Background: The lead region, Po, Pb, Hg, and Pt, shows up the presence of coexisting structures having different deformation and corresponding to different particle-hole configurations in the Shell Model language.
Purpose: We intend to study the importance of configuration mixing in the understanding of the nuclear structure of even-even Po isotopes, where the shape coexistence phenomena are not clear enough.
Method: We study in detail a long chain of polonium isotopes, 190-208Po, using the interacting boson model with configuration mixing (IBM-CM). We fix the parameters of the Hamiltonians through a least-squares fit to the known energies and absolute B(E2) transition rates of states up to 3 MeV.
Results: We obtained the IBM-CM Hamiltonians and we calculate excitation energies, B(E2)'s, electric quadrupole moments, nuclear radii and isotopic shifts, quadrupole shape invariants, wave functions, and deformations.
Conclusions: We obtain a good agreement with the experimental data for all the studied observables and we conclude that shape coexistence phenomenon is hidden in Po isotopes, very much as in the case of the Pt isotopes.
We intend to provide a consistent description of the even-even Hg isotopes, 172-200Hg, using the interacting boson model including configuration mixing. We pay special attention to the description of the shape of the nuclei and to its connection with the shape coexistence phenomenon.
The goal of this contribution is to analyze the connection between shape coexistence and quantum phase transition, two seemingly unrelated phenomena that share common aspects, namely, the rapid change in the ground state structure along an isotope chain or the presence of several minima at the mean-field level. To illustrate the similarities and differences between both phenomena, we will focus in the Pb region, in particular in Pt and Hg isotopes, as well as in Zr isotopes.
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…
We have performed tilted-axis-cranked Hartree-Fock-Bogoliubov calculations for a neutron-rich hafnium isotope ($^{182}$Hf) whose proton and neutron numbers are both in the upper shell region. We study whether the shell effects play a role in producing high-$K$ isomers or highly gamma-deformed states at high spin. In particular, the possibility of shape coexistence and the effect of wobbling motion are discussed.
Total Routhian Surface (TRS) calculations have been performed to investigate shape coexistence and evolution in neutron-deficient krypton isotopes ${}^{72,74,76}$Kr. The ground-state shape is found to change from oblate in ${}^{72}$Kr to prolate in ${}^{74,76}$Kr, in agreement with experimental data. Quadrupole deformations of the ground states and coexisting $0^{+}_{2}$ 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-lyi…
We investigate ground states properties of Pb isotopes located between neutron and proton drip-lines estimated by two-neutron (two-proton) separation energies and Fermi energies within the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc). First, we report some candidates of nuclear shape coexistence in the isotope chain. They are accessed by calculating total energy as a function of the deformation parameter $β_2$, and for the coexistence candidates we take a couple of the deformation region bringing about minima of the energy within energy difference $ΔE < $ 1 MeV. Second,…
We study shape coexistence and low-energy excitation spectra in neutron-deficient Pb isotopes using configuration mixing of angular-momentum and particle-number projected self-consistent mean-field states. The same Skyrme interaction SLy6 is used everywhere in connection with a density-dependent zero-range pairing force.
The phenomena of shape evolution and shape coexistence in even-even $^{88-126}$Zr and $^{88-126}$Mo isotopes is studied by employing covariant density functional theory (CDFT) with density-dependent point coupling parameter sets DD-PCX and DD-PC1, and with separable pairing interaction. The results for rms deviation in binding energies, two-neutron separation energy, the differential variation of two-neutron separation energy, and rms charge radii, as a function of neutron number, are presented and compared with available experimental data. In addition to the oblate-prolate shape coexistence i…
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 discuss shape coexistence in the neutron-deficient Kr72-Kr78 isotopes in the framework of configuration mixing calculations of particle-number and angular-momentum projected axial mean-field states obtained from self-consistent calculations with the Skyrme interaction SLy6 and a density-dependent pairing interaction. While our calculation reproduces qualitatively and quantitatively many of the global features of these nuclei, such as coexistence of prolate and oblate shapes, their strong mixing at low angular momentum, and the deformation of collective bands, the ordering of our calculated …
In this paper, we have considered the interplay between phase transitions and configuration mixing of intruder excitations in the 118-128Te isotopes. A transitional interacting boson model Hamiltonian in both IBM-1 and IBM-2 versions which are based on affine Lie Algebra are employed to describe the evolution from spherical to deformed gamma unstable shapes along the chain of Te isotopes. The excitation energies, B(E0) and B(E2) transition rates are rather well reproduced in comparison with experimental counterparts when the weight of SO(6) limit is increased in Hamiltonian. Also our results s…
We investigate the emergence and evolution of shape coexistence in the neutron-deficient Lead isotopes within the interacting boson model (IBM) plus configuration mixing with microscopic input based on the Gogny energy density functional (EDF). The microscopic potential energy surface obtained from the constrained self-consistent Hartree-Fock-Bogoliubov method employing the Gogny-D1M EDF is mapped onto the coherent-state expectation value of the configuration-mixing IBM Hamiltonian. In this way, the parameters of the IBM Hamiltonian are fixed for each of the three relevant configurations (sphe…
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…
In this paper, we have studied the shapes coexistence in the 180-190Hg isotopes. The SO(6) representation of eigenstates and a transitional Hamiltonian in the Interacting Boson Model are used to consider the evolution from prolate to oblate shapes for systems with total boson number N = 9-12. Parameter free (up to overall scale factors) predictions for energy spectra and quadrupole transition rates are found to be in good agreement with experimental counterparts. The results for the control parameter of transitional Hamiltonian offer a combination of spherical and deformed shapes in these Hg i…
The properties of the low-lying states, especially the relevant shape coexistence in $^{80}$Ge 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 proton bosons', the calculated energy levels of low-lying states, $E2$ transition strengths can reproduce the experimental data very well. Particularly, the first excited state 0$^+_2$ is reproduced quite nicely, which is intimately related with the s…
Background: The Po, Pb, Hg, and Pt region is known for the presence of coexisting structures that correspond to different particle-hole configurations in the Shell Model language or equivalently to nuclear shapes with different deformation. Purpose: We intend to study the configuration mixing phenomenon in the Hg isotopes and to understand how different observables are influenced by it. Method: We study in detail a long chain of mercury isotopes, $^{172-200}$Hg, using the interacting boson model with configuration mixing. The parameters of the Hamiltonians are fixed through a least square fit …
We study the role of intruder states and shape coexistence in the even-even $^{190-206}$Po isotopes, through an interacting boson model with configuration mixing calculation. We analyzed the results in the light of known systematics on various observable in the Pb region, paying special attention to the unperturbed energy systematics and quadrupole deformation. We find that shape coexistence in the Po isotopes behaves in very much the same way as in the Pt isotopes, i.e., it is somehow hidden, contrary to the situation in the Pb and the Hg isotopes.
The ground-state deformations of the Ge isotopes are investigated in the framework of Gogny-Hartree-Fock-Bogoliubov (HFB) and Skyrme Hartree-Fock plus pairing in the BCS approximation. Five different Skyrme parametrizations are used to explore the influence of different effective masses and spin-orbit models. There is generally good agreement for binding energies and deformations (total quadrupole moment, triaxiality) with experimental data where available (i.e., in the valley of stability). All calculations agree in predicting a strong tendency for triaxial shapes in the Ge isotopes with only…
The evolution of the total energy surface and the nuclear shape in the isotopic chain $^{172-194}$Pt are studied in the framework of the interacting boson model, including configuration mixing. The results are compared with a self-consistent Hartree-Fock-Bogoliubov calculation using the Gogny-D1S interaction and a good agreement between both approaches shows up. The evolution of the deformation parameters points towards the presence of two different coexisting configurations in the region 176 $\leq$ A $\leq$ 186.
The shape evolution and shape coexistence phenomena in neutron-rich nuclei at $N\approx60$, including Kr, Sr, Zr, and Mo isotopes, are studied in the covariant density functional theory (DFT) with the new parameter set PC-PK1. Pairing correlations are treated using the BCS approximation with a separable pairing force. Sharp rising in the charge radii of Sr and Zr isotopes at N=60 is observed and shown to be related to the rapid changing in nuclear shapes. The shape evolution is moderate in neighboring Kr and Mo isotopes. Similar as the results of previous Hartree-Fock-Bogogliubov (HFB) calcula…
Low-energy structure of even-even $^{108-116}$Cd isotopes is analyzed using a collective model that is based on the nuclear density functional theory. Spectroscopic properties are computed by solving the triaxial quadrupole collective Hamiltonian, with parameters determined by the constrained self-consistent mean-field calculations within the relativistic Hartree-Bogoliubov method employing a universal energy density functional and a pairing force. The collective Hamiltonian reproduces the observed quadrupole phonon states of vibrational character, which are based on the moderately deformed eq…
Lifetime measurements in neutron-rich odd-A yttrium isotopes ($^{93-99}$Y): Investigation of shape coexistence and the intertwined quantum phase transition
Universität zu Köln（科隆大学）；Nuclear Research Center-Negev（内盖夫核科学研究中心）；Tel Aviv University（特拉维夫大学）；Institut Laue-Langevin（朗之万研究所）；Institute of Nuclear Physics, PAN（波兰科学院核物理研究所）；Institut of Physique des 2 Infinis de Lyon（里昂两无限物理研究所）；INFN Sezione di Milano（意大利国家核物理研究所米兰分部）
Masses, deformations, radii, and single-particle properties of the very neutron-rich Sulfur isotopes are investigated in the framework of the selfconsistent mean-field theory. The stability of the N=28 magic gap around $^{44}$S is discussed.
A method of deriving the Hamiltonian of the interacting boson model, that is based on the microscopic framework of the nuclear energy density functional, is presented. The constrained self-consistent mean-field calculation with a given energy density functional provides potential energy surface within the relevant collective coordinates, which is subsequently mapped onto the expectation value of the interacting-boson Hamiltonian in the boson condensate state. This procedure completely determines the strength parameters of the IBM, and the diagonalization of the mapped Hamiltonian yields excita…
The negative and positive parity low-spin states of the even-odd Tungsten isotopes, 183,185,187W are studied in the frame work of the Interacting Boson-Fermion Approximation (IBFA) model. The fermion that is coupled to the system of bosons is taken to be in the negative parity 2f_7|2, 2f_5\2, 3p_3\2, 3p_1\2 and in the positive parity 1i_13\2 single-particle orbits. The calculated energies of low-spin energy levels of the odd isotopes are found to agree well with the experimental data. Also B(E2) values and spectroscopic factors for single-neutron transfer are calculated and found to be in good…
Recent interest in spectroscopic factors for single-neutron transfer in low-spin states of the even-odd Xenon $^{125,127,129.131}$Xe and even-odd Tellurium, $^{123,125,127,129,131}$Te isotopes stimulated us to study these isotopes within the frame work of the Interacting Boson-Fermion Model. The fermion that is coupled to the system of bosons is taken to be in the positive parity $3s_{1/2}$, $2d_{3/2}$, $2d_{5/2}$, $1g_{7/2}$ and in the negative $1h_{11/2}$ single-particle orbits, the complete 50-82 major shell. The calculated energies of low-spin energy levels of the odd isotopes are found to…
Sr isotopes are located in the mass region $A\approx 100$, where a very quick onset of nuclear deformation exists, being other notable examples of this area Yb, Zr, and Nb nuclei. The presence of the proton subshell closure $Z=40$ allows the existence of particle-hole excitations that produces low-lying intruder bands. Purpose: The goal of this work is the study of the nuclear structure of the even-even $^{92-102}$Sr isotopes through the accurate description of excitation energies, $B(E2)$ transition rates, nuclear radii and two-neutron separation energies. Method: The interacting boson model …
The mass region with A~100 and Z~40 is known to experience a sudden onset of deformation. The presence of the subshell closure $Z=40$ makes feasible to create particle-hole excitations at a moderate excitation energy and, therefore, likely intruder states could be present in the low-lying spectrum. In other words, shape coexistence is expected to be a key ingredient to understand this mass region. The aim of this work is to describe excitation energies, transition rates, radii, and two-neutron separation energies for the even-even 94-110Zr nuclei and, moreover, to obtain information about wave…
The phenomenon of shape coexistence is studied in the Relativistic Hartree-Bogoliubov framework. Standard relativistic mean-field effective interactions do not reproduce the ground state properties of neutron-deficient Pt-Hg-Pb isotopes. It is shown that, in order to consistently describe binding energies, radii and ground state deformations of these nuclei, effective interactions have to be constructed which take into account the sizes of spherical shell gaps.
This work studies the systematic reduced transition probabilities B(E2), intrinsic quadrupole moments and deformation parameters of Pd isotopes with even neutrons from N= 62 to 66. The downward reduced transition probabilities B(E2) from gamma transition 8+ to 6+, 6+ to 4+, 4+ to 2+ and 2+ to 0+ states of even-even 108-112Pd isotopes were calculated by the Interacting Boson Model (IBM-1) and compared with the available previous experimental results. The ratio of the excitation energies of the first 4+ and the first 2+ excited states, R4/2, is also studied for the classification of symmetry of …
The role of configuration mixing in the Pt region is investigated. For this chain of isotopes, the nature of the ground state changes smoothly, being spherical around mass $A\sim 174$ and $A\sim 192$ and deformed around the mid-shell N=104 region. This has a dramatic effect on the systematics of the energy spectra as compared to the systematics in the Pb and Hg nuclei. Interacting Boson Model with configuration mixing calculations are presented for gyromagnetic factors, $α$-decay hindrance factors, and isotope shifts. The necessity of incorporating intruder configurations to obtain an accurate…
In this paper, nuclear structure of yrast state bands of even-even 120-130Te isotopes has been studied under the frame work of interacting boson model (IBM-1). The theoretical energy levels for Z=52, even N = 68-78 up to 12+ state have been obtained by using the MatLab 6.5 computer program. The ratio of the excitation energies of first 4+ and first 2+ excited states, R4/2, has been also calculated for those nuclei. The values of the parameters in the IBM-1 hamiltonian yield the best fit to the experimental energy spectrum. The results are compared with the most recent experimental values where…
The systematics of one-quasiparticle configurations in neutron-rich Sr, Zr, and Mo odd isotopes is studied within the Hartree-Fock-Bogoliubov plus Equal Filling Approximation method preserving both axial and time reversal symmetries. Calculations based on the Gogny energy density functional with both the standard D1S parametrization and the new D1M incarnation of this functional are included in our analysis. The nuclear deformation and shape coexistence inherent to this mass region are shown to play a relevant role in the understanding of the spectroscopic features of the ground and low-lying …
We explore two-particle transfer reactions as a unique probe of the occurence of shape coexistence in shape phase transitions. The (t,p) reactions to the ground state and to excited $0^+$ states are calculated for the isotope chain of even-even Zirconium isotopes starting from stable nuclei up to beyond current experimental limits. Two-particle spectroscopic factors derived from Monte Carlo Shell Model calculations are used, together with the sequential description of the two-particle transfer reaction mechanism. The calculation shows a clear signature for a shape phase transition between $^{9…
A quantitative analysis of the evolution of nuclear shapes and shape phase transitions, including regions of short-lived nuclei that are becoming accessible in experiments at radioactive-beam facilities, necessitate accurate modeling of the underlying nucleonic dynamics. Important theoretical advances have recently been made in studies of complex shapes and the corresponding excitation spectra and electromagnetic decay patterns, especially in the "beyond mean-field" framework based on nuclear density functionals. Interesting applications include studies of shape evolution and coexistence in N …
The description of the of nuclei energy spectra of 0^{+} states has been done involving the degree of their collectivity as a systematics parameter. Within the framework of this approach the parameter of the collectivity is mainly determined by pairs of particles placed on single ''effective'' level and coupled to monopole bosons. Holshtein-Primakoff transformation of these monopole bosons leads to clear physical explanation of the structure of 0^{+} states in terms of ''ideal bosons''. The results may be helpful both for experimentalists and theorists in their investigations of low- lying sta…
The goal of this study is to find an observable that could distinguish between both phenomena, shape coexistence and quantum phase transitions. The selected observable to be analyzed is the two-neutron transfer intensity between the 0+ states in the parent and daughter nuclei. The framework in which the study is done is the Interacting Boson Model (IBM), including its version with configuration mixing (IBM-CM). In order to generate the wave functions of the isotope chains of interest, needed for calculating transfer intensities, previous systematic studies with IBM and IBM-CM are taken without…
Large Scale Shell Model calculations (SM-CI) predict that the region of deformation which comprises the heaviest Chromium and Iron isotopes at and beyond N=40 will merge with a new one at N=50 in an astonishing parallel to the N=20 and N=28 case in the Neon and Magnesium isotopes. We propose a valence space including the full pf-shell for the protons and the full sdg shell for the neutrons; which represents a comeback of the the harmonic oscillator shells in the very neutron rich regime. The onset of deformation is understood in the framework of the algebraic SU3-like structures linked to quad…
The role of intruder configurations in the description of energy spectra and B(E2) values in the Pt region is analyzed. In particular, we study the differences between Interacting Boson Model calculations with or without the inclusion of intruder states in the even Pt nuclei $^{172-194}$Pt. As a result, it shows that for the description of a subset of the existing experimental data, i.e., energy spectra and absolute B(E2) values up to an excitation energy of about 1.5 MeV, both approaches seem to be equally valid. We explain these similarities between both model spaces through an appropriate m…
We investigate the properties of the excited spectra of the even-even isotopes of krypton using a Generalised Bohr Hamiltonian with three different Skyrme functionals. In particular, we investigate the evolution of the low-lying $2^{+}_{1}$ and $4^{+}_{1}$ states and their associated electromagnetic transitions. The model reproduces quite nicely the energy trends apart from ${}^{88}$Kr, where none of the interactions used here are able to grasp a sudden change in the energy spectrum. Additionally, we explore the neutron deficient region ${}^{72-76}$Kr which is a proposed region for shape coexi…
The isotopic evolution of the ground-state nuclear shapes and the systematics of one-quasiproton configurations are studied in neutron-rich odd-A Yttrium and Niobium isotopes. We use a selfconsistent Hartree-Fock-Bogoliubov formalism based on the Gogny energy density functional with two parametrizations, D1S and D1M. The equal filling approximation is used to describe odd-A nuclei preserving both axial and time reversal symmetries. Shape-transition signatures are identified in the N=60 isotopes in both charge radii and spin-parities of the ground states. These signatures are a common character…
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 low-lying structure of neutron-rich odd-mass Kr isotopes is studied within the interacting boson-fermion model (IBFM) based on the Gogny-D1M energy density functional (EDF). The $(β,γ)$-deformation energy surfaces, spherical single-particle energies and occupation probabilities of the odd-mass systems are obtained using the constrained Hartree-Fock-Bogoliubov (HFB) approximation. Those quantities are used as a microscopic input to determine most of the parameters of the IBFM Hamiltonian. The remaining parameters are specifically tailored to the experimental spectrum for each of the studied…
A shape phase transition between stable octupole deformation and octupole vibrations in Th nuclei is analyzed in a microscopic framework based on nuclear density functional theory. The relativistic functional DD-PC1 is used to calculate axially-symmetric quadrupole-octupole constrained energy surfaces. Observables related to order parameters are computed using an interacting-boson Hamiltonian, with parameters determined by mapping the microscopic energy surfaces to the expectation value of the Hamiltonian in the boson condensate. The systematics of constrained energy surfaces and low-energy ex…
The structure of even-even $^{108-116}$Cd isotopes is investigated based on the self-consistent mean-field approach. By mapping the quadrupole-$(β,γ)$ deformation energy surface, obtained from the constrained self-consistent mean-field calculations with a choice of the Skyrme force and pairing property, onto the Hamiltonian of the interacting boson model with configuration mixing, the strength parameters of the Hamiltonian are determined. The low-lying excitation spectra and electric quadrupole transition rates for the considered Cd nuclei are computed by the resultant Hamiltonian, and are com…
