Structure of 190−200Hg within the covariant density functional theory

We study in detail the chain of even even mercury isotopes 190−200Hg using the relativistic point coupling model. A five-dimensional collective Hamiltonian (5DCH) model, with parameters determined by constrained self-consistent mean-field (SCMF) calculations based on the relativistic density-dependent pointcoupling (DD-PC1) energy density functional, and a finite-range pairing interaction is used to calculate the low-energy excitation spectrum and the B(E2) transitions rates of even-even nuclei. The calculations suggest coexisting configurations in 190Hg, increased collectivity in the isotopes 192−198Hg and a more spherical structure in 200Hg.


Introduction
One of the most intriguing subjects of research in nuclear structure studies is the shape evolution in atomic nuclei. The variation of nuclear shapes far from stability depends on the residual interaction of the valence nucleons outside closed shells that can polarize the nuclear core leading to axially deformed ground state configurations or triaxial structures. In a number of cases different shape configurations can coexist.
The phenomena of phase transitions and phase coexistence in even-even nuclei near shell closures have been extensively investigated both theoretically and experimentally (see Refs. [1][2][3][4][5][6] for reviews). In the region of Z = 82 near the neutron midshell N = 104 the phenomena of phase coexistence [6] and phase transitions [7] were first observed in studies of hyperfine structure [8]. Later spectroscopic studies [9][10][11][12][13][14][15][16][17][18] revealed that the structure of those isotopes was defined by intruder prolate deformed configurations coexisting with less deformed oblate ground states. The low-lying excited states of the intruder band exhibit a parabola shape as a function of neutron number, starting from 188 Hg down to the midshell N = 104, with a minimum observed at 182 Hg and going up to 180 Hg and 178 Hg [19,20]. On the other hand, in the heavier transitional isotopes with 190 < A < 200, the observed energy levels of the yrast band remain almost constant. Although, the isotopes between the stable 200 Hg and the beginning of the midshell in 190 Hg have been investigated by different experiments [21][22][23][24][25][26][27][28][29][30][31][32], there are still crucial observables that remain to be measured.
Theoretical studies based on the Gogny [33][34][35], the relativistic mean field (RMF) interactions [36,37], and the Nilsson-Strutinsky method [38] have generally confirmed these experimental findings. A systematic study of * e-mail: vprassa@uth.gr the low-lying states in the lead region has been performed within the number and angular-momentum projected generator coordinate method with axial symmetry, employing the Skyrme energy density functional (EDF) [39]. Excitation energies, electromagnetic transition rates, deformation properties, and ground-state properties relevant to the shape coexistence in Hg isotopes, have been investigated using the interacting boson model (IBM) [40][41][42]. A recent study within the Elliott and the proxy-SU(3) models [43] suggests that the evolution of shape coexistence in the neutron deficient Hg isotopes is accompanied by a merging of the spin-orbit (SO) -like shell with the open harmonic oscillator (HO) shell [43].
In this contribution we present contrained SCMF calculations for even-even 190−200 Hg isotopes within the relativistic Hartree-Bogoliubov [44] method with the densitydependent point-coupling (DD-PC1) [45] energy density functional in the particle-hole channel and a separable pairing force [46] in the particle-particle channel. The DD-PC1 density functional has been successfully applied to various studies of nuclear structure phenomena related to quantum phase transitions [47][48][49][50], shape coexistence [51] and the effect of collective correlations on the ground state and fission properties of superheavy nuclei [52,53].
A five-dimensional collective Hamiltonian (5DCH) with quadrupole deformations as dynamical collective coordinates [54,55] is used to calculate the low-energy excitation spectrum and the B(E2) transitions rates of 190−200 Hg isotopes.
The microscopic self-consistent solutions of deformation-constrained triaxial relativistic Hartree-Bogolyubov (RHB) calculations, the singleparticle wave functions, occupation probabilities, and quasiparticle energies, are used to calculate the Hamiltonian parameters. The moments of inertia are calculated with the Inglis-Belyaev formula [56,57] and the mass parameters with the cranking approximation [58]. The col- lective potential is obtained by subtracting the zero-point energy corrections [58] from the total energy that corresponds to the solution of constrained triaxial SCMF calculations. The resulting collective potential and inertia parameters as functions of the collective coordinates determine the dynamics of the 5DCH. Calculations shown here have been partially presented in [59].

Potential energy surfaces
To illustrate the rapid change of equilibrium shapes in Fig. 1 we present the potential energy surfaces of eveneven 190−200 Hg within the SCMF framework with the DD-PC1 functional and a separable pairing force. Starting with the lighter isotope 190 Hg the energy surface is γ-soft with two minima within an energy difference of 500keV, which indicates a case of shape coexistence of the two different configurations. The more pronounced minimum is oblate deformed at β ≈ 0.15 and the second one is prolate at β ≈ 0.25. In 192 Hg the energy surface is still rather flat in the γ-direction with the equilibrium configuration on The experimental data are taken from Ref. [21].
the oblate side at 0.1 < β < 0.2. The prolate minimum diminishes and only the oblate one is seen in 194−198 Hg. The single oblate minimum becomes less deformed and approaches β = 0 for 200 Hg, which implies a structural change from weakly oblate deformed to nearly spherical states.

Spectroscopic properties
The constrained self-consistent solutions of the relativistic Hartree-Bogoliubov (RHB) equations at each point on the energy surface determine the mass parameters the three moments of inertia and the zero-point energy corrections as functions of the deformation parameters β and γ. The diagonalization of the Hamiltonian yields the excitation spectra and collective wave functions that are used in the calculation of various observables, e.g., electromagnetic transition probabilities B(E2) and electric monopole transition strengths ρ(E0). Physical observables are calculated in the full configuration space and there are no effective charges in the model.
As an example in Fig. 2 we display the low-lying collective spectrum of 196 Hg, in comparison to available data for the excitation energies and reduced electric quadrupole transition probabilities B(E2) in Weisskopf units (W.u.) taken from Refs. [21]. In addition to the yrast ground-state band, in deformed and transitional nuclei excited states are also assigned to (quasi-) β and γ-bands. The comparison with the few existing experimental data shows a  rather reasonable agreement of the excitation energy levels in the yrast band for J π < 6 + . The theoretical reduced electric-quadrupole transition probabilities BE(2) (in W.u.) are generally larger than the data. Although a reasonable agreement within the experimental errors is observed for the first excited state 2 + 1 , the calculated value for the B(E2; 4 + 1 −→ 2 + 1 ) overestimates the experimental value considerably. This indicates that there is probably more mixing between the theoretical states than what can be inferred from the data.
One of the distinct characteristics of shape transitions between axially symmetric rotors, γ-soft rotors and spherical vibrators is the evolution of the ratio R 4/2 of excitation energies of the yrast states 4 + 1 and 2 + 1 . For an axially symmetric rotor R 4/2 = 3.33, values between 2.5 and 2 are typical for a nucleus characterized by a γ -unstable potential and, finally, R 4/2 = 2.0 for a spherical vibrator. In Fig.  3 we plot the theoretical values of R 4/2 as function of the neutron number of even-even 190−200 Hg isotopes in comparison to data taken from Ref. [21]. The calculated R 4/2 ratio starts at 2.32 in 190 Hg increases rapidly to 2.64 in 192 Hg and then decreases gradually to 2.45 in 200 Hg. The experimental values in this region vary slighty around 2.5 as the neutron number increases. The crossing between the 2 + 1 and 4 + 1 normal and intruder states at N = 110 in Hg is probably the reason for the drop of the ratio R 4/2 in 190 Hg, as reported in Ref. [42]. This effect is less pronounced in the Pt isotopes where the ratio is around 3.3 for N = 102 up to N = 106 and then decreases gradually with neutron number to approximately 2.5 for N = 110 − 118 (cf. Ref. [61]). The results disclose the γ-softness of the potential energy surfaces in 190−200 Hg. The strong configuration mixing predicted by the collective Hamiltonian model is probably the cause of the theoretical deviations from data.
the DD-PC1 functional are shown. Theoretical results with the interacting boson model, IBM-2 [40] and the interacting boson model with configuration mixing (CM), IBM-CM [41] are presented for comparison. Our calculations reproduce the general decreasing trend with neutron number, however for all isotopes 190−198 Hg an increased collectivity in the 2 + 1 yrast states is observed compared to data. For the isotopes 192−200 Hg the accuracy of the calculations compared to the experimental values is of the same quality as the one reached with the IBM-2 model in Ref. [40].
The most noticeable discrepancies between the theoretical calculations and the measured values are consistently in the lighter isotope 190 Hg. The potential energy surface of 190 Hg exhibits two minima, a dominant oblate configuration and a prolate one at larger defomation that are degenarate in energy with a rather flat path connecting them going through the triaxial region. The inclusion of dynamical correlations yields an oblate deformed but γ-soft 0 + 1 state and a 0 + 2 state, within an energy gap of 400keV, that is predominantly on the prolate side but with oblate admixtures. The relatively large overlap between the 0 + 1 and 0 + 2 wavefunctions, the large electric monopole transition strength ρ(E0) from the 0 + 2 to the 0 + 1 state (ρ 2 (E0; 0 + 2 −→ 0 + 1 ) × 10 3 = 150) and the large B(E2) values of the interband transitions, suggest a strong mixing between the two configurations and support the hypothesis of shape-coexistence at N = 110 in Hg.

Overview and conclusions
Deformation constrained SCMF calculations have been performed with the relativistic Hartree-Bogoliubov method based on the universal energy density functional DD-PC1 and a separable pairing interaction. The triaxial A quadrupole collective Hamiltonian, with parameters determined by self-consistent constrained triaxial RHB calculations, has been used to calculate the low-energy spectra, B(E2) transitions rates and electric monopole transition strengths ρ(E0) of Hg isotopes at neutron number N = 110 − 120. The calculated excitations energies of the low-lying yrast band in 196 Hg reproduce the experimental values, however the B(E2) reduced transition probabilities for the 2 + 1 and 4 + 1 yrast states overestimate the data. The two low-lying bands based on the lowest excited vibrational state that appear in the energy spectrum support the hypothesis of increased collectivity in the theoretical calculations than what can be deduced from the data. The evolution with neutron number of the ratio R 4/2 validate the above assumptions. In 190 Hg, the triaxial SCMF calculations of the energy surface and the results of the quadrupole collective Hamiltonian model suggest shapecoexistence of a dominant oblate configuration and a more prolate deformed intruder state.
Theoretical and experimental efforts in the region of even-even neutron deficient 190−200 Hg isotopes predict shape transitions from nearly spherical configurations in 200 Hg to γ-softness in 192−198 Hg and shape coexistence in 190 Hg. The results presented in this work verify these findings, demonstrating the potential of the semi-empirical relativistic EDFs including the explicit treatment of collective correlations using a microscopic collective Hamiltonian.