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 cluster structure should have molecular orbitals built on that structure. This is demonstrated through a series of examples that range from $^{13}$C to $^{57}$Ni.
We consider the possibility of identifying nuclei exhibiting the partial dynamical SU(3) symmetry (SU(3)-PDS) as those having excitation energy ratio R(4/2)>3.00 . For this purpose, the level energy spectra of a set of 51 nuclei in the rare earth and actinide regions which presenting an axially deformed prolate rotational structure were analyzed via nuclear partial dynamical SU(3) symmetry in the framework of interacting boson model, to see if the SU(3)-PDS is broadly applicable, and where, how, and in which nuclei it breaks down. Overall, the PDS works very well, the predictions of such inter…
We use the considered axial deformed relativistic mean field theory to perform systematical calculations for Z=112 and 104 isotopic chains with force parameters NL3, NL-SH and NL-Z2 sets. Three deformed chains (oblate, moderate prolate and super-deformed chain) are found for Z=112 and 104 isotopic chains. It is found that there is a chain of super-deformed nuclei which can increase the stability of superheavy nuclei in the Z=112 isotopic chain. Shape coexistence is found for Z=112, 104 isotopic chain and the position is defined. For moderate prolate deformed chains of Z=112 and 104, there is s…
In a nucleus which has two Hartree states deformed in quadrupole symmetry, i.e., prolate state and oblate state, the nuclear residual interaction derived in the present theory beyond the Hartree approximation acts as the restoring force for the spherical symmetry of the nuclear system to be recovered so that the deformed nucleus makes collective tunneling transitions between prolate states and oblate states. We derive the Hamiltonian that is effective not only for the Hartree states but also for the collective tunneling transitions between the Hartree states. Solving the secular equation for t…
We study the breathing oscillations in bose-fermi mixtures with Yb isotopes in the largely prolate deformed trap, which are realized by Kyoto group. We choose the three combinations of the Yb isotopes, Yb170-Yb171, Yb170-Yb173 and Yb174-Yb173, whose boson-fermion interactions are weakly repulsive, strongly attractive and strongly repulsive. The collective oscillations in the deformed trap are calculated in the dynamical time-development approach, which is formulated with the time-dependent Gross-Pitaevskii and the Vlasov equations. We analyze the results in the time-development approach with t…
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.
Using a simple model based on the knowledge of spherical and deformed Woods-Saxon potentials, it is shown that the recent observation of halo phenomena in $^{29}$F can be interpreted as an evidence for the prolate deformation of the ground state of $^{29}$F. The prolate deformation is the result of the shell structure, which is unique in one-neutron resonant levels, in particular near degeneracy of the neutron 1$f_{7/2}$ and 2$p_{3/2}$ resonant levels, together with the strong preference of prolate shape by the proton number $Z$ = 9. On the other hand, in oxygen isotopes spherical shape is so …
A new approach to the old problem of the predominance of prolate deformations among well deformed nuclei is proposed within the shell model framework. The parameter space is explored using the ensemble of random rotationally-invariant interactions. Subsets with rotational energy ratio $E(4^{+})/E(2^{+})$ and the rigid-rotor relation between the quadrupole moment $Q(2^{+})$ and the transition probability $B(E2;2^{+}\to 0^{+}))$ are found exhibiting prolate predominance. We identify matrix elements of the effective forces responsible for the predominance of prolate deformation.
The observed almost complete dominance of prolate over oblate deformations in the ground states of deformed even-even nuclei is related to the splitting of high $\ell$ "surface" orbits in the Nilsson diagram: on the oblate side the occurrence of numerous strongly avoided crossings which reduce the fanning out of the low $Λ$ orbits, while on the prolate side the same interactions increase the fanning out. It is further demonstrated that the prolate dominance is rather special for the restricted particle number of available nuclei and is not generic for finite systems with mean-field potentials …
The rapid increase of computational power over the last several years has allowed detailed microscopic investigations of the structure of many nuclei in terms of Relativistic Mean Field theories as well as in the framework of the no-core Shell Model. In heavy deformed nuclei, in which microscopic calculations remain a challenge, algebraic models based on the SU(3) symmetry offer specific predictions, parameter-independent in several cases, directly comparable to experimental data. Two different approximate models for heavy deformed nuclei based on the SU(3) symmetry, the pseudo-SU(3) and the p…
We analyze the ability of the three different Liquid Drop Mass (LDM) formulas to describe nuclear masses for nuclei in various deformation regions. Separating the 2149 measured nuclear species in eight sets with similar quadrupole deformations, we show that the masses of prolate deformed nuclei are better described than those of spherical ones. In fact, the prolate deformed nuclei are fitted with an RMS smaller than 750 keV, while for spherical and semi-magic species the RMS is always larger than 2000 keV. These results are found to be independent of pairing. The macroscopic sector of the Dufl…
We show that the Liquid Drop Model is best suited to describe the masses of prolate deformed nuclei than of spherical nuclei. To this end three Liquid Drop Mass formulas are employed to describe nuclear masses of eight sets of nuclei with similar quadrupole deformations. It is shown that they are able to fit the measured masses of prolate deformed nuclei with an RMS smaller than 750 keV, while for the spherical nuclei the RMS is, in the three cases, larger than 2000 keV. The RMS of the best fit of the masses of semi-magic nuclei is also larger than 2000 keV. The parameters of the three models …
Prolate-oblate shape competition and impact on charge radii in Bk isotopes
School of Physics, Zhengzhou University（郑州大学物理学院）；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核技术国家重点实验室）
The origin of the dominance of prolate shapes over oblate ones of the ground states of atomic nuclei is investigated with the Nilsson-Strutinsky method. The number of prolate nuclei among all the deformed even-even nuclei is calculated as a function of the strengths of the spin-orbit and the l^2 terms of the Nilsson potential. The latter simulates a square-well like radial profile of the mean potential. The ratio of prolate nuclei is 86% with the standard strengths corresponding to the actual atomic nuclei. By weakening the spin-orbit potential, the ratio oscillates strongly, having a local mi…
We study the breathing oscillations in bose-fermi mixtures in the axially-symmetric deformed trap of prolate, spherical and oblate shapes, and clarify the deformation dependence of the frequencies and the characteristics of collective oscillations. The collective oscillations of the mixtures in deformed traps are calculated in the scaling method. In largely-deformed prolate and oblate limits and spherical limit, we obtain the analytical expressions of the collective frequencies. The full calculation shows that the collective oscillations become consistent with the analytically-obtained frequen…
Relationship between quantum shell structure and classical periodic orbits is briefly reviewed on the basis of semi-classical trace formula. Using the spheroidal cavity model, it is shown that three-dimensional periodic orbits, which are born out of bifurcation of planar orbits at large prolate deformations, generate the superdeformed shell structure.
Recent particle-particle-$γ$ coincident measurements on a $^{28}\rm Si+{}^{28}Si$ resonance have suggested ''vanishing spin alignments''. New analyses for spin alignments by using a molecular model are reported. Different aspects between di-nuclear systems with prolate deformed nuclei and those with oblate deformed nuclei are clarified.
The nearest neighbor spacing distribution of levels of deformed even-even nuclei classified according to their quadrupole deformation parameter is investigated. The results suggest that the oblate deformed nuclei have more regular spectra than prolate ones.
Shell structure of low-lying neutron resonant levels in axially-symmetric quadrupole-deformed potentials is discussed, which seems analogous to that of weakly-bound neutrons. As numerical examples, nuclei slightly outside the neutron-drip-line, $^{39}_{12}$Mg$_{27}$ and $^{21}_{6}$C$_{15}$, are studied. For the lowest resonance I obtain $I^π$ = $Ω^π$ = 5/2$^{-}$ for $^{39}$Mg which is likely to be prolately deformed, while $I^π$ = $Ω^π$ = 1/2$^{+}$ may be assigned to the nucleus $^{21}$C which may be oblately deformed. Consequently, $^{21}$C will not be observed as a recognizable resonant stat…
Finite amplitude method based on the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc-FAM) is developed and applied to study isoscalar giant monopole resonance in exotic nuclei. Validation of the numerical implementation is examined for $^{208}\textrm{Pb}$. The isoscalar giant monopole resonances for even-even calcium isotopes from $^{40}\textrm{Ca}$ to the last bound neutron-rich nucleus $^{80}\textrm{Ca}$ are calculated, and a good agreement with the available experimental centroid energies is obtained for $^{40-48}\textrm{Ca}$. For the exotic calcium isotopes, e.g., $^{68…
Deformed shell structures in nuclear mean-field potentials are systematically investigated as functions of deformation and surface diffuseness. As the mean-field model to investigate nuclear shell structures in a wide range of mass numbers, we propose the radial power-law potential model, V \propto r^α, which enables a simple semiclassical analysis by the use of its scaling property. We find that remarkable shell structures emerge at certain combinations of deformation and diffuseness parameters, and they are closely related to the periodic-orbit bifurcations. In particular, significant roles …
Approximate analytical solutions in closed form are obtained for the 5-dimensional Bohr Hamiltonian with the Woods-Saxon potential, taking advantage of the Pekeris approximation and the exactly soluble one-dimensional extended Woods-Saxon potential with a dip near its surface. Comparison to the data for several gamma-unstable and prolate deformed nuclei indicates that the potential can describe well the ground state and gamma-1 bands of many prolate deformed nuclei corresponding to large enough "well size" and diffuseness, while it fails in describing the beta-1 bands, due to its lack of a har…
Magnetic dipole(M1) transitions of N=11 nuclei, $^{17}$C and $^{21}$Ne are investigated by using shell model and deformed Skyrme Hartree-Fock+blocked BCS wave functions. Shell model calculations predict well observed energy spectra and magnetic dipole transitions in $^{21}$Ne, while the results are rather poor to predict these observables in $^{17}$C. In the deformed HF calculations, the ground states of two nuclei are shown to have large prolate deformations close to $β_2$=0.4. It is also pointed out that the first $K^π=1/2^+$ state in $^{21}$Ne is prolately deformed, while the first $K^π=1/2…
The evolution of shapes and low-energy shape coexistence is analyzed in neutron-deficient Nd and Sm nuclei, using a five-dimensional quadrupole collective Hamiltonian (5DCH). Deformation energy surfaces, calculated with the relativistic energy density functional PC-PK1 and a separable finite-range pairing interaction, exhibit a transition from spherical shapes near $N=80$, to $γ-$soft shapes, and to prolate deformed minima in lighter isotopes. The corresponding 5DCH model calculation, based on the self-consistent mean-field potentials, reproduces the empirical isotopic trend of characteristic …
The shape of a nucleus is one of fundamental nuclear properties. We perform a systematic investigation of bubble nuclei that also exhibit shape coexistence in Hf, W, Os, Pt and Hg even-even isotopes using the deformed relativistic Hartree-Bogoliubov theory in continuum. For a systematic study, we first consider nuclear bubble structures and shape coexistence separately. We confirm that deformations and pairing correlations hinder bubble structures by comparing our results with those from relativistic continuum Hartree-Bogoliubov theory that assumes spherical symmetry in nuclei. We then predict…
In order to describe the exotic nuclear structure in unstable odd-$A$ or odd-odd nuclei, the deformed relativistic Hartree Bogoliubov theory in continuum has been extended to incorporate the blocking effect due to the odd nucleon. For a microscopic and self-consistent description of pairing correlations, continuum, deformation, blocking effects, and the extended spatial density distribution in exotic nuclei, the deformed relativistic Hartree Bogoliubov equations are solved in a Woods-Saxon basis in which the radial wave functions have a proper asymptotic behavior at large $r$. The formalism an…
The deformations of multi-$\la$ hypernuclei corresponding to even-even core nuclei ranging from $^8$Be to $^{40}$Ca with 2, 4, 6, and 8 hyperons are studied in the framework of the deformed Skyrme-Hartree-Fock approach. It is found that the deformations are reduced when adding 2 or 8 $\la$ hyperons, but enhanced when adding 4 or 6 $\la$ hyperons. These differences are attributed to the fact that $\la$ hyperons are filled gradually into the three deformed $p$ orbits, of which the [110]1/2$^-$ orbit is prolately deformed and the degenerate [101]1/2$^-$ and [101]3/2$^-$ orbits are oblately deform…
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 …
The effect of an octupole term in a quadrupole deformed single particle potential is studied from the classical and quantum mechanical view point. Whereas the problem is nonintegrable, the quantum mechanical spectrum nevertheless shows some shell structure in the superdeformed prolate case for particular, yet fairly large octupole strengths; for spherical or oblate deformation the shell structure disappears. This result is associated with classical periodic orbits which are found by employing the removal of resonances method; this approximation method allows determination of the shape of the o…
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…
We study the properties of the spin dependent one body density in momentum space for odd--A polarized deformed nuclei within the mean field approximation. We derive analytic expressions connecting intrinsic and laboratory momentum distributions. The latter are related to observable transition densities in {\bf p}--space that can be probed in one nucleon knock--out reactions from polarized targets. It is shown that most of the information contained in the intrinsic spin dependent momentum distribution is lost when the nucleus is not polarized. Results are presented and discussed for two prolate…
We have investigated the Gamow-Teller transitions from ${}^{22}\textrm{Ne}$ to ${}^{22}\textrm{Na}$ by applying the isospin projected antisymmetrized molecular dynamics combined with generator coordinate method. We have found that the GT strength from $^AZ(J^πT)={}^{22}\textrm{Ne}(0_1^+1)$ is fragmented into two final states ${}^{22}\textrm{Na}(1_{1,2}^+0)$, which belong to $K=0$ and $K=1$ bands constructed by a prolately deformed $^{20}$Ne core with a $S=1$ proton-neutron ($pn$) pair. Coupling of the intrinsic-spin of the $pn$ pair with the core deformation play an important role in the GT fr…
Temperature and angular momentum induced shape changes in the well deformed 100 Nb have been investigated within the theoretical framework of Statistical theory combined with triaxially deformed Nilson potential and Strutinsky prescription. Two shape coexistence, one in the ground state of 104 Nb between oblate and triaxial shapes and another one between oblate and rarely seen prolate non-collective shapes in excited hot rotating 100 Nb at the mid spin values around 14-16h are reported for the first time. The level density parameter indicates the influence of the shell effects and changes dras…
The structures of the ground and excited states of sd-shell nuclei: Ca-40 and Ar-36 were studied with a method of antisymmetrized molecular dynamics. Recently observed rotational bands were described with deformed intrinsic states which are dominated by excited configurations such as 4p-4h, 8p-8h and 4p-8h states. The results showed the coexistence of various kinds of exotic shapes with oblate, prolate and large prolate deformatios in the low-energy region of Ca-40. Possible cluster aspects in heavy sd-shell nuclei were implied in these exotic shapes.
The neutron drop is firstly investigated in an axially symmetric harmonic oscillator (ASHO) field, whose potential strengths of different directions can be controlled artificially. The shape of the neutron drop will change from spherical to oblate or prolate according to the anisotropy of the external field. With the potential strength increasing in the axial direction, the neutron prefers to occupy the orbital perpendicular to the symmetry axis. On the contrary, the neutron likes to stay in the orbital parallel to the symmetry axis when the potential strength increases in the radial direction…
A deformed relativistic Hartree-Bogoliubov theory in continuum has been developed for the study of neutron halos in deformed nuclei and the halo phenomenon in deformed weakly bound nuclei is investigated. Magnesium and neon isotopes are studied and some results are presented for the deformed neutron-rich and weakly bound nuclei 44Mg and 36Ne. The core of the former 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 existence of halos in deformed nuclei and for the occurr…
Gamow-Teller (GT) strength distributions of Mg isotopes are investigated within a framework of the deformed quasi-particle random phase approximation(DQRPA). We found that the N=20 shell closure in $^{28 \sim 34}$Mg was broken by the prolate shape deformation originating from the {\it fp}-intruder states. The shell closure breaking gives rise to a shift of low-lying GT excited states into high-lying states. Discussions regarding the shell evolution trend of single particle states around N=20 nuclei are also presented with the comparison to other approaches.
A microscopic theory of linear response based on the Vlasov equation is extended to systems having spheroidal equilibrium shape. The solution of the linearized Vlasov equation, which gives a semiclassical version of the random phase approximation, is studied for electrons moving in a deformed equilibrium mean field. The deformed field has been approximated by a cavity of spheroidal shape, both prolate and oblate. Contrary to spherical systems, there is now a coupling among excitations of different multipolarity induced by the interaction among constituents. Explicit calculations are performed …
We have studied positive-parity states of $^{28}$Si using antisymmetrized molecular dynamics (AMD) and multi-configuration mixing (MCM) with constrained variation. Applying constraints to the cluster distance and the quadrupole deformation of the variational calculation, we have obtained basis wave functions that have various structures such as $α$-$^{24}$Mg and $^{12}$C-$^{16}$O cluster structures as well as deformed structures. Superposing those basis wave functions, we have obtained a oblate ground state band, a $β$ vibration band, a normal-deformed prolate band, and a superdeformed band. I…
Elastic electron scattering from oriented odd-Aaxially deformed nuclei is studied in the plane-wave Born approximation. The nuclear structure is described within a microscopic selfconsistent Skyrme deformed Hartree-Fock approximation with pairing correlations. The interference form factors between monopole and quadrupole Coulomb terms that characterize the nuclear response with aligned nuclear targets are shown to increase or decrease the unpolarized cross-section, depending on the sign of the quadrupole deformation. This feature provides valuable information on the nuclear deformation that ca…
It is argued that there exist natural shell model spaces optimally adapted to the operation of two variants of Elliott' SU3 symmetry that provide accurate predictions of quadrupole moments of deformed states. A selfconsistent Nilsson-like calculation describes the competition between the realistic quadrupole force and the central field, indicating a {\em remarkable stability of the quadruplole moments}---which remain close to their quasi and pseudo SU3 values---as the single particle splittings increase. A detailed study of the $N=Z$ even nuclei from $^{56}$Ni to $^{96}$Cd reveals that the reg…
We investigate the evolution of structure in the zirconium isotopes where one of the most complex situations encountered in nuclear physics occurs. We demonstrate the role of two concurrent types of quantum phase transitions, sharing a common critical point. The first type, involves an abrupt crossing of coexisting normal and intruder configurations. The second type, involves a gradual shape-phase transition within the intruder configuration, changing from weakly-deformed to prolate-deformed and finally to gamma-unstable. Evidence for this scenario is provided by a detailed comparison with exp…
The effect of an axially symmetric hexadecapole term is investigated in a strongly deformed quadrupole potential. While the system is nonintegrable and shows significant chaotic behaviour classically, the quantum mechanical treatment not only produces a general smoothing effect with regard to chaos but even yields a pronounced shell structure at certain hexadecapole strength parameter values for oblate and prolate deformation.
We study weakly-bound deformed nuclei based on the coordinate-space Skyrme Hartree-Fock-Bogoliubov approach, in which a large box is employed for treating the continuum and surface diffuseness. Approaching the limit of core-halo deformation decoupling, calculations found an exotic "egg"-like structure consisting of a spherical core plus a prolate halo in $^{38}$Ne, in which the resonant continuum plays an essential role. Generally the halo probability and the decoupling effect in heavy nuclei are reduced compared to light nuclei, due to denser level densities around Fermi surfaces. However, de…
The Deformation Dependent Mass (DDM) Kratzer model is constructed by considering the Kratzer potential in a Bohr Hamiltonian, in which the mass is allowed to depend on the nuclear deformation, and solving it by using techniques of supersymmetric quantum mechanics (SUSYQM), involving a deformed shape invariance condition. Analytical expressions for spectra and wave functions are derived for separable potentials in the cases of gamma-unstable nuclei, axially symmetric prolate deformed nuclei, and triaxial nuclei, implementing the usual approximations in each case. Spectra and B(E2) transition ra…
Electron-capture rates in the medium-mass nuclei $^{48}$Ti, $^{56}$Ni, $^{60}$Zn, and $^{64}$Ge within deformed quasiparticle random-phase approximation
Hanyang University（汉阳大学）；Soongsil University（崇实大学）；RIKEN, Nishina Center for Accelerator-Based Science（理化学研究所西村加速器科学中心）；University of Aizu（会津大学）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Beihang University（北京航空航天大学）；The University of Tokyo（东京大学）；National Astronomical Observatory of Japan（日本国立天文台）
Critical Point Symmetries (CPS) appear in regions of the nuclear chart where a rapid change from one symmetry to another is observed. The first CPSs, introduced by F. Iachello, were E(5), which corresponds to the transition from vibrational [U(5)] to gamma-unstable [O(6)] behaviour, and X(5), which represents the change from vibrational [U(5)] to prolate axially deformed [SU(3)] shapes. These CPSs have been obtained as special solutions of the Bohr collective Hamiltonian. More recent special solutions of the same Hamiltonian, to be described here, include Z(5) and Z(4), which correspond to max…
We study the spherical to prolate-deformed shape transition in $^{144-158}$Sm and $^{146-160}$Gd isotopes with modern calculations beyond the mean field with the Gogny D1S force. We compare the results with the shape-phase transition predicted by the collective hamiltonian model and with the experimental data. Our calculations do not support the existence of a first order phase transition in these isotopic chains in the viewpoint of the Bohr hamiltonian neither the interpretation of the nuclei N=90 as critical points.
Isomeric states in the nuclei along the rapid proton capture process path are studied by the projected shell model. Emphasis is given to two waiting point nuclei Se-68 and Kr-72 that are characterized by shape coexistence. Energy surface calculations indicate that the ground state of these nuclei corresponds to an oblate-deformed minimum, while the lowest state at the prolate-deformed minimum can be considered as a shape isomer. Due to occupation of the orbitals with large K-components, states built upon two-quasiparticle excitations at the oblate-deformed minimum may form high $K$-isomers. Th…
Analytical expressions for spectra and wave functions are derived for a Bohr Hamiltonian, describing the collective motion of deformed nuclei, in which the mass is allowed to depend on the nuclear deformation. Solutions are obtained for separable potentials consisting of a Davidson potential in the beta variable, in the cases of gamma-unstable nuclei, axially symmetric prolate deformed nuclei, and triaxial nuclei, implementing the usual approximations in each case. The solution, called the Deformation Dependent Mass (DDM) Davidson model, is achieved by using techniques of supersymmetric quantu…
