ interacting boson model (IBM), that is, onto the energy expectation value in the boson condensate state, the Hamiltonian parameters are determined. The study is based on the global relativistic energy density functional DD-PC1. The resulting IBM Hamiltonian is used to calculate excitation spectra and …
``Beat'' patterns are shown to appear in the octupole bands of several actinides and rare earths, their appearance being independent from the formula used in order to isolate and demonstrate them. It is shown that the recent formalism, making use of discrete approximations to derivatives of the transition energies (or of the energy levels) gives results consistent with the traditional formulae. In both regions it is seen that the first vanishing of the staggering occurs at higher values of the angular momentum I in nuclei exhibiting higher staggering at low I. Since these nuclei happen to be g…
High spin states of ^{32}S$ and ^{56}Ni are investigated by means of the cranking Hartree-Fock method with the Gogny interaction without imposing a restriction on the axial reflection symmetry. It was found that a non-axial octupole deformation of the Y_{31} type becomes important in the yrast states of ^{32}S. A similar effect is predicted for the nucleus ^{56}Ni.
The Delta I = 1 staggering (odd--even staggering) in octupole bands of light actinides is found to exhibit a ``beat'' behaviour as a function of the angular momentum I, forcing us to revise the traditional belief that this staggering decreases gradually to zero and then remains at this zero value. Various algebraic models (spf-Interacting Boson Model, spdf-IBM, Vector Boson Model, Nuclear Vibron Model) are shown to predict in their su(3) limits constant staggering for this case, being thus unable to describe the ``beat'' behaviour. An explanation of the ``beat'' behaviour is given in terms of …
We present a computational methodology for a theory of the lowest octupole excitations applicable to all even-even nuclei beyond the lightest. The theory is the well-known generator-coordinate extension (GCM) of the Hartree-Fock-Bogoliubov self-consistent mean field theory (HFB). We use the discrete-basis Hill-Wheel method (HW) to compute the wave functions with an interaction from the Gogny family of Hamiltonians. Comparing to the compiled experimental data on octupole excitations, we find that the performance of the theory depends on the deformation characteristics of the nucleus. For nondef…
Low-energy positive and negative parity collective states in the equilibrium minimum, and the dynamics of induced fission of actinide nuclei are investigated in a unified theoretical framework based on the generator coordinate method (GCM) with the Gaussian overlap approximation (GOA). The collective potential and inertia tensor, both at zero and finite temperature, are computed using the self-consistent multidimensionally constrained relativistic mean field (MDC-RMF) model, based on the energy density functional DD-PC1. Pairing correlations are treated in the BCS approximation with a separabl…
The constrained Hartree-Fock-Bogoliubov method is used with the Gogny interaction D1S to calculate potential energy surfaces of fissioning nuclei ${}^{226}$Th and ${}^{256,258,260}$Fm up to very large deformations. The constraints employed are the mass quadrupole and octupole moments. In this subspace of collective coordinates, many scission configurations are identified ranging from symmetric to highly asymmetric fragmentations. Corresponding fragment properties at scission are derived yielding fragment deformations, deformation energies, energy partitioning, neutron binding energies at sciss…
We propose a collective model formalism which describes the strong parity shift observed in low-lying spectra of nuclei with octupole deformations together with the fine rotational band structure developed at higher angular momenta. The parity effect is obtained by the Schroedinger equation for oscillations of the reflection asymmetric (octupole) shape between two opposite orientations in an angular momentum dependent double-well potential. The rotational structure is obtained by a collective quadrupole-octupole rotation Hamiltonian. The model scheme reproduces the complicated beat staggering …
The ground state octupole correlation energy is computed with the D1M variant of the Gogny force in different theoretical frameworks and analyzed in detail. First I consider the correlation energy gained at the mean field level by breaking reflection symmetry. Next I consider the energy gain coming from symmetry (parity) restoration and finally I analyze the ground state correlation energy after configuration mixing with axially symmetric octupole states. The impact of the latter on theoretical binding energies indicates that octupole correlations do not affect in a significant way the trend a…
Theoretical results for giant resonances in the three doubly magic exotic nuclei $^{78}$Ni, $^{100}$Sn and $^{132}$Sn are obtained from Hartree-Fock (HF) plus Random Phase Approximation (RPA) calculations using the D1S parametrization of the Gogny two-body effective interaction. Special attention is paid to full consistency between the HF field and the RPA particle-hole residual interaction. The results for the exotic nuclei, on average, appear similar to those of stable ones, especially for quadrupole and octupole states. More exotic systems have to be studied in order to confirm such a trend…
We have developed multi-dimensional constrained covariant density functional theories (MDC-CDFT) for finite nuclei in which the shape degrees of freedom β_{λμ} with even μ, e.g., β_{20}, β_{22}, β_{30}, β_{32}, β_{40}, etc., can be described simultaneously. The functional can be one of the following four forms: the meson exchange or point-coupling nucleon interactions combined with the non-linear or density-dependent couplings. For the pp channel, either the BCS approach or the Bogoliubov transformation is implemented. The MDC-CDFTs with the BCS approach for the pairing (in the following label…
We analyze the octupole deformations and the related collective excitations in medium-heavy and heavy nuclei based on the microscopic framework of the nuclear energy density functional theory. Constrained self-consistent mean-field calculation with a given energy density functional is performed to provide for each nucleus a potential energy surface with axial quadrupole and octupole shape degrees of freedom. Spectroscopic properties are computed by means of the interacting-boson Hamiltonian, which is determined by mapping the fermionic potential energy surface onto the bosonic counterpart. The…
The cranked relativistic Hartree-Bogoliubov (CRHB) theory has been applied for a systematic study of pairing and rotational properties of actinides and light superheavy nuclei. Pairing correlations are taken into account by the Brink-Booker part of finite range Gogny D1S force. For the first time in the covariant density functional theory (CDFT) framework the pairing properties are studied via the quantities (such as three-point $Δ^{(3)}$ indicators) related to odd-even mass staggerings. The investigation of the moments of inertia at low spin and the $Δ^{(3)}$ indicators shows the need for an …
We study the interplay of quadrupole and octupole degrees of freedom in the structure of the isotope $^{144}$Ba. A symmetry conserving configuration mixing method (SCCM) based on a Gogny energy density functional (EDF) has been used. The method includes particle number, parity and angular momentum restoration as well as axial quadrupole and octupole shape mixing within the generator coordinate method. Predictions both for excitation energies and electromagnetic transition probabilities are in good agreement with the most recent experimental data.
We report on the results of the calculations of the low energy excitation patterns for three Zirconium isotopes, viz. $^{80}$Zr$_{40}$, $^{96}$Zr$_{56}$ and $^{110}$Zr$_{70}$, reported by other authors to be doubly-magic tetrahedral nuclei (with tetrahedral magic numbers $Z$=40 and $N$=40, 56 and 70). We employ the realistic Gogny effective interactions using three variants of their parametrisation and the particle-number, parity and the angular-momentum projection techniques. We confirm quantitatively that the resulting spectra directly follow the pattern expected from the group theory consid…
For the first time the potential energy surfaces of actinide nuclei in the $(β_{20}, β_{22}, β_{30})$ deformation space are obtained from a multi-dimensional constrained covariant density functional theory. With this newly developed theory we are able to explore the importance of the triaxial and octupole shapes simultaneously along the whole fission path. It is found that besides the octupole deformation, the triaxiality also plays an important role upon the second fission barriers. The outer barrier as well as the inner barrier are lowered by the triaxial deformation compared with axially sy…
We discuss the octupole deformation properties of the recently proposed Barcelona-Catania-Paris (BCP) energy density functionals for two sets of isotopes, those of radium and barium, where it is believed that octupole deformation plays a role in the description of the ground state. The analysis is carried out in the mean field framework (Hartree- Fock- Bogoliubov approximation) by using the axially symmetric octupole moment as a constraint. The main ingredients entering the octupole collective Hamiltonian are evaluated and the lowest lying octupole eigenstates are obtained. In this way we rest…
The phenomenon of reflection-asymmetric nuclear shapes is relevant to nuclear stability, nuclear spectroscopy, nuclear decays and fission, and the search for new physics beyond the standard model. Global surveys of ground-state octupole deformation, performed with a limited number of models, suggest that the number of pear-shaped isotopes is fairly limited across the nuclear landscape. We carry out global analysis of ground-state octupole deformations for particle-bound even-even nuclei with $Z \leq 110$ and $N \leq 210$ using nuclear density functional theory (DFT) with several non-relativist…
In the framework of the symplectic extension of the Interacting Vector Boson Model (IVBM) a good description of the first excited positive and negative parity bands of the nuclei in the rare earth and the actinide region is achieved. The bands investigated in the model are extended to very high angular momenta as a result of their consideration as "yrast" bands with respect to the symplectic classification of the basis states. The analysis of the eigenvalues of the model Hamiltonian reveals the presence of an interaction between these bands. Due to this iteraction the Delta(L)=1 staggering eff…
Cluster radioactivity is the emission of a fragment heavier than $α$ particle and lighter than mass 50. The range of clusters observed in experiments goes from $^{14}$C to $^{32}$Si while the heavy mass residue is always a nucleus in the neighborhood of the doubly-magic $^{208}$Pb nucleus. Cluster radioactivity is described in this paper as a very asymmetric nuclear fission. A new fission valley leading to a decay with large fragment mass asymmetry matching the cluster radioactivity products is found. The mass octupole moment is found to be more convenient than the standard quadrupole moment a…
The evolution of quadrupole and octupole collectivity in neutron-deficient Xe, Ba, Ce, and Nd nuclei near the "octupole magic" neutron number $N=56$ is investigated within the mapped $sdf$-IBM framework. Microscopic input is obtained via quadrupole and octupole constrained Hartree-Fock-Bogoliubov calculations, based on the parametrization D1M of the Gogny energy density functional. Octupole-deformed mean-field ground states are predicted for Ba and Ce isotopes near $N=56$. Excitation energies of positive- and negative-parity states as well as electric transition rates are computed with wave fu…
The onset of octupole deformation and its impact on related spectroscopic properties is studied in even-even neutron-rich lanthanide isotopes Xe, Ba, Ce, and Nd with neutron number $86\leqslant N\leqslant 94$. Microscopic input comes from the Hartree-Fock-Bogoliubov approximation with constrains on the axially symmetric quadrupole and octupole operators using the Gogny-D1M interaction. At the mean-field level, reflection asymmetric ground states are predicted for isotopes with neutron number around $N=88$. Spectroscopic properties are studied by diagonalizing the interacting boson model Hamilt…
We propose a collective Hamiltonian which incorporates the standard quadrupole terms, octupole terms classified according to the irreducible representations of the octahedron group, a quadrupole-octupole interaction, as well as a term for the bandhead energy linear in K (the projection of angular momentum on the body-fixed z-axis). The energy is subsequently minimized with respect to K for each given value of the angular momentum I, resulting in K values increasing with I within each band, even in the case in which K is restricted to a set of microscopically plausible values. We demonstrate th…
In the present study, we have investigated the impact of the tensor force on fission paths, in particular the symmetric and asymmetric barriers in 230 Th, 226 Th, 222 Th and 216 Th isotopes which display an asymmetric to symmetric fission transition. This analysis has been performed within the HFB approach with (Q 20 ,Q 30 ,Q 40 ) as collective variable constraints, using the D1ST2a Gogny+tensor term interaction and comparing to the standard D1S Gogny interaction results. The effects from the tensor term on the potential energy surface landscape, and especially on barrier heights and its topol…
A high-resolution $(p,t)$ experiment at $E_{p}$= 24 MeV was performed to populate low-spin states in the actinide nucleus ${}^{240}$Pu. The Q3D magnetic spectrograph of the Maier-Leibnitz Laboratory (MLL) in Munich (Germany) was used to identify the ejected tritons via $dE/E$ particle identification with its focal-plane detection system. Angular distributions were measured at nine different Q3D angles to assign spin and parity to the excited states based on a comparison with coupled-channels DWBA calculations. In total, 209 states have been excited in $^{240}$Pu up to an excitation energy of 3…
Reflection asymmetric, octupole shapes in nuclei are a prominent aspect of nuclear structure, and have been recurrently studied over the decades. Recent experiments using radioactive-ion beams have provided evidence for stable octupole shapes. A variety of nuclear models have been employed for the related theoretical analyses. We review recent studies on the nuclear octupole shapes and collective excitations within the interacting boson model. A special focus is placed on the microscopic formulation of this model by using the mean-field method that is based on the framework of nuclear density …
Multi-dimensional constrained covariant density functional theories were developed recently. In these theories, all shape degrees of freedom β_{λμ} deformations with even μare allowed, e.g., β_{20}, β_{22}, β_{30}, β_{32}, β_{40}, β_{42}, β_{44}, and so on and the CDFT functional can be one of the following four forms: the meson exchange or point-coupling nucleon interactions combined with the non-linear or density-dependent couplings. In this contribution, some applications of these theories are presented. The potential energy surfaces of actinide nuclei in the (β_{20}, β_{22}, β_{30}) deform…
The fission barriers of the nuclei 254Fm, 256Fm, 258Fm, 258No and 260Rf are investigated in a fully microscopic way up to the scission point. The analysis is based on the constrained Hartree-Fock-Bogoliubov theory and Gogny's D1S force. The quadrupole, octupole and hexadecapole moments as well as the number of nucleons in the neck region are used as constraints. Two fission paths, corresponding to the bimodal fission, are found. The decrease with isotope mass of the half-life times of heavy Fm isotopes is also explained.
The low-lying spectra including the first few excited positive and negative parity bands of some heavy even-even nuclei from the rare earth and actinide mass regions are investigated within the framework of the symplectic Interacting Vector Boson Model with Sp(12,$R$) dynamical symmetry group. Symplectic dynamical symmetries allow the change of the number of excitation quanta or phonons building the collective states providing for larger representation spaces and richer subalgebraic structures to incorporate more complex nuclear spectra. The theoretical predictions for the energy levels and th…
Impacts of hexadecapole correlations in actinide nuclei
Institute for Medical Research and Occupational Health（医学研究与职业健康研究所）；Hokkaido University（北海道大学）；Nazarbayev University（纳扎尔巴耶夫大学）；Universidad Autónoma de Madrid（马德里自治大学）；Universidad Politécnica de Madrid（马德里理工大学）
We investigate the systematics of fission barriers in superheavy elements in the range Z = 108-120 and N = 166-182. Results from two self-consistent models for nuclear structure, the relativistic mean-field (RMF) model as well as the non-relativistic Skyrme-Hartree-Fock approach are compared and discussed. We restrict ourselves to axially symmetric shapes, which provides an upper bound on static fission barriers. We benchmark the predictive power of the models examining the barriers and fission isomers of selected heavy actinide nuclei for which data are available. For both actinides and super…
