The validity of the rotational formula used to compute E1 and E3 transition strengths in even-even nuclei is analyzed within the Generator Coordinate Method framework based on mean field wave functions. It turns out that those nuclei with spherical or near spherical shapes the E1 and E3 strengths computed with this formula are strongly underestimated and a sound evaluation of them requires angular momentum projected wave functions. Results for several isotopic chains with proton number equal or near magic numbers are analyzed and compared with experimental data. The use of angular momentum projected wave functions greatly improves the agreement with the scarce experimental data.
Long-lived opposite-parity states and the onset of octupole collectivity in atomic nuclei
San Diego State University（圣地亚哥州立大学）；Ho Chi Minh City University of Education（胡志明市教育大学）；Institute for Basic Science（基础科学研究院）；Tel Aviv University（特拉维夫大学）
Octupole correlations in the low-energy collective states of neutron-rich nuclei with the neutron number $N\approx56$ are studied within the interacting boson model (IBM) that is based on the nuclear density functional theory. The constrained self-consistent mean-field (SCMF) calculations using a universal energy density functional and a pairing interaction provide the potential energy surfaces in terms of the axially-symmetric quadrupole and octupole deformations for the even-even nuclei $^{86-94}$Se, $^{88-96}$Kr, $^{90-98}$Sr, $^{92-100}$Zr, and $^{94-102}$Mo. The SCMF energy surface is the…
A model is presented that simultaneously describes shape coexistence and quadrupole and octupole collective excitations within a theoretical framework based on the nuclear density functional theory and the interacting boson model. An optimal interacting-boson Hamiltonian that incorporates the configuration mixing between normal and intruder states, as well as the octupole degrees of freedom, is identified by means of self-consistent mean-field calculations using a universal energy density functional and a pairing interaction, with constraints on the triaxial quadrupole and the axially-symmetri…
Fission of $^{180}$Hg produces mass asymmetric fragments which are expected to be influenced by deformed shell-effects at N=56 in the heavy fragment and Z=34 in the light fragment [G. Scamps and C. Simenel, arXiv:1904.01275 (2019)]. To investigate both shell-effects and to determine which one has the main influence on the asymmetry in the region of the $^{180}$Hg, we produce a systematic of Constraint-Hartree-Fock calculations in nuclei with similar N/Z ratio than the $^{178}$Pt. It is found that N=56 determines the asymmetry of systems in this region of the nuclear chart.
The model of coherent quadrupole and octupole motion (CQOM) is applied to describe non-yrast split parity-doublet spectra in odd-mass nuclei. The yrast levels are described as low-energy rotation-vibration modes coupled to the ground single-particle (s.p.) state, while the non-yrast parity-doublet structures are obtained as higher-energy rotation-vibration modes coupled to excited s.p. states. It is shown that the extended model scheme describes both the yrast and non-yrast quasi parity-doublet spectra and the related B(E1) and B(E2) transition rates in different regions of heavy odd-A nuclei.…
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 …
The model, introduced in a previous paper, for the description of the octupole and quadrupole degrees of freedom in conditions close to the axial symmetry, is used to describe the negative-parity band based on the first octupole vibrational state in nuclei close to the critical point of the U(5) to SU(3) phase transition. The situation of 150Nd and 152Sm is discussed in detail. The positive parity levels of these nuclei, and also the in-band E2 transitions, are reasonably accounted for by the X(5) model. With simple assumptions on the nature of the octupole vibrations, it is possible to descri…
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 …
Origin of octupole deformation softness in atomic nuclei
Center for Exotic Nuclear Studies, Institute for Basic Science (IBS)（基础科学研究所）；Department of Physics, Ho Chi Minh City University of Education（胡志明市教育大学物理系）；Department of Theoretical Physics, Faculty of Physics and Engineering Physics, University of Science（理科大学物理与工程物理学院理论物理系）；Vietnam National University, Ho Chi Minh City（胡志明市越南国立大学）；Rare Isotope Science Project, Institute for Basic Science（基础科学研究所稀有同位素科学项目）；School of Physics and Astronomy, Tel Aviv University（特拉维夫大学物理与天文学学院）
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…
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 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…
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…
Octupole vibrations are studied in some doubly magic nuclei using the time-dependent Hartree-Fock (TDHF) theory with a Skyrme energy density functional. Through the use of the linear response theory, the energies and transition amplitudes of the low-lying vibrational modes for each of the nuclei were determined. Energies were found to be close to experimental results. However, transition amplitudes, quantified by the deformation parameter $β_3$, are underestimated by TDHF. A comparison with single-particle excitations on the Hartree-Fock ground-state shows that the collective octupole vibratio…
A systematic investigation of octupole deformed nuclei is presented for even-even systems with $Z\leq 106$ located between the two-proton and two-neutron drip lines. For this study we use five most up-to-date covariant energy density functionals of different types, with a non-linear meson coupling, with density dependent meson couplings, and with density-dependent zero-range interactions. Pairing correlations are treated within relativistic Hartree-Bogoliubov (RHB) theory based on an effective separable particle-particle interaction of finite range. This allows us to assess theoretical uncerta…
[Background] Predictions of spectroscopic properties of low-lying states are critical for nuclear structure studies. Theoretical methods can be particularly involved for odd-mass nuclei because of the interplay between the unpaired nucleon and collective degrees of freedom. Only a few models have been developed for systems in which octupole collective degrees of freedom play a role. [Purpose] We aim to predict spectroscopic properties of odd-mass nuclei characterized by octupole shape deformation, employing a model that describes single-particle and collective degrees of freedom within the sam…
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 …
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…
There is sparse direct experimental evidence that atomic nuclei can exhibit stable pear shapes arising from strong octupole correlations. In order to investigate the nature of octupole collectivity in radium isotopes, electric octupole ($E3$) matrix elements have been determined for transitions in $^{222,228}$Ra nuclei using the method of sub-barrier, multi-step Coulomb excitation. Beams of the radioactive radium isotopes were provided by the HIE-ISOLDE facility at CERN. The observed pattern of $E$3 matrix elements for different nuclear transitions is explained by describing $^{222}$Ra as pear…
We have performed a systematic study of the ground state for even-even 1,002 nuclei in which 28 octupole-deformed nuclei are found. The interplay between the spatial deformation and the pairing correlation, plays important roles in nuclear structure. Our model is based on the Skyrme Hartree-Fock plus BCS model represented in the three-dimensional Cartesian coordinate space which can describe any kind of nuclear shape. The quadrupole and octupole deformed nuclei appear in the mass region with characteristic neutron and proton numbers which are consistent with previous studies. In our results, t…
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…
A solvable model is proposed for the description of octupole phonons in closed-shell nuclei, formulated in terms of shell-model par\-ticle--hole excitations. With some simple assumptions concerning single-particle energies and two-body interactions, closed expressions are derived for the energy and wave function of the octupole phonon. In particular, it is shown that the components of the octupole phonon are proportional to Wigner $3j$ coefficients. This analytic wave function is proven to be exactly valid in light nuclei, which have $LS$ shell closures that coincide with those of the three-di…
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.…
We show that the potential in the radial equation in the model of coherent quadrupole-octupole motion (CQOM) in nuclei generates a sequence of superpotentials and subsequent series of effective potentials which satisfy the shape-invariance condition and correspond to a SUSY-QM hierarchy of Hamiltonians. On this basis we suggest that the original CQOM level scheme possesses a generic supersymmetric structure of the spectrum inherent for the coherent quadrupole-octupole mode. We outline the mechanism in which the real quadrupole-octupole spectra in even-even and odd-even nuclei deviate from the …
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…
Nuclei with an octupole deformation have a non-zero electric octupole moment, electric dipole moment (EDM), Schiff moment and magnetic quadrupole moment (MQM) in the intrinsic frame which rotates with the nucleus. In a state with definite angular momentum in the laboratory frame, these moments are forbidden by parity (P) and time reversal invariance (T) conservation, meaning their expectation values vanish due to nuclear rotation. However, nuclei with an octupole deformation have doublets of close opposite parity rotational states with the same spin, which are mixed by T,P-odd nuclear forces. …
Quadrupole and octupole deformation energy surfaces, low-energy excitation spectra and transition rates in fourteen isotopic chains: Xe, Ba, Ce, Nd, Sm, Gd, Rn, Ra, Th, U, Pu, Cm, Cf, and Fm, are systematically analyzed using a theoretical framework based on a quadrupole-octupole collective Hamiltonian (QOCH), with parameters determined by constrained reflection-asymmetric and axially-symmetric relativistic mean-field calculations. The microscopic QOCH model based on the PC-PK1 energy density functional and $δ$-interaction pairing is shown to accurately describe the empirical trend of low-ener…
Nuclei with static intrinsic octupole deformation or a soft octupole vibrational mode lead to strongly enhanced collective nuclear Schiff? moments. Interaction between electrons and these Schiff moments produce enhanced time reversal (T) and parity (P) violating electric dipole moments (EDM) in atoms and molecules. Corresponding experiments may be used to test CP-violation theories predicting T,P-violating nuclear forces and to search for axions. Nuclear octupole deformations are predicted in many short lived isotopes. This paper investigates octupole deformations in stable and very long lifet…
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…
