Relativistic Hartree-Bogoliubov Calculations Research Articles

Matter density distributions of Ne20,22 and Mg24,26 extracted through proton elastic scattering at 0.8 GeV

Reported small-angle differential cross sections of proton elastic scattering off $^{20,22}\mathrm{Ne}$ and $^{24,26}\mathrm{Mg}$ at 0.8 GeV were analyzed with the Glauber model. Matter density distributions of $^{20,22}\mathrm{Ne}$ and $^{24,26}\mathrm{Mg}$ were determined based on the two-parameter Fermi density model, and the corresponding root-mean-square point-matter radii are 2.891(52) fm, 2.895(104) fm, 2.935(20) fm, and 2.946(21) fm. The occupation number effects of the $2s$ orbital on inner density were probed by the matter density difference between $^{22}\mathrm{Ne}$ and $^{24}\mathrm{Mg}$. Combined with the experimental occupation numbers, the relativistic Hartree-Bogoliubov calculations with the DD-ME2 and PC-PK1 interactions describe the obtained density difference trend, and indicate that there may be a bubble structure in $^{24}\mathrm{Mg}$.

Microscopic analysis of octupole shape phase transitions and critical points in neutron rich actinides

Octupole constrained energy surfaces, and spectroscopic observables of four isotopic chains of: Cm, Cf, Fm and No with neutron numbers 186 N200 are analysed using a collective quadrupole - octupole Hamiltonian (QOCH). The parameters of the Hamiltonian are determined by axially reflection-asymmetric relativistic Hartree-Bogoliubov calculations based on the energy density functional DD-PC1, and a finite-range pairing interaction. The theoretical results suggest quantum phase transitions from non-octupole to octupole deformed shapes and to octupole vibrations with increasing neutron number. 288Cm is possibly close to the critical point of a simultaneous phase transition from spherical to prolate deformed and from non-octupole to stable octupole deformed configurations.

Time-dependent Dirac equation applied to one-proton radioactive emission

Relativistic energy-density functional (REDF) theory has been developed and utilized for self-consistent mean-field calculations of atomic nuclei. The proton-emitting radioactivity can provide a suitable reference to improve the predicting ability of REDF especially on the proton-drip line. One needs to consider the quantum tunneling effect, which plays an essential role in nucleon-emitting radioactive processes. However, the relativistic quantum tunneling has been less investigated compared with the nonrelativistic case. This work is devoted to a theoretical evaluation of one-proton ($1p$) radioactivity based on the relativistic Dirac formalism. For this purpose, I develop the time-dependent (TD) Dirac-spinor calculation to simulate the $1p$ emission. By utilizing the relativistic Hartree-Bogoliubov (RHB) calculation with the DD-PCX parameters, single-proton potentials for the time-dependent Dirac spinor are determined. The TD-Dirac calculation is applied to the $1p$ emissions from the $^{37}\mathrm{Sc}$ and $^{39}\mathrm{Sc}$ nuclei, which can be well approximated as the valence proton and the proton-close-shell cores. The sensitivity of $1p$-emission energy and decaying width to the mass number is demonstrated. Remarkable sensitivity exists due to the size of the system, which affects the nuclear part of potentials and energy levels, whereas the Coulomb barrier is common with the same atomic number. The calculated $1p$ energy and decaying lifetime are roughly consistent to the experimental limitation. The present TD-Dirac calculation is expected as applicable widely to proton-rich nuclides in order to improve the REDF by utilizing the $1p$-emission data.

Neutron 3s1/2 occupation change across the stable tin isotopes investigated using isotopic analysis of proton scattering at 295 MeV

The isotopic systematics of Sn isotopes in the range A=116--124 were investigated by combining the nuclear structure and reaction calculations for the analysis of Sn(p,p) reactions at 295 MeV. The Sn(p,p) reactions were calculated employing relativistic impulse approximation (RIA) with theoretical densities obtained for the Sn isotopes from relativistic Hartree-Bogoliubov (RHB) calculations with the DD-ME2 interaction and nonrelativistic Skyrme Hartree-Fock-Bogoliubov (SHFB) calculations with the SKM* interaction. Calculation using the DD-ME2 density reproduced the experimental data for $^{122}$Sn(p,p) but overestimated the $^{116}$Sn(p,p) and $^{118}$Sn(p,p) cross sections at backward angles. In the isotopic analysis of the cross section ratio $R(\sigma)$ of $^{122}$Sn to $^{116}$Sn, a calculation using the SKM* density reproduced the peak amplitudes of $R(\sigma)$ obtained from the experimental cross sections, whereas a calculation using the DD-ME2 density did not. The ratio $R(\sigma)$ was found to be sensitive to the isotopic change of the neutron $3s_{1/2}$ occupation through the isotopic difference of the surface neutron density around r=4--5 fm. Isotopic analysis indicated that a rapid increase of the $3s_{1/2}$ occupation from $^{116}$Sn to $^{122}$Sn obtained by the DD-ME2 calculation is unlikely. This result derived from the 295 MeV Sn(p,p) cross section data is consistent with the direct measurement of the neutron occupations in Sn isotopes by neutron transfer and pick-up reactions.

Mass relations of mirror nuclei in terms of Coulomb energies based on relativistic continuum Hartree-Bogoliubov calculations

In this paper we study mass relations of mirror nuclei in terms of Coulomb energies calculated by the microscopic relativistic continuum Hartree-Bogoliubov theory. An additional term corresponding to the Nolen-Schiffer anomaly is assumed to take the same form as the well-known empirical formula. The resultant root-mean-square deviations of these mass relations are typically less than 100 keV. Based on these relations, 61 unknown proton-rich nuclear masses are predicted for $18\ensuremath{\leqslant}A\ensuremath{\leqslant}87$ with positive two-proton separation energies and are tabulated in the Supplemental Material of this paper.

Two-quasiparticle K isomers within the covariant density functional theory

Two-quasiparticle excitations of medium mass nuclei with well-defined axial deformation are studied within the covariant density functional framework. The evolution of high-$K$ isomers is analyzed in a self-consistent axially symmetric relativistic Hartree-Bogoliubov calculation using the blocking approximation. The occurrence of the ${6}^{+}$ and ${8}^{\ensuremath{-}}$ low-energy high-$K$ isomers in the region from Er to Pb ($68\ensuremath{\le}Z\ensuremath{\le}82,\phantom{\rule{0.28em}{0ex}}98\ensuremath{\le}N\ensuremath{\le}112$) is evaluated and compared to available data. The importance of the quasiparticle spectrum in the energy evolution of the high-$K$ states is discussed in detail.

Proton drip line nuclei in the Relativistic Hartree-Bogoliubov model

The Relativistic Hartree Bogoliubov (RHB) theory is applied to the description of ground-state properties of proton-rich odd-Z nuclei. The model predicts the location of the proton drip-line, the ground-state quadrupole deformations and one-proton separation energies at and beyond the drip-line, the deformed single-particle Orbitals occupied by the odd valence proton, and the corresponding spectroscopic factors. The results of fully self-consistent RHB calculations are compared with experimental data on deformed and transitional proton emitters and the predictions of various mass models.

Extension of the nuclear landscape to hyperheavy nuclei

The properties of hyperheavy nuclei and the extension of nuclear landscape to hyperheavy nuclei are extensively studied within covariant density functional theory. Axial reflection symmetric and reflection asymmetric relativistic Hartree-Bogoliubov (RHB) calculations are carried out. The role of triaxiality is studied within triaxial RHB and triaxial relativistic mean field + BCS frameworks. With increasing proton number beyond Z~130 the transition from ellipsoidal-like nuclear shapes to toroidal ones takes place. The description of latter shapes requires the basis which is typically significantly larger than the one employed for the description of ellipsoidal-like shapes. Many hyperheavy nuclei with toroidal shapes are expected to be unstable towards multifragmentation. However, three islands of stability of spherical hyperheavy nuclei have been predicted for the first time in Ref. [1]. Proton and neutron densities, charge radii, neutron skins and underlying shell structure of the nuclei located in the centers of these islands have been investigated in detail. Large neutron shell gaps at N=228, 308 and 406 define approximate centers of these islands in neutron number. On the contrary, large proton gap appear only at Z=154 in the (Z~156, N~310) island. As a result, this is the largest island of stability of spherical hyperheavy nuclei found in the calculations. The calculations indicate the stability of the nuclei in these islands with respect of octupole and triaxial distortions. Fission barriers in neutron-rich superheavy nuclei are studied in triaxial RHB framework; the impact of triaxiality on the heights of fission barriers is substantial in some parts of this region. Based on the results obtained in the present work, the extension of nuclear landscape to hyperheavy nuclei is provided.

Prediction of high-K isomeric states in transactinide nuclei close to N=162

Transactinide nuclei around neutron number N=162 display axially deformed equilibrium shapes. In the present study we are particularly interested in the occurrence of high-K isomers in the axially deformed isotopes of Rf (Z=104), Sg (Z=106), Hs (Z=108), and Ds (Z=110), with neutron number N=160-166 and the effect of the N=162 closure on the structure and distribution of two-quasiparticle (2qp) states. The evolution of high-K isomers is analyzed in a self-consistent axially symmetric relativistic Hartree-Bogoliubov calculation using the blocking approximation with time-reversal symmetry breaking.

Systematic studies of the influence of single-particle resonances on neutron halo and skin in the relativistic-mean-field and complex-momentum-representation methods

The newly developed relativistic-mean-field and complex-momentum-representation (RMF-CMR) method is used to explore the exotic structure for the neutron-rich Ca, Ni, Sn, and Pb isotopes. The results are compared with the relativistic Hartree-Bogoliubov (RHB) calculations as well as the available data. The calculated root mean square (rms) radius of the neutron and proton suggests a halo or giant halo structure in the Ca isotopes with neutron number more than $N=40$, and a thick neutron skin in the extremely neutron-rich Ni, Sn, and Pb isotopes. The extracted two-neutron separation energies agree the experimental data as well as the RHB calculations. The conclusions are supported by the available single particle levels, the valence nucleon occupations, the contributions of resonant and weakly bound levels to the neutron rms radius, the density distributions of the occupation levels, and the total neutron, proton, and matter densities. Especially, it is found that the unusually increasing radius comes mainly from the contributions of the weakly bound levels with lower orbital angular momentum, and the diffuse density distributions, like the halo, originate mostly from the resonant levels with lower orbital angular momentum in the vicinity of the continuum threshold. The predication on the structure of halo and skin in the neutron-rich Ca, Ni, Sn, and Pb isotopes are of referential value for experiment.

Deformed band structures in neutron-rich Pm152–158 isotopes

High spin band structures of neutron-rich $^{152-158}$Pm isotopes have been obtained from the measurement of prompt $\gamma$-rays of isotopically identified fragments produced in fission of $^{238}$U+$^{9}$Be and detected using the VAMOS++ magnetic spectrometer and EXOGAM segmented Clover array at GANIL and also from the high statistics $\gamma$-$\gamma$-$\gamma$ and $\gamma$-$\gamma$-$\gamma$-$\gamma$ data from the spontaneous fission of $^{252}$Cf using Gammasphere. The excited states in $^{157}$Pm and those above the isomers in even-A Pm isotopes $^{152,154,156,158}$Pm have been identified for the first time. The spectroscopic information on the rotational band structures in odd-A Pm isotopes has been extended considerably to higher spins and the possibility of the presence of reflection asymmetric shapes is explored. The configuration assignments are based on the results of Cranked Relativistic Hartree-Bogoliubov calculations. From the systematics of bands in odd-A Pm isotopes and weak population of opposite parity bands, octupole deformed shapes in neutron rich Pm isotopes beyond $N=90$ seem unlikely to be present.

Microscopic description of structural evolution in Pd, Xe, Ba, Nd, Sm, Gd and Dy isotopes

Aiming to understand the role of triaxiality and the evolution of the ground state nuclear shapes, we have carried out a microscopic study for a series of chains of Pd, Xe, Ba, Nd, Sm, Gd, and Dy isotopes. This is done within the self-consistent Relativistic–Hartree–Bogoliubov (RHB) formalism, and supported by the Triaxial Projected Shell Model (TPSM) approach. Pairing interaction separable in the momentum space with DD-ME2 force parameter is used to generate the potential energy surfaces (PESs) under the axial and triaxial symmetry. Shape evolution manifest themself in very clear manner in almost all the isotopic chains. Properties of the global mimima have been found to be in good agreement with the available experimental data. Relatively flat PESs, and γ-soft nature, have been suggested 108,110Pd, 132,134Xe and 134Ba as good candidates for E(5) symmetry, while 102Pd is not found suitable for E(5) symmetry. The PESs with a bump, and rigidity against triaxial variable (γ) suggested 150Nd, 152Sm and 154Gd to be good candidates while 150Sm and 156Dy are poor candidates of X(5) critical-point symmetry. The findings of the present RHB calculations supported by TPSM are qualitatively in good agreement with the experimental and other theoretical calculations.

Resonant-continuum relativistic mean-field plus BCS in complex momentum representation

To develop methods for open quantum systems is one of the most important tasks in theoretical studies. We develop the resonant-continuum relativistic mean-field theory in complex momentum representation with the BCS approximation for pairing correlations in weakly bound nuclei. The bound states and resonant states are treated on the same footing and the physical resonant states are considered self-consistently in the present calculations. The Zr isotopes are chosen as illustrative examples. The calculated binding energies, two-neutron separation energies, and root-mean-square radii are in excellent agreement with the available data as well as the relativistic Hartree-Bogoliubov calculations. Especially, several resonant states lying near the continuum threshold are found to play important roles in the formation of exotic phenomena and support the prediction of giant halo in the Zr isotopes.

Quadrupole and octupole collectivity and cluster structures in neon isotopes

The lowest positive- and negative-parity bands of $^{20}$Ne and neutron-rich even-even Ne isotopes are investigated using a theoretical framework based on energy density functionals. Starting from a self-consistent relativistic Hartree-Bogoliubov calculation of axially-symmetric and reflection-asymmetric deformation energy surfaces, the collective symmetry-conserving states are built using projection techniques and the generator coordinate method. Overall a good agreement with the experimental excitation energies and transition rates is obtained. In particular, the model provides an accurate description of the excitation spectra and transition probabilities in $^{20}$Ne. The contribution of cluster configurations to the low-energy states is discussed, as well as the transitional character of the ground state. The analysis is extended to $^{22}$Ne and the shape-coexisting isotope $^{24}$Ne, and to the drip-line nuclei $^{32}$Ne and $^{34}$Ne. The role of valence neutrons in the formation of molecular-type bonds between clusters is discussed.

Spectroscopy of the neutron-rich actinide nucleusU240following multinucleon-transfer reactions

Background: Nuclear structure information for the neutron-rich actinide nuclei is important since it is the benchmark for theoretical models that provide predictions for the heaviest nuclei. Purpose: gamma-ray spectroscopy of neutron-rich heavy nuclei in the actinide region. Method: Multinucleon-transfer reactions in Zn-70 + U-238 and Xe-136 + U-238 have been measured in two experiments performed at the INFN Legnaro, Italy. In the Zn-70 experiment the high-resolution HPGe Clover Array (CLARA) coupled to the magnetic spectrometer PRISMA was employed. In the Xe-136 experiment the high-resolution Advanced Gamma Tracking Array (AGATA) was used in combination with PRISMA and the Detector Array for Multinucleon Transfer Ejectiles (DANTE). Results: The ground-state band (g.s. band) of U-240 was measured up to the 20(+) level and a tentative assignment was made up to the (24(+)) level. Results from gamma gamma coincidence and from particle coincidence analyses are shown. Moments of inertia (MoI) show a clear upbend. Evidence for an extended first negative-parity band of U-240 is found. Conclusions: A detailed comparison with latest calculations shows best agreement with cranked relativistic Hartree-Bogoliubov (CRHB) calculations for the g.s. band properties. The negative-parity band shows the characteristics of a K-pi = 0 band based on an octupole vibration.

Analytical continuation from bound to resonant states in the Dirac equation with quadrupole-deformed potentials

Background: Resonances with pronounced single-particle characteristics are crucial for quantitative descriptions of exotic nuclei near and beyond the drip lines, and often impact halo formation and nucleon decay processes. Since the majority of nuclei are deformed, the interplay between deformation and orbital structure near threshold can lead to improved descriptions of exotic nuclei.Purpose: Develop a method to study single-particle resonant orbital structure in the Dirac equation with a quadrupole-deformed Woods-Saxon potential. Determine the structure evolution of bound and resonant levels with deformation in this scheme, and examine the impact on halo formation in loosely bound systems, with a focus on the recent halo candidate nucleus $^{37}\mathrm{Mg}$.Method: Analytical continuation of the coupling constant (ACCC) method is developed on the basis of the Dirac equation with a deformed Woods-Saxon potential. The scalar and vector terms in the deformed potential are determined by the energies of the valence neutron and nearby orbitals, which are extracted from a self-consistent relativistic Hartree-Bogoliubov (RHB) calculation with the PC-PK1 density functional.Results: We compare the energies and widths of resonant orbitals in the recent halo nucleus candidate $^{37}\mathrm{Mg}$ using the ACCC method based on the Dirac coupled-channel equations with those determined from the scattering phase shift (SPS) method. It is found that the results from the two methods agree well for narrow resonances, whereas the SPS method fails for broad resonances. Nilsson levels for bound and resonant orbitals from the ACCC method are calculated over a wide range of deformations and show some decisive hints of halo formation in $^{37}\mathrm{Mg}$.Conclusions: In our ACCC model for deformed potentials in the coupled-channel Dirac equations, the crossing of the configuration $1/2[321]$ and $5/2[312]$ orbitals at a deformation of approximately 0.5 enhances the probability to occupy the $1/2[321]$ orbital coming from $2{p}_{3/2}$ thereby explaining the recent observation of a $\mathit{p}$-wave one-neutron halo configuration in $^{37}\mathrm{Mg}$. The resonant $1/2[301]$ configuration plays a crucial role in halo formation in the magnesium isotopes beyond $A=40$ for a wide range of deformations larger than 0.2.

High-Kisomers in transactinide nuclei close toN=162

We extend our recent study of shape evolution, collective excitation spectra, and decay properties of transactinide nuclei [V. Prassa, T. Nik\ifmmode \check{s}\else \v{s}\fi{}i\ifmmode \acute{c}\else \'{c}\fi{}, and D. Vretenar, Phys. Rev. C 88, 044324 (2013)], based on the microscopic framework of relativistic energy density functionals, to two-quasiparticle (2qp) excitations in the axially deformed Rf, Sg, Hs, and Ds isotopes, with neutron number $N=160--166$. The evolution of high-$K$ isomers is analyzed in a self-consistent axially symmetric relativistic Hartree-Bogoliubov calculation using the blocking approximation with time-reversal symmetry breaking. The occurrence of a series of low-energy high-$K$ isomers is predicted, in particular the ${9}_{\ensuremath{\nu}}^{\ensuremath{-}}$ in the $N=160$ and $N=166$ isotopes, and the ${12}_{\ensuremath{\nu}}^{\ensuremath{-}}$ in the $N=164$ nuclei. The effect of the $N=162$ deformed-shell closure on the excitation of 2qp states is discussed. In the $N=162$ isotones we find a relatively low density of 2qp states, with no two-neutron states below 1.6 MeV excitation energy and two-proton states at $\ensuremath{\approx}0.5$ MeV higher energy than the lowest 2qp states in neighboring isotopes. This is an interesting result that can be used to characterise the occurrence of deformed shell gaps in very heavy nuclei.

Density functional theory studies of cluster states in nuclei

The framework of nuclear energy density functionals is applied to a study of the formation and evolution of cluster states in nuclei. The relativistic functional DD-ME2 is used in triaxial and reflection-asymmetric relativistic Hartree-Bogoliubov calculations of relatively light $N = Z$ and neutron-rich nuclei. The role of deformation and degeneracy of single-nucleon states in the formation of clusters is analysed, and interesting cluster structures are predicted in excited configurations of Be, C, O, Ne, Mg, Si, S, Ar and Ca $N = Z$ nuclei. Cluster phenomena in neutron-rich nuclei are discussed, and it is shown that in neutron-rich Be and C nuclei cluster states occur as a result of molecular bonding of $\alpha$-particles by the excess neutrons, and also that proton covalent bonding can occur in $^{10}$C.

Global performance of covariant energy density functionals: Ground state observables of even-even nuclei and the estimate of theoretical uncertainties

Covariant density functional theory is a modern theoretical tool for the description of nuclear structure phenomena. The current investigation aims at the global assessment of the accuracy of the description of the ground state properties of even-even nuclei. We also estimate theoretical uncertainties defined here as the spreads of predictions within four covariant energy density functionals in known regions of the nuclear chart and their propagation towards the neutron drip line. Large-scale axial relativistic Hartree-Bogoliubov calculations are performed for all $Z\ensuremath{\le}104$ even-even nuclei between the two-proton and two-neutron drip lines with four modern covariant energy density functionals such as NL3*, DD-ME2, DD-ME$\ensuremath{\delta}$, and DD-PC1. The physical observables of interest include the binding energies, two-particle separation energies, charge quadrupole deformations, isovector deformations, charge radii, neutron skin thicknesses, and the positions of the two-proton and two-neutron drip lines. The predictions for the two-neutron drip line are also compared in a systematic way with the ones obtained in nonrelativistic models.

Effect of pairing correlations on nuclear low-energy structure: BCS and general Bogoliubov transformation

Low-lying nuclear states of Sm isotopes are studied in the framework of a collective Hamiltonian based on covariant energy density functional theory. Pairing correlations are treated by both BCS and Bogoliubov methods. It is found that the pairing correlations deduced from relativistic Hartree-Bogoliubov (RHB) calculations are generally stronger than those deduced by relativistic mean field plus BCS ($\mathrm{RMF}+\mathrm{BCS}$) with the same pairing force. By simply renormalizing the pairing strength, the diagonal part of the pairing field is changed in such a way that the essential effects of the off-diagonal parts of the pairing field neglected in the $\mathrm{RMF}+\mathrm{BCS}$ calculations can be recovered, and consequently the low-energy structure is in a good agreement with the predictions of the RHB model.