Five-dimensional Collective Hamiltonian Research Articles

Microscopic analysis of structural evolution in the Pt-Hg region

The phenomena of shape-phase transitions and shape coexistence in neutron deficient even-even Pt and Hg isotopes are investigated, using a five-dimensional collective Hamiltonian (5DCH) based on covariant density-functional theory. The triaxial deformation energy surfaces in Pt isotopes display a transition from prolate (188Pt) to triaxial or oblate (190-198Pt), and to near spherical (198Pt) shapes. The calculations suggest coexisting configurations in 190Hg, γ-soft potential energy surfaces in 192-198Hg and a more spherical structure in 200Hg. The corresponding 5DCH model calculations confirm the structural evolution in this region and suggest more increased collectivity than what can be deduced from the data.

Shape evolution of Hg isotopes within the covariant density functional theory

The shape evolution in the neutron-deficient Hg region is investigated within the covariant density functional framework. We study in detail the chain of even–even mercury isotopes 190−200Hg using the relativistic point coupling model. The low-energy excitation spectrum and the B(E2) transitions rates of even–even nuclei are obtained as solutions of a five-dimensional collective Hamiltonian (5DCH) model, with parameters determined by constrained self-consistent mean-field calculations based on the relativistic energy density functional DD-PC1, and a finite-range pairing interaction. The calculations suggest a very interesting structure evolution with coexisting configurations for 190Hg, increased collectivity for the isotopes 192−198Hg and a more spherical structure for 200Hg.

Microscopic analysis of prolate-oblate shape phase transition and shape coexistence in the Er-Pt region

The shape transition and possible occurrence of low-energy shape coexistence and rigid triaxial deformation are analyzed in the six even-even Er, Yb, Hf, W, Os, and Pt isotopic chains with the neutron number $102\ensuremath{\le}N\ensuremath{\le}124$, using a five-dimensional collective Hamiltonian (5DCH) based on covariant density-functional theory. The potential-energy surfaces display a transition from prolate to oblate or triaxial, and then to near spherical shapes as the neutron number increases. The corresponding 5DCH model calculations reproduce the empirical isotopic trend of the characteristic collective observables and confirm the overall shape transition in this region. It is emphasized that a rapid shape transition between prolate and oblate shapes is predicted in Er and Yb isotopic chains while it becomes smooth for higher-$Z$ isotopic chains and signature for rigid triaxial deformation is found in the transitional isotopes, e.g., $^{194}\mathrm{W}$ and $^{192\text{--}196}\mathrm{Os}$ by analyzing the energy staggering and probability density distribution in the $\ensuremath{\gamma}$ bands. Finally, the calculated low-lying spectra for $^{184}\mathrm{Er}$ and $^{186}\mathrm{Yb}$ demonstrate a remarkable multishape coexistence of medium-deformed oblate, medium- and large-deformed prolate shapes in both nuclei.

Structure of 190-200Hg within the covariant density functional theory

We study in detail the chain of even - even mercury isotopes 190-200Hg using the relativistic point coupling model. A five-dimensional collective Hamiltonian (5DCH) model, with parameters determined by constrained self-consistent mean-field (SCMF) calculations based on the relativistic density-dependent pointcoupling (DD-PC1) energy density functional, and a finite-range pairing interaction is used to calculate the low-energy excitation spectrum and the B(E2) transitions rates of even-even nuclei. The calculations suggest coexisting configurations in 190Hg, increased collectivity in the isotopes 192-198Hg and a more spherical structure in 200Hg.

Nuclear chart in covariant density functional theory with dynamic correlations: From oxygen to tin * * Supported by the National Key R&D Program of China (2018YFA0404400, 2017YFE0116700) and the National Natural Science Foundation of China (11621131001, 11875075, 11935003, 11975031)

Nuclear masses of even-even nuclei with the proton number (O to Sn isotopes) from the proton drip line to neutron drip line are investigated using the triaxial relativistic Hartree-Bogoliubov theory with the relativistic density functional PC-PK1. Further, the dynamical correlation energies (DCEs) associated with the rotational motion and quadrupole-shaped vibrational motion are taken into account by the five-dimensional collective Hamiltonian (5DCH) method. The root-mean-square deviation with respect to the experimental masses reduces from 2.50 to 1.59 MeV after the consideration of DCEs. The inclusion of DCEs has little influence on the position of drip lines, and the predicted numbers of bound even-even nuclei between proton and neutron drip lines from O to Sn isotopes are 569 and 564 with and without DCEs, respectively.

β and γ bands in N=88 , 90, and 92 isotones investigated with a five-dimensional collective Hamiltonian based on covariant density functional theory: Vibrations, shape coexistence, and superdeformation

A comprehensive systematic study is made for the collective β and γ bands in even-even isotopes with neutron numbers N=88 to 92 and proton numbers Z=62(Sm) to 70 (Yb). Data, including excitation energies, B(E0) and B(E2) values, and branching ratios from previously published experiments are collated with new data presented for the first time in this study. The experimental data are compared to calculations using a five-dimensional collective Hamiltonian (5DCH) based on the covariant density functional theory (CDFT). A realistic potential in the quadrupole shape parameters V(β,γ) is determined from potential energy surfaces (PES) calculated using the CDFT. The parameters of the 5DCH are fixed and contained within the CDFT. Overall, a satisfactory agreement is found between the data and the calculations. In line with the energy staggering S(I) of the levels in the 2γ+ bands, the potential energy surfaces of the CDFT calculations indicate γ-soft shapes in the N=88 nuclides, which become γ rigid for N=90 and N=92. The nature of the 02+ bands changes with atomic number. In the isotopes of Sm to Dy, they can be understood as β vibrations, but in the Er and Yb isotopes the 02+ bands have wave functions with large components in a triaxial superdeformed minimum. In the vicinity of Sm152, the present calculations predict a soft potential in the β direction but do not find two coexisting minima. This is reminiscent of Sm152 exhibiting an X(5) behavior. The model also predicts that the 03+ bands are of two-phonon nature, having an energy twice that of the 02+ band. This is in contradiction with the data and implies that other excitation modes must be invoked to explain their origin.20 MoreReceived 4 July 2018Revised 5 June 2019DOI:https://doi.org/10.1103/PhysRevC.100.044324©2019 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCollective levelsCollective modelsElectromagnetic transitionsLow & intermediate energy heavy-ion reactionsNuclear density functional theoryNuclear spin & parityNucleon induced nuclear reactionsProperties150 ≤ A ≤ 189Nuclear Physics

Effective field theory for triaxially deformed nuclei

Effective field theory (EFT) is generalized to investigate the rotational motion of triaxially deformed even-even nuclei. A Hamiltonian, called the triaxial rotor model (TRM), is obtained up to next-to-leading order (NLO) within the EFT formalism. Its applicability is examined by comparing with a five-dimensional collective Hamiltonian (5DCH) for the description of the energy spectra of the ground state and $\gamma$ band in Ru isotopes. It is found that by taking into account the NLO corrections, the ground state band in the whole spin region and the $\gamma$ band in the low spin region are well described. The results presented here indicate that it should be possible to further generalize the EFT to triaxial nuclei with odd mass number.

Global analysis of quadrupole shape invariants based on covariant energy density functionals

Coexistence of different geometric shapes at low energies presents a universal structure phenomenon that occurs over the entire chart of nuclides. Studies of the shape coexistence are important for understanding the microscopic origin of collectivity and modifications of shell structure in exotic nuclei far from stability. The aim of this work is to provide a systematic analysis of characteristic signatures of coexisting nuclear shapes in different mass regions, using a global self-consistent theoretical method based on universal energy density functionals and the quadrupole collective model. The low-energy excitation spectrum and quadrupole shape invariants of the two lowest $0^{+}$ states of even-even nuclei are obtained as solutions of a five-dimensional collective Hamiltonian (5DCH) model, with parameters determined by constrained self-consistent mean-field calculations based on the relativistic energy density functional PC-PK1, and a finite-range pairing interaction. The theoretical excitation energies of the states: $2^+_1$, $4^+_1$, $0^+_2$, $2^+_2$, $2^+_3$, as well as the $B(E2; 0^+_1\to 2^+_1)$ values, are in very good agreement with the corresponding experimental values for 621 even-even nuclei. Quadrupole shape invariants have been implemented to investigate shape coexistence, and the distribution of possible shape-coexisting nuclei is consistent with results obtained in recent theoretical studies and available data. The present analysis has shown that, when based on a universal and consistent microscopic framework of nuclear density functionals, shape invariants provide distinct indicators and reliable predictions for the occurrence of low-energy coexisting shapes. This method is particularly useful for studies of shape coexistence in regions far from stability where few data are available.

Five-dimensional collective Hamiltonian with the Gogny force: An ongoing saga

We provide a sample of analyses for nuclear spectroscopic properties based on the five-dimensional collective Hamiltonian (5DCH) implemented with the Gogny force. The very first illustration is dating back to the late 70's. It is next followed by others, focusing on shape coexistence, shape isomerism, superdeformation, and systematics over the periodic table. Finally, the inclusion of Thouless-Valatin dynamical contributions to vibrational mass parameters is briefly discussed as a mean of strengthening the basis of the 5DCH theory.

Novel triaxial structure in low-lying states of neutron-rich nuclei aroundA≈100

Background: In recent years, the study of triaxiality in the low-lying states of atomic nuclei with transition character or shape coexistence has been of great interest. Previous studies indicate that the neutron-rich nuclei in the $A\ensuremath{\sim}100$ mass region with $Z\ensuremath{\sim}40,N\ensuremath{\sim}60$ serve as good grounds for examining the role of triaxiality in nuclear low-lying states.Purpose: The aim of this work is to provide a microscopic study of low-lying states for nuclei in the $A\ensuremath{\sim}100$ mass regions and to examine in detail the role of triaxiality in the shape-coexistence phenomena and the variation of shape with the isospin and spin values at the beyond mean-field level.Method: The starting point of our method is a set of relativistic mean-field plus BCS wave functions generated with a constraint on triaxial deformations $(\ensuremath{\beta},\ensuremath{\gamma})$. The excitation energies and electric multipole transition strengths of low-lying states are calculated by solving a five-dimensional collective Hamiltonian (5DCH) with parameters determined by the mean-field wave functions.Results: The low-lying states of Mo isotopes and of $N=60$ isotones in the $A\ensuremath{\sim}100$ mass region are calculated. The results indicate that triaxiality is essential to reproduce the data of excitation energies and electric quadrupole transition strengths in low-lying states and plays an important role in the shape evolution as a function of nucleon number. However, the decrease of nuclear collectivity with the increase of angular momentum in neutron-rich Mo isotopes has not been reproduced.Conclusions: The evolution of nuclear collectivity in the low-lying states of neutron-rich nuclei in the $A\ensuremath{\sim}100$ mass region as a function of nucleon number is governed by the novel triaxial structure. However, the mechanism that governs the variation of nuclear shape with spin in Mo isotopes remains unclear and deserves further investigation by taking into account the effects other than the collective motions.

Triaxially deformed relativistic point-coupling model forΛhypernuclei: A quantitative analysis of the hyperon impurity effect on nuclear collective properties

The impurity effect of hyperon on atomic nuclei has received a renewed interest in nuclear physics since the first experimental observation of appreciable reduction of $E2$ transition strength in low-lying states of hypernucleus $^{7}_\Lambda$Li. Many more data on low-lying states of $\Lambda$ hypernuclei will be measured soon for $sd$-shell nuclei, providing good opportunities to study the $\Lambda$ impurity effect on nuclear low-energy excitations. We carry out a quantitative analysis of $\Lambda$ hyperon impurity effect on the low-lying states of $sd$-shell nuclei at the beyond-mean-field level based on a relativistic point-coupling energy density functional (EDF), considering that the $\Lambda$ hyperon is injected into the lowest positive-parity ($\Lambda_s$) and negative-parity ($\Lambda_p$) states. We adopt a triaxially deformed relativistic mean-field (RMF) approach for hypernuclei and calculate the $\Lambda$ binding energies of hypernuclei as well as the potential energy surfaces (PESs) in $(\beta, \gamma)$ deformation plane. We also calculate the PESs for the $\Lambda$ hypernuclei with good quantum numbers using a microscopic particle rotor model (PRM) with the same relativistic EDF. The triaxially deformed RMF approach is further applied in order to determine the parameters of a five-dimensional collective Hamiltonian (5DCH) for the collective excitations of triaxially deformed core nuclei. Taking $^{25,27}_{\Lambda}$Mg and $^{31}_{\Lambda}$Si as examples, we analyse the impurity effects of $\Lambda_s$ and $\Lambda_p$ on the low-lying states of the core nuclei...

Global study of beyond-mean-field correlation energies in covariant energy density functional theory using a collective Hamiltonian method

We report the first global study of dynamic correlation energies (DCEs) associated with rotational motion and quadrupole shape vibrational motion in a covariant energy density functional (CEDF) for 575 even-even nuclei with proton numbers ranging from $Z=8$ to $Z=108$ by solving a five-dimensional collective Hamiltonian, the collective parameters of which are determined from triaxial relativistic mean field plus BCS calculation using the PC-PK1 force. After taking into account these beyond mean-field DCEs, the root-mean-square (rms) deviation with respect to nuclear masses is reduced significantly down to 1.14 MeV, which is smaller than those of other successful CEDFs: NL3* (2.96 MeV), DD-ME2 (2.39 MeV), $\mathrm{DD}\text{\ensuremath{-}}\mathrm{ME}\ensuremath{\delta}$ (2.29 MeV), and DD-PC1 (2.01 MeV). Moreover, the rms deviation for two-nucleon separation energies is reduced by $\ensuremath{\sim}34%$ in comparison with the cranking prescription.

Spectroscopy of 74Ge: From soft to rigid triaxiality

The first heavy ion fusion–evaporation reaction study for 74Ge has been performed through the reaction channel Zn70(Li7,2np)Ge74 at beam energies of 30 and 35 MeV. Previously known yrast band is extended to higher spins and five new collective bands are established. Based on comparison with the neighboring 72,76Ge isotopes, an intermediate pattern of energy staggering S(I) is observed in the γ band of 74Ge. The collective structure of 74Ge, including the excitation energies and transition probabilities of ground-state band and γ band, is reproduced by the state-of-the-art five-dimensional collective Hamiltonian (5DCH) model constructed from the covariant density functional. By including the 72,76,78Ge isotopes, systematical investigation of the structure evolution in Ge isotopes is performed. Based on the systematic comparisons and analysis, the triaxial evolution with spin in 74Ge is revealed and 74Ge is found to be the crucial nucleus marking the triaxial evolution from soft to rigid in Ge isotopes.

Beyond relativistic mean-field studies of low-lying states in neutron-deficient krypton isotopes

Neutron-deficient krypton isotopes are of particular interest due to the coexistence of oblate and prolate shapes in low-lying states and the transition of ground-state from one dominate shape to another as a function of neutron number. A detailed interpretation of these phenomena in neutron-deficient Kr isotopes requires the use of a method going beyond a mean-field approach that permits to determine spectra and transition probabilities. The aim of this work is to provide a systematic calculation of low-lying state in the even-even 68-86Kr isotopes and to understand the shape coexistence phenomenon and the onset of large collectivity around N=40 from beyond relativistic mean-field studies. The starting point of our method is a set of relativistic mean-field+BCS wave functions generated with a constraint on triaxial deformations (beta, gamma). The excitation energies and electric multipole transition strengths of low-lying states are calculated by solving a five-dimensional collective Hamiltonian (5DCH) with parameters determined by the mean-field wave functions. To examine the role of triaxiality, a configuration mixing of both particle number (PN) and angular momentum (AM) projected axially deformed states is also carried out within the exact generator coordinate method (GCM) based on the same energy density functional. The energy surfaces, the excitation energies of 0^+_2, 2^+_1, 2^+_2 states, as well as the E0 and E2 transition strengths are compared with the results of similar 5DCH calculations but with parameters determined by the non-relativistic mean-field wave functions, as well as with the available data...

Impurity effect of Lambda hyperon on collective excitations of nuclear core in MgΛ25

Taking the ground state rotational band in $^{24}$Mg as an example, we investigate the impurity effect of $\Lambda$ hyperon on collective excitations of atomic nuclei in the framework of non-relativistic energy density functional theory. To this end, we take into account correlations related to the restoration of broken symmetries and fluctuations of collective variables by solving the eigenvalue problem of a five-dimensional collective Hamiltonian for quadrupole vibrational and rotational degrees of freedom. The parameters of the collective Hamiltonian are determined with constrained mean-field calculations for triaxial shapes using the SGII Skyrme force. We compare the low-spin spectrum for $^{24}$Mg with the spectrum for the same nucleus inside $^{25}_{\Lambda}$Mg. It is found that the $\Lambda$ hyperon stretches the ground state band and reduces the $B(E2:2^+_1 \rightarrow 0^+_1)$ value by $\sim 9%$, mainly by softening the potential energy surface towards the spherical shape, even though the shrinkage effect on the average proton radius is only $\sim0.5%$.

New parametrization for the nuclear covariant energy density functional with a point-coupling interaction

A new parametrization PC-PK1 for the nuclear covariant energy density functional with nonlinear point-coupling interaction is proposed by fitting to observables for 60 selected spherical nuclei, including the binding energies, charge radii and empirical pairing gaps. The success of PC-PK1 is illustrated in its description for infinite nuclear matter and finite nuclei including the ground-state and low-lying excited states. Particularly, PC-PK1 improves the description for isospin dependence of binding energy along either the isotopic or the isotonic chains, which makes it more reliable for application in exotic nuclei. The predictive power of PC-PK1 is also illustrated for the nuclear low-lying excitation states in a five-dimensional collective Hamiltonian in which the parameters are determined by constrained calculations for triaxial shapes.

Beyond the relativistic mean-field approximation. III. Collective Hamiltonian in five dimensions

The framework of relativistic energy density functionals is extended to include correlations related to the restoration of broken symmetries and fluctuations of collective variables. A new implementation is developed for the solution of the eigenvalue problem of a five-dimensional collective Hamiltonian for quadrupole vibrational and rotational degrees of freedom, with parameters determined by constrained self-consistent relativistic mean-field calculations for triaxial shapes. The model is tested in a series of illustrative calculations of potential energy surfaces and the resulting collective excitation spectra and transition probabilities of the chain of even-even gadolinium isotopes.

Description of the even-mass Hg and Pt isotopes by a microscopic collective hamiltonian

The collective hamiltonian of the deformed nucleus is microscopically calculated in a large basis of deformed Woods-Saxon single-particle states. The potential energy ( V def) and the inertia parameters for rotation ( J k ) and vibration ( B ββ , B γγ , B βγ ) are obtained as functions of the intrinsic quadrupole deformation (β, γ), using the Strutinsky shell-correction method and the cranking formula, respectively. The five-dimensional collective hamiltonian is solved in a numerical method. The model is employed to calculate the collective states of 196Hg. We give the theoretical results of the potential energy, the inertia parameters, the collective energy spectrum and wave functions, transition strengths for E2, M1, E0 and the spectroscopic quadrupole and magnetic moments. Good agreement with the experiment is obtained in this model without adjustment of any model parameter. We also give the potential energies and spectra for the neighbouring even mass nuclei, 192–200Hg and 190–198Pt.