Prolate-oblate shape phase transition is an interesting topic in nuclear structure, which is useful for understanding the intrinsic interactions between nucleons. Recently, the interacting boson model with $SU(3)$ higher-order interactions was proposed, in which the prolate shape and the oblate shape are not described in a mirror symmetric way. This asymmetric description seems more realistic. The level evolutions, $B(E2)$ values and other important indicators showing the prolate-oblate asymmetric transitions are investigated in detail, and realistic structure evolutions from $^{180}$Hf to $^{200}$Hg are compared. A key finding is that, the average deformation of the prolate shape is nearly twice the one of the oblate shape. These results, together with the successful description of the $B(E2)$ anomaly in $^{168,170}$Os, $^{172}$Pt, the $γ$-soft properties of $^{196}$Pt, $^{82}$Kr and the normal states of $^{110}$Cd, support the validity of the new model.
Using a new approximate analytic parameter-free proxy-SU(3) scheme, we make simple predictions of shape observables for deformed nuclei, namely gamma and beta deformation variables, the global feature of prolate dominance and the locus of the prolate-oblate shape transition. The predictions are compared with empirical results.
Using a new approximate analytic parameter-free proxy-SU(3) scheme, we make simple predictions for the global feature of prolate dominance in deformed nuclei and the locus of the prolate-oblate shape transition and compare these with empirical data.
The yrast lines in Kr isotopes with $N=42$, 44, and 46 are investigated in a beyond mean field framework with both prolate-oblate coexistence and quasiparticle alignment taken into account. Quasiparticle orbitals with high-$j$ and low-$Ω$ on the oblate side are shown to be responsible for the sharp backbending observed in $^{82}$Kr, by driving the yrast shape from prolate to oblate. This suggests that quasiparticle alignment may not be neglected in the investigation of the shape evolution along the yrast line.
By means of the adiabatic self-consistent collective coordinate method and the pairing-plus-quadrupole interaction, we have obtained the self-consistent collective path connecting the oblate and prolate local minima in 68Se and 72Kr for the first time. The self-consistent collective path is found to run approximately along the valley connecting the oblate and prolate local minima in the collective potential energy landscape. This result of calculation clearly indicates the importance of triaxial deformation dynamics in oblate-prolate shape coexistence phenomena.
By means of the adiabatic self-consistent collective coordinate method and the pairing plus quadrupole interaction, we have obtained the selfconsistent collective path connecting the oblate and prolate local minima in 68Se for the first time. Along the collective path, the triaxial deformation parameter (gamma) changes between 0 and 60 (degree) keeping the axially symmetric deformation parameter (beta) approximately constant, indicating the importance of triaxial deformation dynamics in the oblate-prolate shape coexistence phenomena.
A critical point symmetry for the prolate to oblate shape phase transition is introduced, starting from the Bohr Hamiltonian and approximately separating variables for $γ=30^{\rm o}$. Parameter-free (up to overall scale factors) predictions for spectra and B(E2) transition rates are found to be in good agreement with experimental data for 194-Pt, which is supposed to be located very close to the prolate to oblate critical point, as well as for its neighbours (192-Pt, 196-Pt).
Nuclear shape transition and oblate-prolate coexistence in $N=Z$ nuclei are investigated within the configuration space ($2p_{3/2}$, $1f_{5/2}$, $2p_{1/2}$, and $1g_{9/2}$). We perform shell model calculations for $^{60}$Zn, $^{64}$Ge, and $^{68}$Se and constrained Hartree-Fock (CHF) calculations for $^{60}$Zn, $^{64}$Ge, $^{68}$Se, and $^{72}$Kr, employing an effective pairing plus quadrupole residual interaction with monopole interactions. The shell model calculations reproduce well the experimental energy levels of these nuclei. From the analysis of potential energy surface in the CHF calcu…
We investigate shape coexistence in a rotating nucleus. We concentrate on the interesting case of 72-Kr which exhibits an interesting interplay between prolate and oblate states as a function of angular momentum. The calculation uses the local harmonic version of the method of self-consistent adiabatic large-amplitude collective motion. We find that the collective behaviour of the system changes with angular momentum and we focus on the role of non-axial shapes.
Using a new approximate analytic parameter-free proxy-SU(3) scheme, simple predictions for the global feature of prolate dominance and for the locus of the prolate-oblate shape transition have been made and compared with empirical data. Emphasis is placed on the mechanism leading to the breaking of the particle-hole symmetry, which is instrumental in shaping up these predictions. It turns out that this mechanism is based on the SU(3) symmetry and the Pauli principle alone, without reference to any specific Hamiltonian.
From a viewpoint of oblate-prolate symmetry and its breaking, we adopt the quadrupole collective Hamiltonian to study dynamics of triaxial deformation in shape coexistence phenomena. It accommodates the axially symmetric rotor model, the $γ$-unstable model, the rigid triaxial rotor model and an ideal situation for the oblate-prolate shape coexistence as particular cases. Numerical solutions of this model yield a number of interesting suggestions. (1) The relative energy of the excited 0+ state can be a signature of the potential shape along the $γ$ direction. (2) Specific E2 transition probabi…
Relativistic mean field theory with the NL3 force is used for producing potential energy surfaces (PES) for series of isotopes suggested as exhibiting critical point symmetries. Relatively flat PES are obtained for nuclei showing the E(5) symmetry, while in nuclei corresponding to the X(5) case, PES with a bump are obtained. The PES corresponding to the Pt chain of isotopes suggest a transition from prolate to oblate shapes at 186-Pt.
The adiabatic selfconsistent collective coordinate method is applied to an exactly solvable multi-O(4) model which simulates nuclear shape coexistence phenomena. Collective mass and dynamics of large amplitude collective motions in this model system are analysed, and it is shown that the method can well describe the tunneling motions through the barrier between the prolate and oblate local minima in the collective potential. Emergence of the doublet pattern is well reproduced.
We consider the possibility that a quantum-mechanical off-center effect may be behind the deformed oblate and prolate shapes of nuclei in nuclear physics. In solid state physics, finite off-center displacements result from the mixing of electronic states through their coupling to vibrational (phonon) modes of appropriate symmetries. This is an example of fermion-boson interaction which may materialize in nuclear physics as well in the form of a coupling of nucleons to the pi-meson field. We carry out calculations to substantiate the proposal.
Anomalous isotope shifts in the chain of light Hg isotopes are investigated by using the Hartree-Fock-Bogoliubov method with the Skyrme SIII, SkI3 and SLy4 forces. The sharp increase in the mean-square radius of the odd mass $^{181-185}$Hg isotopes is well explained in terms of the transition from an oblate to a prolate shape in the ground state of these isotopes. We discuss the polarization energy of time-odd mean-field terms in relation to the blocked level by the odd neutron.
The oblate-prolate shape coexisting/mixing phenomena in proton-rich 68,70,72Se are investigated by means of the adiabatic self-consistent collective coordinate (ASCC) method. The one-dimensional collective path and the collective Hamiltonian describing the large-amplitude shape vibration are derived in a fully microscopic way. The excitation spectra, B(E2) and spectroscopic quadrupole moments are calculated by requantizing the collective Hamiltonian and solving the collective Schroedinger equation. The basic properties of the coexisting two rotational bands in low-lying states of these nuclei …
We present a symmetry-based approach for prolate-oblate and spherical-prolate-oblate shape coexistence, in the framework of the interacting boson model of nuclei. The proposed Hamiltonian conserves the SU(3) and $\overline{\rm SU(3)}$ symmetry for the prolate and oblate ground bands and the U(5) symmetry for selected spherical states. Analytic expressions for quadrupole moments and $E2$ rates involving these states are derived and isomeric states are identified by means of selection rules.
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 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 …
A shape phase transition is demonstrated to occur in 190W by applying the Projected Shell Model, which goes beyond the usual mean-field approximation. Rotation alignment of neutrons in the high-j, i_{13/2} orbital drives the yrast sequence of the system, changing suddenly from prolate to oblate shape at angular momentum 10$\hbar$. We propose observables to test the picture.
We investigate the prolate-to-oblate shape phase transitions in the neutron-rich Pt, Os and Ir nuclei in the mass $A\approx 190$ region. The Hamiltonian of the interacting boson-fermion model, used to describe the odd-mass $^{185-199}$Pt, $^{185-193}$Os and $^{185-195}$Ir isotopes, is partially constructed by using as a microscopic input the results of constrained self-consistent mean-field calculations within the Hartree-Fock-Bogoliubov method with the Gogny force. The remaining few parameters are adjusted to experimental data in the odd systems. In this way the calculations reasonably descri…
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…
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 …
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…
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.
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…
Classical periodic orbits responsible for emergence of the superdeformed shell structures for single-particle motions in spheroidal cavities are identified and their relative contributions to the shell structures are evaluated. Both prolate and oblate superdeformations (axis ratio about 2:1) as well as prolate hyperdeformation (axis ratio about 3:1) are investigated. Fourier transforms of quantum spectra clearly show that three-dimensional periodic orbits born out of bifurcations of planar orbits in the equatorial plane become predominant at large prolate deformations, while butterfly-shaped p…
We investigate the mixing of different shapes in the A~70 region using the adiabatic self-consistent collective method in a rotating nuclei. The calculation is done for 68^Se and 72-78^Kr which are known to show oblate-prolate shape coexistence at low angular momentum. A pairing-plus-quadrupole Hamiltonian, which is known to reproduce reproduce the structure nuclei in this mass region, is used. We calculate the collective path between the oblate and prolate minima and discuss how the collective behaviour of the system changes with increasing angular momentum.
The shape coexistence in the nuclei $^{182-192}$Pb is analyzed within the Hartree-Fock-Bogoliubov approach with the effective Gogny force. A good agreement with the experimental energies is found for the coexisting spherical, oblate and prolate states. Contrary to the established interpretation, it is found that the low-lying prolate and oblate $0^+$ states observed in this mass region are predominantly characterized by neutron correlations whereas the protons behave as spectators rather than playing an active role.
We study the oblate-prolate shape mixing in the low-lying states of proton-rich Kr isotopes using the five-dimensional quadrupole collective Hamiltonian. The collective Hamiltonian is derived microscopically by means of the CHFB (constrained Hartree-Fock-Bogoliubov) + Local QRPA (quasiparticle random phase approximation) method, which we have developed recently on the basis of the adiabatic self-consistent collective coordinate method. The results of the numerical calculation show the importance of large-amplitude collective vibrations in the triaxial shape degree of freedom and rotational eff…
A symmetry-based approach for describing shape-coexistence, is presented in the framework of the interacting boson model of nuclei. It involves a construction of a number-conserving Hamiltonian which preserves the dynamical symmetry of selected bands associated with each shape, while breaking the symmetries in other states. The resulting structure embodies multiple partial dynamical symmetries. The procedure is applied to prolate-oblate and spherical-prolate-oblate coexistence, at and slightly away from the critical points.
Microscopic dynamics of the oblate-prolate shape coexistence/mixing phenomena in 68Se and 72Kr are studied by means of the adiabatic self-consistent collective coordinate (ASCC) method in conjunction with the pairing-plus-quadrupole (P+Q) Hamiltonian including the quadrupole pairing interaction. Quantum collective Hamiltonian is constructed, and excitation spectra, spectroscopic quadrupole moments and quadrupole transition properties are evaluated. The effect of the time-odd pair field on the collective mass (inertia function) of the large-amplitude vibration and the rotational moments of iner…
The consequences of the short range nature of the nucleon-nucleon interaction, which forces the spatial part of the nuclear wave function to be as symmetric as possible, on the pseudo-SU(3) scheme are examined through a study of the collective deformation parameters beta and gamma in the rare earth region. It turns out that beyond the middle of each harmonic oscillator shell possessing an SU(3) subalgebra, the highest weight irreducible representation (the hw irrep) of SU(3) has to be used, instead of the irrep with the highest eigenvalue of the second order Casimir operator of SU(3) (the hC i…
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 prolate-shape predominance of the nuclear ground-state deformation by calculating the masses of more than two thousand even-even nuclei using the Strutinsky method, modified by Kruppa, and improved by us. The influences of the surface thickness of the single-particle potentials, the strength of the spin-orbit potential, and the pairing correlations are investigated by varying the parameters of the Woods-Saxon potential and the pairing interaction. The strong interference between the effects of the surface thickness and the spin-orbit potential is confirmed to persist for six sets …
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…
The adiabatic self-consistent collective coordinate (ASCC) method is applied to the pairing-plus-quadrupole (P + Q) model Hamiltonian including the quadrupole pairing, and the oblate-prolate shape coexistence phenomena in proton-rich nuclei, 68Se and 72Kr, are investigated. It is shown that the collective path connecting the oblate and prolate local minima runs along a triaxial valley in the beta-gamma plane. Quantum collective Hamiltonian is constructed and low-lying energy spectra and E2 transition probabilities are calculated for the first time using the ASCC method. Basic properties of the…
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…
On the basis of the adiabatic self-consistent collective coordinate method, we develop an efficient microscopic method of deriving the five-dimensional quadrupole collective Hamiltonian and illustrate its usefulness by applying it to the oblate-prolate shape coexistence/mixing phenomena in proton-rich 68,70,72Se. In this method, the vibrational and rotational collective masses (inertial functions) are determined by local normal modes built on constrained Hartree-Fock-Bogoliubov states. Numerical calculations are carried out using the pairing-plus-quadrupole Hamiltonian including the quadrupole…
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.
Systematic analysis of deformations in neutron-rich N=14 isotones was done based on the method of antisymmetrized molecular dynamics. The property of the shape coexistence in $^{28}$Si, which is known to have the oblate ground state and the prolate excited states, was successfully described. The results suggest that the shape coexistence may occur also in neutron-rich N=14 nuclei as well as $^{28}$Si. It was found that the oblate neutron shapes are favored because of the spin-orbit force in most of N=14 isotones. $Q$ moments and $E2$ transition strengths in the neutron-rich nuclei were discuss…
Exclusive cross sections for the $^{43}$P$(-1p)^{42}$Si reaction to the lowest $0^+$ and $2^+$ states, measured at NSCL with GRETINA and the S800, are interpreted in terms of a two-level mixing (collective) model of oblate and prolate co-existing shapes. Using the formalism developed for deformed nuclei we calculate the spectroscopic amplitudes and exclusive cross-sections in the strong coupling limit, where for $^{43}$P the schematic wavefunction includes the coupling of the Nilsson [211]$\frac{1}{2}$ proton orbit. Good agreement with the experimental data is obtained when the amplitude of th…
The interacting boson model with configuration mixing, with parameters derived from the self-consistent mean-field calculation employing the microscopic Gogny energy density functional, is applied to the systematic analysis of the low-lying structure in Hg isotopes. Excitation energies, electromagnetic transition rates, deformation properties, and ground-state properties of the $^{172-204}$Hg nuclei are obtained by mapping the microscopic deformation energy surface onto the equivalent IBM Hamiltonian in the boson condensate. These results point to the overall systematic trend of the transition…
