Shape Coexistence and Band Termination in Doubly Magic Nucleus ^40Ca

Shape coexistence and band structure near yrast line of the Z ＝ N doubly magic nucleus 40Ca have been investigated by the configuration-dependent cranked Nilsson-Strutinsky approach. The observed normal deformed and superdeformed bands are explained and the terminating states are confirmed by the calculations. The transition quadrupole moment Qt of the calculated superdeformed band is in good agreement with the observed one at high spin. There is shape coexistence within the same configuration. Possible normal deformed and superdeformed bands with rotation around the intermediate axis in several interesting configurations of40Ca are discussed. Possible favored superdeformed band terminations in 38Ca and 38Ar are predicted. The experimental results in 38Ar are discussed simply.

Description of the Shape Coexistence in 98Mo with IBM2

We investigate the properties of the low-lying states and the relevant shape dynamics of ^98 Mo within the frame- work of the proton-neutron interacting boson model （IBM2）. By considering the relative energy of the d proton boson to be different from that of the neutron bosons, the low-lying levels and the key observable B（E2） transition branching ratios are calculated. The characteristic feature of the energy spectrum and the most crucial available structure indicator indicate that the substantial mixing between the spherical-vibrational and γ-unstable shapes in 9SMo. The calculation results of the overall deformation in ^98Mo are almost the same for both the ground and the first excited 02^＋ states, showing a weak deformation. While the triaxiality parameter indicates that the mostly triaxial shape with some oblate for the ground state, and the triaxial shape with some prolate for the excited 02^＋ state, being equilibrium shapes of spherical-vibrational and γ-unstable in ^98 Mo.

Shape Coexistence for ^179Hg in Relativistic Mean-Field Theory

The potential energy surface of179 Hg is traced and the multi-shape coexistence phenomenon in that nucleus is studied within the relativistic mean-field theory with quadrupole moment constraint. The calculation results of binding energies and charge radii of mercury isotopes are in good agreement with the experimental data.

Description of Shape Coexistence and Mixed-Symmetry States in ^(96)Mo Using IBM-2

We investigate the level structure and E2,M1 electromagnetic transition properties in an even-even^（96）Mo nucleus within the framework of the proton-neutron interacting boson model（IBM-2）.The calculated results of the IBM-2 can reproduce the recent new experimental data on^（96）Mo both qualitatively and quantitatively.It is found that both shape coexistence and mixed-symmetry states in^（96）Mo can be simultaneously described very well with the IBM-2 by taking into account that the relative energy of d neutron bosons is different from that of proton bosons.

Possible shape coexistence in Ne isotopes and the impurity effect of Λ hyperon

The possible shape coexistence in even-even Ne isotopes and the impurity effects of the sΛ and pΛ hyperons are explored employing the multidimensionally constrained relativistic-mean-field(MDC-RMF) model with the PK1 parameter set for the N N interaction and PK1-Y1 for the ΛN interaction. The quadrupole deformation potential energy surfaces(PESs), nuclear deformations, nuclear radii, binding energies, and density distributions of the hypernuclei and core nuclei are examined. The possible shape coexistence in ^(24,26,28)Ne is predicted with small energy differences of 140, 336, and 128 keV, respectively, between the two local energy minima. Different impurity effects of the sΛand pΛ hyperons are revealed. The sΛ hyperon exhibits clear shrinkage effects, which reduce the nuclear size and facilitate a spherical nuclear shape. The prolate pΛhyperon on the 1/2^(-)[110] orbital renders the nuclear shape more prolate, while the oblate pΛ hyperon on the 3/2^(-)[101] or 1/2^(-)[101] orbital renders the nuclei more oblate. Moreover, the Λ hyperon can increase the probabilities of the shape coexistence by reducing the energy differences between the two local energy minima, although the shape coexistence may disappear owing to the vanishment of one energy minimum on the flat energy surface.

Two-neutron transfer reactions as a tool to study the interplay between shape coexistence and quantum phase transitions

The atomic mass table presents zones where the structure of the states changes rapidly as a function of the neutron or proton number.Among them,notable examples are the A≈100 Zr region,the Pb region around the neutron midshell(N=104),and the N≈90 rare-earth region.The observed phenomena can be understood in terms of either shape coexistence or quantum phase transitions.The objective of this study is to find an observable that can distinguish between both shape coexistence and quantum phase transitions.As an observable to be analyzed,we selected the two-neutron transfer intensity between the 0+states in the parent and daughter nuclei.The framework used for this study is the Interacting Boson Model(IBM),including its version with configuration mixing(IBM-CM).To generate wave functions of isotope chains of interest needed for calculating transfer intensities,previous systematic studies using IBM and IBM-CM were used without changing the parameters.The results of two-neutron transfer intensities are presented for Zr,Hg,and Pt isotopic chains using IBM-CM.Moreover,for Zr,Pt,and Sm isotopic chains,the results are presented using IBM with only a single configuration,i.e.,without using configuration mixing.For Zr,the two-neutron transfer intensities between the ground states provide a clear observable,indicating that normal and intruder configurations coexist in the low-lying spectrum and cross at A=98→100.This can help clarify whether shape coexistence induces a given quantum phase transition.For Pt,in which shape coexistence is present and the regular and intruder configurations cross for the ground state,there is almost no impact on the value of the two-neutron transfer intensity.Similar is the situation with Hg,where the ground state always has a regular nature.For the Sm isotope chain,which is one of the quantum phase transition paradigms,the value of the two-neutron transfer intensity is affected strongly.

Shape coexistence and evolution in neutron-def icient krypton isotopes

Total Routhian Surface （TRS） calculations have been performed to investigate shape coexistence and evolution in neutron-deficient krypton isotopes 72,74,76Kr. The ground-state shape is found to change from oblate in 72Kr to prolate in 74,76Kr, in agreement with experimental data. Quadrupole deformations of the ground states and coexisting 0＋ states as well as excitation energies of the latter are also well reproduced. While the general agreement between calculated moments of inertia and those deduced from observed spectra confirms the prolate nature of the low-lying yrast states of all three isotopes （except the ground state of 72Kr）, the deviation at low spins suggests significant shape mixing. The role of triaxiality in describing shape coexistence and evolution in these nuclei is finally discussed.

Triaxiality and shape coexistence in the A ～ 30 “island of inversion”nuclei

Understanding the properties of nuclei inside "island of inversion" is still an interesting issue. Based on a simple Nilsson model with a new set of isospin-dependent parameters, and with non-axial deformations considered, we have performed three-dimensional potential-energy-surface calculations for Ne, Na, Mg and Al isotopes that are claimed to be in or nearby the A～30 island of inversion". It is found that shape coexistence and triaxial deformation(or softness) exist in these nuclei. Large deformations are obtained by the improved Nilsson parameters, which explains the observed large electric quadrupole transition probabilities. The large deformations happening in30 Ne,31Na,32 Mg and33Al indicate the quenching of the spherical N = 20 neutron shell gap. The calculations of nuclear binding and two-neutron separation energies have been also improved with the isospin-dependent parameters and the inclusion of the non-axial deformation degree of freedom.

Possible shape coexistence in odd-A Ne isotopes and the impurity effects ofΛhyperons

In this study, shape evolution and possible shape coexistence are explored in odd-A Ne isotopes in the framework of the multidimensionally constrained relativistic-mean-field(MDC-RMF)model. By introducing ~sΛ and phyperons, the impurity effects on the nuclear shape, energy, size, and density distribution are investigated.For the NN interaction, the PK1 parameter set is adopted, and for the ΛN interaction, the PK1-Y1 parameter set is used. The nuclear ground state and low-lying excited states are determined by blocking the unpaired odd neutron in different orbitals around the Fermi surface. Moreover, the potential energy curves(PECs), quadrupole deformations,nuclear r.m.s. radii, binding energies, and density distributions for the core nuclei as well as the corresponding hy pernuclei are analyzed. By examining the PECs, possibilities for shape coexistence inNe and a triple shape coexistence inNe are found. In terms of the impurity effects ofΛhyperons, as noted for even-even Ne hypernuclear isotopes, the shyperon exhibits a clear shrinkage effect, which reduces the nuclear size and results in a more spherical nuclear shape. The phyperon occupying the 1/2~-[110]orbital is prolate, which causes the nuclear shape to be more prolate, and the phyperon occupying the 3/2~-[101] orbital displays an oblate shape, which drives the nuclei to be more oblate.

Shape coexistence in 76Se within the neutron–proton interacting boson model

We have investigated the low-lying energy spectrum and electromagnetic transition strengths in even-even ^(76)Se using the proton-neutron interacting boson model(IBM-2).The theoretical calculation for the energy levels and E2 and M1 transition strengths is in good agreement with the experimental data.Specifically,the excitation energy and E2 transition of 0_(2)^(+) state,which is intimately associated with shape coexistence,can be accurately reproduced.The analysis on lowlying states and the key structure indicators R_(1),R_(2),R_(3) and R_(4) and M1 transitions indicates that there is a coexistence between spherical shape and γ-soft shape in ^(76)Se.

A unified description of shape coexistence and mixed-symmetry states in^98Mo using IBM-2

Level structure and electromagnetic transitions in^98Mo have been investigated on the basis of the proton-neutron interacting boson model(IBM-2)by considering the energy difference between neutron bosonενand proton bosonεπ.The results are compared with the recent experimental data and it is observed that they are in good agreement.In particular,the strongest M1 transition from 25^+state to 22^+can be well reproduced,from which one can determine the 25^+as an mixed-symmetry(MS)state.We have calculated the electric monopole strengthρ^2(E0,02^+→01^+),and our result agrees with the experimental one.The calculation indicates that shape coexistence and MS states are simultaneously well described using IBM-2.

Shape coexistence and α-decay chains of ^293LV

Two recently observed ^293Lv （Z = 116） α-decay chains [Eur. Phys. J. A 48, 62 （2012）] are investigated in the framework of covariant density functional theory with PC-PK1, where the pairing correlations are treated by the Bardeen-Cooper-Schrieffer method with a density-independent zerorange force. From the calculated potential energy curves, it is found that two minima always occur, with one having an almost spherical shape and the other exhibiting a large deformed prolate shape. Originating from the ground state and the shape-isomeric state of ^293Lv, the two observed a-decay chains are constructed and the calculated Qα values are found to be in good agreement with the data.

Shape coexistence close to N=50 in the neutron-rich isotope ^(80)Ge investigated by IBM-2

The properties of the low-lying states especially the relevant shape coexistence in 80Ge, close to one of most neutron-rich doubly magic nuclei at N 50 and Z = 28, have been investigated within the framework of the proton-neutron interacting model （IBM-2）. Based on the fact that the relative energy of the d neutron boson is different from that of the proton boson, the calculated energy levels of low-lying states and E2 transition strengths can reproduce the experimental data very well. Particularly, the first excited state 0＋, which is intimately related to the shape coexistence phenomenon, is reproduced quite nicely. The p2（E0,02＋→01＋） transition strength is also predicted. The experimental data and theoretical results indicate that both collective spherical and γ-soft vibration structures coexist in 80Ge.

Description of the shape coexistence in neutron-deficient ^(74,76)Kr with IBM2

The shape deformation and shape coexistence in ^(74,76) Kr isotopes are investigated within the framework of the proton-neutron interacting boson model(IBM2). By considering the relative energy of the d proton boson to be different from that of the neutron boson, the low-lying energy spectrum is in good agreement with experimental results both qualitatively and quantitatively. In particular, the low-lying 0_2^+ states associated with the shape-coexistence phenomenon are reproduced quite well. The calculated key sensitive quantities of B(E2) transition branch ratios are fairly consistent with the experimental data except for R_4. The predicated deformation parameter is very similar for the ground states in ^(74)Kr and ^(76)Kr, showing good agreement with the experimental result,and the calculated deformation parameter for the second 0^+ state in ^(74)Kr is close to the experimental data. The calculated results of the triaxiality parameter indicated an almost purely prolate shape for the ground state of ^(76)Kr and a mostly prolate shape with a little triaxiality for the ground state of ^(74)Kr. The calculations also show an oblate triaxial shape for the second 0^+ state in ^(76)Kr and maximum triaxiality for the second 0^+ state in ^(74)Kr. These results confirm the importance of the triaxial deformation for the description of such shape coexistence.

Coexistence of Prolate and Oblate Shapes in ^(98)Sr Nuclei Using IBM2

The properties of the low-lying states and the shape coexistence in 98Sr are investigated within the framework of the proton-neutron interacting boson model （IBM2）. By considering the relative energy of the d proton boson to be different from that of the neutron boson, it is found that the calculated energy levels and the B（E2） transition strengths agree with the experimental data perfectly. Particularly, the second 0＋ state, which is associated with the shape coexistence phenomenon and has the lowest energy E（O＋） among all known even-even nuclei, is reproduced very well. The behavior of the calculated quadrupole shape invariants is consistent with the experimental results.

Shape Co-Existence in the Transitional Nuclide ^(157)yb

High-spin states in 157Yb have been populated in the 144Sm（160, 3n）157yb fusion- evaporation reaction at a beam energy of 85 MeV, and two rotational bands have been established for the first time. Within the framework of the triaxial particle-rotor model, the energy spectra and single-particle configurations of 157Yb are investigated. The calculated energy spectra agree well with the experimental data. The newly observed vf7/2 band, and the previously known vi13/2 band in 157Yb, are also discussed by means of Total-Routhian-Surface methods. The structural characters observed in 157Yb provide evidence for the shape coexistence of three distinct shapes： prolate, triaxial and oblate. At higher spins, both the vf7/2 band and the vi13/2 band in 157Yb undergo a shape evolution with sizable alignments occurring.

Investigation on the deformation of Ne and Mg isotope chains within relativistic mean-field model

Ne and Mg isotope chains are investigated based on constrained calculations in the framework of a deformed relativistic mean-field （RMF） model with the NL075 parameter set. The calculated quadrupole deformation and binding energy are compared with other theoretical results as well as the available experimental data. It shows that the calculated deformations of Ne and Mg with the NL075 are more accurate than those obtained with the NL-SH. It is predicted that ^19,29,32Ne and ^20,31Mg maybe have a triaxial deformation and ^25-28Ne and ^27-30Mg exhibit a shape coexistence probably. The closure effect of neutron number N=8 for ^20Mg is predicted to be very weak.

Evidence for coexisting prolate and near-oblate shapes at high spin in 125Cs

Excited states in the odd-proton nucleus 125Cs were investigated by means of in-beam β-ray spectroscopy. The πg7/2 band is observed to fork into a △I = 1 coupled band and a △I = 2 decoupled band at high spins. To assign the possible configurations of these two bands, experimental B（M1）/B（E2） ratios and signature splittings have been evaluated for the △I = 1 band, and calculations based on the geometrical model, cranked shell model and total Routhian surfaces model have been performed. They are suggested to be a near-oblate band built on the （πg7/2/d5/2） （vh11/2）2 configuration and a prolate band built on the πg7/2 （πh11/2）2 configuration, respectively.

Description of the properties of the low-lying energy states in ^(100)Mo with IBM2

The properties of the low-lying energy states for the l00 Mo isotope is investigated within the framework of the proton-neutron interacting model IBM2. By considering the relative energy of the d proton boson to be different from that of the neutron boson and taking into account the dipole interacting among like-boson Lπ·Lπ and Lπ·Lπ, the low-lying energy spectrum is reproduced well. Particularly, the relative position of the energies for 2＋1, 0＋2, 2＋2 and 4＋1 states shifted correctly fit the experimental data. The electromagnetic properties, including the key observable B（E2） reduced transition branching ratios and the E2 reduced matrix elements of the experimental data, are well described. Our calculations show possible shape coexistence in the l00Mo nucleus.

Possible candidate nuclei for chirality-parity quartet bands

The potential energy surfaces of the even-even 68-92Se, 112-150Ba, and 208-230Ra isotopes are calculated using the macroscopic-microscopic method in a multidimensional space {αλ,μ} including quadrupole （λ=2, μ=0, 2） and octupole （λ=3, μ=0, 1, 2, 3） degrees of freedom. The calculated results show that the even-even isotopes 92Se, 112,114,144-150Ba and 220-228Ra can exhibit the coexistence of triaxial and octupole deformations, thereby leading to simultaneous chiral and reflected symmetry breaking. Therefore, chirality-parity quartet bands are expected in these and their neighboring odd-A/odd-odd nuclei.