Single-particle Resonant States Research Articles

Pseudospin symmetry in resonant states in deformed nucleus 154Dy

It is known that pseudospin symmetry plays a crucial role in formation of many physical phenomena. By combining the relativistic mean field theory with the complex momentum representation method, the pseudospin symmetry in the single particle resonant states in the deformed nucleus [Formula: see text]Dy is investigated through the energy and width splittings, the quadrupole deformation parameter, the radial density distributions and occupation probabilities of the pseudospin doublets. Near the continuum threshold, the pseudospin symmetry is well reserved in both bound and resonant states. The energy and width splittings of pseudospin doublets in resonant states exhibit correlations with the deformation and quantum numbers. The good pseudospin symmetry is expected with lower pseudo-orbital angular momentum projection [Formula: see text] and the main quantum number N. In general, an increase in deformation tends to weaken the quality of the pseudospin symmetry. The understanding of the evolution of the pseudospin doublets in the resonant states has been deepened by studying the pseudospin symmetry in the deformed nuclei.

Pseudospin symmetry in resonant states in deformed nuclei

Pseudospin symmetry (PSS) plays critical roles in the formation of many physical phenomena, and has been studied extensively from spherical nuclei to deformed nuclei. However, there remains an open question about the PSS in resonant states in deformed systems. For the first time, we investigate the PSS in resonant states in deformed nuclei by solving the Dirac equation in the complex-momentum representation for a potential with quadrupole deformation at the first order obtained from relativistic mean-field (RMF) calculations. Taking 170Yb as an example, we explore the PSS in the single-particle resonant states by examining the energies, widths and the density distribution. Compared with the bound states, the pseudospin splittings between the resonant pseudospin doublets are found to be small, and the radial density distributions of pseudospin doublets look very similar, which indicates that the PSS of resonant states is well reserved in deformed nuclei. Moreover, we have comprehensively studied the dependencies of the PSS on nuclear deformations and the shape of potential, which are helpful to recognize the evolution of resonant pseudospin doublets in weakly-bound deformed nuclei and understand their exotic properties.

Single-particle resonant states with Green’s function method * * Partly supported by the Physics Research and Development Program of Zhengzhou University (32410217) and the National Natural Science Foundation of China (11505157, 11875225)

This study employs the relativistic mean field theory with the Green's function method to study the single-particle resonant states. In contrast to our previous work [Phys. Rev. C, 90: 054321 (2014)], the resonant states are identified by searching for the poles of Green's function or the extremes of the density of states. This new approach is highly effective for all kinds of resonant states, no matter whether they are broad or narrow. The dependence on the space size for the resonant energies, widths, and the density distributions in the coordinate space has been checked and was found to be very stable. Taking 120Sn as an example, four new broad resonant states , , , and were observed, and the accuracy for the width of the very narrow resonant state was highly improved to MeV. Further, our results are very close to those obtained using the complex momentum representation method and the complex scaling method.

Pseudospin and spin symmetries in single particle resonant states in Pb isotopes

Based on the newly developed RMF-CMR theory by combining relativistic mean field (RMF) with complex momentum representation (CMR), we have explored the single particle resonant states, and their pseudospin and spin symmetries for Pb isotopes. The available energies and widths decrease monotonously with the increasing mass number in accompany with clear shell structure including the appearance of magic number N=184. The extracted pseudospin and spin splittings become smaller for the neutron-richer nuclei. The extent of the splittings depends on the quantum numbers and position of the states lying in mean field. Good pseudospin and spin symmetries are expected in the neutron-rich nuclei and the states with lower orbital momentum and lying near the continuum threshold. Especially, it is found that both lower (upper) components of the every pseudospin (spin) doublet look similar for not only bound states but also resonant states, which confirms the presence of the pseudospin (spin) symmetry although the corresponding energy splitting is slightly large.

Probing the resonance of Dirac particle in the relativistic point-coupling model by complex-momentum-representation method

The complex-momentum-representation (CMR) method is applied to investigate the single-particle resonant states in the relativistic point-coupling model. Taking Sn isotopes as examples, the resonant energies and widths are systematically calculated and compared with those from the Green’s function method. It is found that the resonant states are predicted accurately with the CMR method even for those lying near the continuum threshold. The predicted resonant energies and widths generally decrease with the increase of neutron number, which are mainly induced by the different potentials of Sn isotopes. It is also found that the wave functions in the momentum space are relatively localized for the resonant states especially for the broad resonances, while they have long tails in the coordinate space, which indicates the advantage of CMR method in studying the properties of resonant states.

Effects of mean-field and pairing correlations on the Bogoliubov quasiparticle resonance

The quasiparticle resonances are investigated by examining three kinds of quasiparticle spectra, i.e., the density of quasiparticle states, the occupation number density, and the pair number density in the continuum Skyrme Hartree-Fock-Bogoliubov theory with the Greens function method.Taking the weakly bound nucleus $^{66}$Ca as an example, the quasiparticle resonant energies and widths extracted from these three kinds of quasiparticle spectra are compared.For the narrow resonances, the extracted resonant energy and the width are consistent with each other. However, it is difficult to use the density of quasiparticle states to identify the broad resonances due to the background of nonresonant continuum.By switching off the pairing potential and/or the Hartree-Fock (HF) potential respectively in the calculation of these quasiparticle spectra, the roles of HF mean-field and pairing correlations in the quasiparticle resonances are demonstrated clearly. It turns out that all the quasiparticle resonances corresponding to the deeply bound, weakly bound and positive-energy single-particle resonant states, are mainly contributed by the HF potential.The pairing potential helps to slightly increase the resonant energy and the width. However, the pairing potential is important to make the nucleons occupy the low-lying nonresonant continuum states near the threshold and take part in the pairing correlations here, especially for the partial waves with small angular momentum $\ell$.

Isospin asymmetry of the pseudospin symmetry in nuclear resonant states

The role of Coulomb and $ \rho$ potentials in the asymmetry of the neutron and proton pseudospin splittings in nuclear resonant states are investigated. To this end, energies and widths of single-particle resonant states in the Sn isotopes are determined within the framework of relativistic mean-field theory combined with a method based on the analytic continuation of the coupling-constant method. We study the effect of these potentials on the energy splittings and widths splittings of the neutron and proton pseudospin partners in correlation with the mass number. Compared with the effect of the Coulomb potential, the $ \rho$ potential gives the main contribution to the isospin asymmetry of the pseudospin energy splittings and width splittings in the resonant states except for the proton rich nuclei. The pseudospin energy symmetry and width symmetry are not always better realized for neutrons than for protons in the resonant states. The Coulomb potential and the $ \rho$ potential always act in an opposite role in influencing the isospin asymmetry of the pseudospin symmetry in the resonant states.

Green's function method for the spin and pseudospin symmetries in the single-particle resonant states

We investigate the spin and pseudospin symmetry in the single-particle resonant states by solving the Dirac equation containing a Woods-Saxon potential with Green's function method. Taking double-magic nucleus $^{208}$Pb as an example, three spin doublets $3d$, $2h$, and $1j$ and three pseudospin doublets $3\tilde{p}$, $1\tilde{i}$, and $1\tilde{j}$ are obtained for the single-neutron resonant states. By analyzing the energy splittings, we find that the threshold effect plays an important role in resonant pseudospin doubles. Besides, there is a reversed level structure of pseudospin doublets in the continuum. Differently, all the width splittings of either the spin doublets or the pseudospin doublets are systematically positive and the splittings are very small except $1\tilde{j}$ doublet. Further studies show that the splittings of the energies and widths for the resonant (pseudo)spin doublets are independent. Besides, the similarity properties of the wave functions of the spin and pseudospin doublets still maintain well in resonant states.

Green's function relativistic mean field theory forΛhypernuclei

The relativistic mean field theory with the Green's function method is extended to study $\mathrm{\ensuremath{\Lambda}}$ hypernuclei. Taking the hypernucleus $_{\mathrm{\ensuremath{\Lambda}}}^{61}\mathrm{Ca}$ as an example, the single-particle resonant states for $\mathrm{\ensuremath{\Lambda}}$ hyperons are investigated by analyzing the density of states, and the corresponding energies and widths are given. Different behaviors are observed for the resonant states, i.e., the distributions of the very narrow $1{f}_{5/2}$ and $1{f}_{7/2}$ states are very similar to bound states while those of the wide $1{g}_{7/2}$ and $1{g}_{9/2}$ states are like scattering states. Besides, the impurity effect of $\mathrm{\ensuremath{\Lambda}}$ hyperons on the single-neutron resonant states is investigated. For most of the resonant states, both the energies and widths decrease with adding more $\mathrm{\ensuremath{\Lambda}}$ hyperons due to the attractive $\mathrm{\ensuremath{\Lambda}}N$ interaction. Finally, the energy level structure of $\mathrm{\ensuremath{\Lambda}}$ hyperons in the Ca hypernucleus isotopes with mass number $A=53\ensuremath{-}73$ are studied; obvious shell structure and small spin-orbit splitting are found for the single-$\mathrm{\ensuremath{\Lambda}}$ spectrum.

Green's function method in covariant density functional theory

This article provides a brief review on the Green's function covariant density functional theory (GF-CDFT), and its appli-cation for the nuclear structure. By reviewing the research status of exotic nuclei and the problems of traditional theory in treating continuum, the Green's function method in nuclear physics is introduced. The GF-CDFT theory is then presented in detail, including a short introduction to the Green's function method, Green's function relativistic mean field (GF-RMF) theory, and Green's function relativistic continuum Hartree-Bogoliubov (GF-RCHB) theory, in which the detailed equa-tions in solving the radial Dirac equation and RCHB equation by Green's function method are given. As to applications, GF-RMF theory is applied to study the single-particle resonant states, in which the energies and widths for the resonant states of 120 Sn are provided; GF-RCHB theory is applied to study the giant halos in the neutron-rich Zr isotopes, in which the extended density distribution as well as the energies and widths of the resonant states playing important roles for giant halos are well described.

Green's function method for single-particle resonant states in relativistic mean field theory

Relativistic mean field theory is formulated with the Green's function method in coordinate space to investigate the single-particle bound states and resonant states on the same footing. Taking the density of states for free particles as a reference, the energies and widths of single-particle resonant states are extracted from the density of states without any ambiguity. As an example, the energies and widths for single-neutron resonant states in ${}^{120}\mathrm{Sn}$ are compared with those obtained by the scattering phase-shift method, the analytic continuation in the coupling constant approach, the real stabilization method, and the complex scaling method. Excellent agreements with these methods are found for the energies and widths of single-neutron resonant states.

Relativistic extension of the complex scaling method for resonant states in deformed nuclei

The complex scaling method is extended to the relativistic framework for describing deformed nuclei and the theoretical formalism is presented in detail. How to expose the resonant states is demonstrated, and the applicability and efficiency of the extended method are confirmed. The energies and widths of the single-particle resonant states in nuclei with $A=31$ are determined, and the structure of single-particle levels for bound and resonant states is shown to be similar to that in the nonrelativistic calculations. Especially, the present formalism has yielded richer numerical results for the resonant levels than those obtained by the coupled-channel method in relativistic and nonrelativistic calculations.

Recent progresses on the pseudospin symmetry in single particle resonant states

The pseudospin symmetry (PSS) is a relativistic dynamical symmetry directly connected with the small component of the nucleon Dirac wave function. Much effort has been made to study this symmetry in bound states. Recently, a rigorous justification of the PSS in single particle resonant states was achieved by examining the asymptotic behaviors of the radial Dirac wave functions: The PSS in single particle resonant states in nuclei is conserved exactly when the attractive scalar and repulsive vector potentials have the same magnitude but opposite sign. Several issues related to the exact conservation and breaking mechanism of the PSS in single particle resonances were investigated by employing spherical square well potentials in which the PSS breaking part can be well isolated in the Jost function. A threshold effect in the energy splitting and an anomaly in the width splitting of pseudospin partners were found when the depth of the square well potential varies from zero to a finite value.

Pseudospin symmetry in single-particle resonances in spherical square wells

The pseudospin symmetry (PSS) has been studied extensively for bound states. Recently we justified rigorously that the PSS in single particle resonant states is exactly conserved when the attractive scalar and repulsive vector potentials of the Dirac Hamiltonian have the same magnitude but opposite sign [PRL 109, 072501 (2012)]. To understand more deeply the PSS, we focus on several issues related to the exact conservation and breaking mechanism of the PSS in single particle resonances. In particular, we are interested in how the energy and width splittings of PS partners depend on the depth of the scalar and vector potentials. We investigate the asymptotic behaviors of radial Dirac wave functions. Spherical square well potentials are employed in which the PSS breaking part in the Jost function can be well isolated. By examining the zeros of Jost functions corresponding to small components of the radial Dirac wave functions, general properties of the PSS are analyzed. By examining the Jost function, the occurrence of intruder orbitals is explained and it is possible to trace continuously the PSS partners from the PSS limit to the case with a finite potential depth. The dependence of the PSS in resonances as well as in bound states on the potential depth is investigated systematically. We find a threshold effect in the energy splitting and an anomaly in the width splitting of pseudospin partners when the depth of the single particle potential varies from zero to a finite value. The conservation and the breaking of the PSS in resonant states and bound states share some similar properties. The appearance of intruder states can be explained by examining the zeros of Jost functions. Origins of the threshold effect in the energy splitting and the anomaly in the width splitting of PS partners, together with many other problems, are still open and should be further investigated.

Pseudospin Symmetry in Single Particle Resonant States

The pseudospin symmetry (PSS) is a relativistic dynamical symmetry connected with the small component of the Dirac spinor. The origin of PSS in single particle bound states in atomic nuclei has been revealed and studied extensively. By examining the zeros of Jost functions corresponding to the small components of Dirac wave functions and phase shifts of continuum states, we show that the PSS in single particle resonant states in nuclei is conserved when the attractive scalar and repulsive vector potentials have the same magnitude but opposite sign. The exact conservation and the breaking of the PSS are illustrated for single particle resonances in spherical square-well and Woods-Saxon potentials.

Research on the single-particle resonant states by the complex scaling method

In the framework of the relativistic mean field (RMF) theory, the single-particle resonant states for spherical nuclei are studied by the complex scaling method. Taking 122Zr for example, we demonstrae the calculated details, and the obtained energies, widths, and wave functions for all the possiable resonant states. The results are in good agreement with those from the analytic continuation in the coupling constant method and the scattering phase shift method. Furthermore, we investigate the resonant states for all the Zr isotopes. The results are in satisfactory agreement with those from the scattering phase-shift method.

Application of the complex scaling method in relativistic mean-field theory

We develop the complex scaling method within the framework of the relativistic mean-field (RMF) model. With the self-consistent nuclear potentials from the RMF model, the complex scaling method is used to study single-particle resonant states in spherical nuclei. As examples, the energies and widths of low-lying neutron resonant states in $^{120}\mathrm{Sn}$ are obtained. The results are compared with those from the real stabilization method, the scattering phase-shift method, and the analytic continuation in the coupling constant approach and satisfactory agreements are found.

SINGLE-PARTICLE RESONANCES IN Ca ISOTOPES

Single-particle resonant states in Ca isotopes are studied systematically by real stabilization method (RSM) in coordinate space within the framework of the self-consistent relativistic mean field (RMF) theory. Phase shifts are obtained by scattering phase shift method. The resonant parameters (the energies, widths) are extracted by fitting energy and phase shift. Wave functions of resonances are obtained by matching conditions of bound and scattering states. Taking 60 Ca as an example, results are compared with corresponding results obtained from the analytic continuation in the coupling constant approach and the scattering phase shift method. Satisfied agreements are found. The rules of resonant parameters changing in Ca isotopes are also analyzed.

Relativistic description of single-particle resonances via phase shift analysis

Single-particle resonant states in spherical nuclei are studied by the real stabilization method in coordinate space within the framework of self-consistent relativistic mean field theory. Taking 122Zr as an example, the resonant parameters, including the energies and widths are extracted by fitting energy and phase shift. Good agreement with the previous calculations has been found. The details of single-particle resonant states are analysed.

Gamow-Hartree-Fock-Bogoliubov method: Representation of quasiparticles with Berggren sets of wave functions

Single-particle resonant states, also called Gamow states, as well as bound and scattering states of complex energy form a complete set, the Berggren completeness relation. It is the building block of the recently introduced Gamow shell model, where weakly bound and resonant nuclear wave functions are expanded with a many-body basis of Slater determinants generated by this set of single-particle states. However, Gamow states have never been studied in the context of Hartree-Fock-Bogoliubov theory, except in the Bardeen-Cooper-Schriefer (BCS) approximation, where both the upper and lower components of a quasiparticle wave function are assumed to possess the same radial dependence with that of a Gamow state associated with the Hartree-Fock potential. Hence, an extension of the notion of Gamow state has to be effected in the domain of quasiparticles. It is shown theoretically and numerically that bound, resonant and scattering quasiparticles are well defined and form a complete set, by which bound Hartree-Fock-Bogoliubov ground states can be constructed. It is also shown that the Gamow-Hartree-Fock single-particle basis can be used to solve the Gamow-Hartree-Fock-Bogoliubov problem. As an illustration, the proposed method is applied to neutron-rich nickel isotopes close to the neutron drip-line.