We present an efficient method for calculating strength functions using the finite amplitude method (FAM) for deformed superfluid heavy nuclei within the framework of the nuclear density functional theory. We demonstrate that FAM reproduces strength functions obtained with the fully self-consistent quasi-particle random-phase approximation (QRPA) at a fraction of computational cost. As a demonstration, we compute the isoscalar and isovector monopole strength for strongly deformed configurations in $^{240}$Pu by considering huge quasi-particle QRPA spaces. Our approach to FAM, based on Broyden's iterative procedure, opens the possibility for large-scale calculations of strength distributions in well-bound and weakly bound nuclei across the nuclear landscape.
Background: To access selected excited states of nuclei, within the framework of nuclear density functional theory, the quasiparticle random phase approximation (QRPA) is commonly used. Purpose: We present a computationally efficient, fully self-consistent framework to compute the QRPA transition strength function of an arbitrary multipole operator in axially-deformed superfluid nuclei. Methods: The method is based on the finite amplitude method (FAM) QRPA, allowing fast iterative solution of QRPA equations. A numerical implementation of the FAM-QRPA solver module has been carried out for defo…
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.
Properties of strongly interacting, two-component finite Fermi systems are discussed within the recently developed coordinate-space Hartree-Fock-Bogoliubov (HFB) code {\hfbax}. Two illustrative examples are presented: (i) weakly bound deformed Mg isotopes, and (ii) spin-polarized atomic condensates in a strongly deformed harmonic trap.
To study shape fluctuations of nuclei in transitional regions, the collective Hamiltonian method has often been employed. We intend to construct the quadrupole collective Hamiltonian with the collective inertial functions given by the local quasiparticle random-phase approximation (QRPA) based on the Skyrme energy density functional. For this purpose, we first construct a practical framework of Skyrme QRPA for triaxial nuclear shapes with the finite amplitude method (FAM). We show quadrupole strength functions for a triaxial superfluid nucleus $^{188}$Os and the Thouless-Valatin rotational mom…
The five-dimensional quadrupole collective model based on energy density functionals (EDF) has often been employed to treat long-range correlations associated with shape fluctuations in nuclei. Our goal is to derive the collective inertial functions in the collective Hamiltonian by the local quasiparticle random phase approximation (QRPA) that correctly takes into account time-odd mean-field effects. Currently, practical framework to perform the QRPA calculation with the modern EDFs on the $(β,γ)$ deformation space is not available. Toward this goal, we develop an efficient numerical method to…
The deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) has been proved one of the best models to probe the exotic structures in deformed nuclei. In DRHBc, the potentials and densities are expressed in terms of the multipole expansion with Legendre polynomials, the dependence on which has only been touched for light nuclei so far. In this paper, taking a light nucleus $^{20}$Ne and a heavy nucleus $^{242}$U as examples, we investigated the dependence on the multipole expansion of the potentials and densities in DRHBc. It is shown that the total energy converges well with the e…
The structure of deformed single-particle wave functions in the vicinity of zero energy limit is studied using a schematic model with a quadrupole deformed finite square-well potential. For this purpose, we expand the single-particle wave functions in multipoles and seek for the bound state and the Gamow resonance solutions. We find that, for the $K^π=0^{+}$ states, where $K$ is the $z$-component of the orbital angular momentum, the probability of each multipole components in the deformed wave function is connected between the negative energy and the positive energy regions asymptotically, alt…
A fully symmetry unrestricted Time-Dependent Density Functional Theory extended to include pairing correlations is used to calculate properties of the isovector giant dipole resonances of the deformed open-shell nuclei 172Yb (axially deformed), 188Os (triaxially deformed), and 238U (axially deformed), and to demonstrate good agreement with experimental data on nuclear photo-absorption cross-sections for two different Skyrme force parametrizations of the energy density functional: SkP and SLy4.
Background: The major challenge for nuclear theory is to describe and predict global properties and collective modes of atomic nuclei. Of particular interest is the response of the nucleus to a time-dependent external field that impacts the low-energy multipole and beta-decay strength. Purpose: We propose a method to compute low-lying collective modes in deformed nuclei within the finite amplitude method (FAM) based on the quasiparticle random-phase approximation (QRPA). By using the analytic property of the response function, we find the QRPA amplitudes by computing the residua of the FAM amp…
The systematic study of isoscalar (IS) and isovector (IV) giant quadrupole responses (GQR) in normal and superfluid nuclei presented in [G. Scamps and D. Lacroix, Phys. Rev. 88, 044310 (2013)] is extended to the case of axially deformed and triaxial nuclei. The static and dynamical energy density functional based on Skyrme effective interaction are used to study static properties and dynamical response functions over the whole nuclear chart. Among the 749 nuclei that are considered, 301 and 65 are respectively found to be prolate and oblate while 54 do not present any symmetry axis. For these …
We study the conditions under which the nucleons inside a deformed nucleus can undergo chaotic motion. To do this we perform self-consistent calculations in semiclassical approximation utilizing a multipole-multipole interaction of the Bohr-Mottelson type for quadrupole and octupole deformations. For the case of harmonic and non-harmonic static potentials, we find that both multipole deformations lead to regular motion of the collective coordinate, the multipole moment of deformation. However, despite this regular collective motion, we observe chaotic single particle dynamics.
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.
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.
Core-particle coupling models are made viable by assuming that core properties such as matrix elements of multipole and pairing operators and excitation spectra are known independently. From the completeness relation, it is seen, however, that these quantities are themselves algebraic functions of the calculated core-particle amplitudes. For the deformed rare-earth nucleus 158Gd, we find that these sum rules are well-satisfied for the ground state band, implying that we have found a self-consistent solution of the non-linear Kerman-Klein equations.
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 discuss the effect of pairing on two-neutron space correlations in deformed nuclei. The spatial correlations are described by the pairing tensor in coordinate space calculated in the HFB approach. The calculations are done using the D1S Gogny force. We show that the pairing tensor has a rather small extension in the relative coordinate, a feature observed earlier in spherical nuclei. It is pointed out that in deformed nuclei the coherence length corresponding to the pairing tensor has a pattern similar to what we have found previously in spherical nuclei, i.e., it is maximal in the interior…
We extend effective field theory to the case of spontaneous symmetry breaking in genuinely finite quantum systems such as small superfluid systems, molecules or atomic nuclei, and focus on deformed nuclei. In finite superfluids, symmetry arguments alone relate the spectra of systems with different particle numbers. For systems with non-spherical intrinsic ground states such as atomic nuclei or molecules, symmetry arguments alone yield the universal features of the low-lying excitations as vibrations that are the heads of rotational bands. The low-lying excitations in deformed nuclei differ fro…
Dipole response in deformed halo nuclei $^{42}\mathrm{Mg}$ and $^{44}\mathrm{Mg}$
State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University（北京大学物理学院核物理与技术国家重点实验室）；School of Physics, Engineering and Technology, University of York（约克大学物理、工程与技术学院）；Center for Theoretical Physics, China Institute of Atomic Energy（中国原子能科学研究院理论物理中心）
We use the quasiparticle random phase approximation with a few Skyrme density functionals to calculate strength functions in the Jpi = 0+, 1-, and 2+ channels for even Ca, Ni, and Sn isotopes, from the proton drip line to the neutron drip line. We show where and how low-lying strength begins to appear as N increases. We also exhibit partial energy-weighted sums of the transition strength as functions of N for all nuclei calculated, and transition densities for many of the interesting peaks. We find that low-energy strength increases with N in all multipoles, but with distinctive features in ea…
In extracting deformation parameters from multipole moments for deformed nuclei, one commonly uses the formulas which are based on a sharp-cut density distribution. We discuss a possible ambiguity for this procedure and clarify the role of diffuseness parameter of the density distribution. For this purpose, we use a deformed Woods-Saxon density as well as a density distribution obtained from the self-consistent relativistic mean-field (RMF) model. We show that the formula using a root-mean-square radius instead of a sharp-cut radius requires a large correction due to a finite surface diffusene…
A separable form of pairing interaction in the $^{1}S_{0}$ channel has been introduced and successfully applied in the description of both static and dynamic properties of superfluid nuclei. By adjusting the parameters to reproduce the pairing properties of the Gogny force in nuclear matter, this separable pairing force is successful in depicting the pairing properties of ground states and vibrational excitations of spherical nuclei on almost the same footing as the original Gogny force. In this article, we extend these investigations for Relativistic Hartree Bogoliubov theory in deformed nucl…
Background : Several recent experiments report significant low-energy isoscalar monopole strength, below the giant resonance, in various nuclei. In light $α$-conjugate nuclei, these low-energy resonances were recently interpreted as cluster vibration modes. However, the nature of these excitations in neutron-rich nuclei remain elusive. Purpose : The present work provides a systematic analysis of the low-energy monopole strength in isotopic chains, from Neon to Germanium, in order to monitor and understand its nature and conditions of emergence. Methods : We perform covariant quasiparticle rand…
Ab initio approaches to the nuclear many-body problem have seen their reach considerably extended over the past decade. However, collective excitations have been scarcely addressed so far due to the prohibitive cost of solving the corresponding equations of motion. Here, a numerically efficient method to compute electromagnetic response functions at zero- and finite-temperature in superfluid and deformed nuclei from an ab initio standpoint is presented and applied to $^{16}$O, $^{28}$Si, $^{46}$Ti and $^{56}$Fe. This work opens the path to systematic ab initio calculations of nuclear responses…
The aim of this work is to develop the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) theory based on the point-coupling density functionals and extend it to provide a unified description for all even-even nuclei in the nuclear chart by overcoming all possible challenges. The nuclear superfluidity is considered via Bogoliubov transformation. Densities and potentials are expanded in terms of Legendre polynomials to include the axial deformation degrees of freedom. Sophisticated relativistic Hartree-Bogoliubov equations in coordinate space are solved in the DiracWoods-Saxon…
Shape deformations and charge radii, basic properties of atomic nuclei, are influenced by both the global features of the nuclear force and the nucleonic shell structure. As functions of proton and neutron number, both quantities show regular patterns and, for nuclei away from magic numbers, they change very smoothly from nucleus to nucleus. In this paper, we explain how the local shell effects are impacting the statistical correlations between quadrupole deformations and charge radii in well-deformed even-even Er, Yb, and Hf isotopes. This implies, in turn, that sudden changes in correlations…
New superheavy nuclei are often identified through their characteristic $α$-decay energies, which requires accurate calculations of $Q_α$ values. While many $Q_α$ predictions are available, little is known about their uncertainties, and this makes it difficult to carry out extrapolations to yet-unknown systems. This work aims to analyze several models, compare their predictions to available experimental data, and study their performance for the unobserved $α$-decay chains of $^{296}$120 and $^{298}$120, which are of current experimental interest. Our quantified results will also serve as a ben…
We analyze the behavior of a gas of classical particles moving in a two-dimensional "nuclear" billiard whose multipole-deformed walls undergo periodic shape oscillations. We demonstrate that a single particle Hamiltonian containing coupling terms between the particles' motion and the collective coordinate induces a chaotic dynamics for any multipolarity, independently on the geometry of the billiard. The absence of coupling terms allows us to recover qualitatively the "wall formula" predictions. We also discuss the dissipative behavior of the wall motion and its relation with the order-to-chao…
In order to understand the origin of one-body dissipation in nuclei, we analyze the behavior of a gas of classical particles moving in a two-dimensional cavity with nuclear dimensions. This "nuclear" billiard has multipole-deformed walls which undergo periodic shape oscillations. We demonstrate that a single particle Hamiltonian containing coupling terms between the particles' motion and the collective coordinate induces a chaotic dynamics for any multipolarity, independently on the geometry of the billiard. If the coupling terms are switched off the "wall formula" predictions are recovered. W…
The Quasi-SU(3) symmetry was uncovered in full pf and sdg shell-model calculations for both even-even and odd-even nuclei. It manifests itself through a dominance of single-particle and quadrupole-quadrupole terms in the Hamiltonian used to describe well-deformed nuclei. A practical consequence of the quasi-SU(3) symmetry is an efficient basis truncation scheme. In a recent work was shown that when this type of Hamiltonian is diagonalized in an SU(3) basis, only a few irreducible represntations (irreps) of SU(3) are needed to describe the Yrast band, the leading S = 0 irrep augmented with the …
From closed shells to open shells: Coupled-cluster calculations of atomic nuclei
Johannes Gutenberg-Universität Mainz（美因茨约翰内斯·古腾堡大学）；Facility for Rare Isotope Beams, Michigan State University（密歇根州立大学稀有同位素束设施）；Oak Ridge National Laboratory（橡树岭国家实验室）；Université Libre de Bruxelles（布鲁塞尔自由大学）；KU Leuven（荷语鲁汶大学）；Helmholtz-Institut Mainz（美因茨亥姆霍兹研究所）；CEA, Université Paris-Saclay（法国原子能和替代能源委员会，巴黎萨克雷大学）；University of Tennessee（田纳西大学）
The nuclear level densities and level-density parameters in fissioning nuclei at their saddle points of fission barriers - $a_{f}$, as well as those for neutron - $a_{n}$, proton - $a_{p}$ , and $α$-particle - $a_α$ emission residues at the ground states are calculated for isotopic chains of superheavy nuclei with $Z$=112-120. The calculations are performed with the superfluid formalism using the single-particle energies obtained from the diagonalization of the deformed Woods-Saxon potential. Spectra were generated at global minima of the adiabatic potential energy surfaces, found by the multi…
