With the relativistic Coulomb wave function boundary condition, the energies, widths and wave functions of the single proton resonant orbitals for $^{17}$Ne are studied by the analytical continuation of the coupling constant (ACCC) approach within the framework of the relativistic mean field (RMF) theory. Pairing correlations and contributions from the single-particle resonant orbitals in the continuum are taken into consideration by the resonant Bardeen-Cooper-Schrieffer (BCS) approach, in which constant pairing strength is used. It can be seen that the fully self-consistent calculations with NL3 and NLSH effective interactions mostly agree with the latest experimental measurements, such as binding energies, matter radii, charge radii and densities. The energy of $π$2s$_{1/2}$ orbital is slightly higher than that of $\pi1d_{5/2}$ orbital, and the occupation probability of the $(π$2s$_{1/2})^2$ orbital is about 20%, which are in accordance with the shell model calculation and three-body model estimation.
Effects of resonant single-particle (s.p.) states on the pairing correlations are investigated by an exact treatment of the pairing Hamiltonian on the Gamow shell model basis. We introduce the s.p. states with complex energies into the Richardson equations. The solution shows the property that the resonant s.p. states with large widths are less occupied. The importance of many-body correlations between bound and resonant prticle pairs is shown.
A proper treatment of the resonant continuum is to take account of not only the energy of the resonant state, but also its width. The effect of the resonant states on pairing correlations is presented based on the relativistic mean field theory plus Bardeen-Cooper-Schrieffer(BCS) approximation with a constant pairing strength. The study is performed in an effective Lagrangian with the parameter set NL3 for neutron rich even-even Ni isotopes. The results show that the contribution of the proper treatment of the resonant continuum to pairing correlations for those nuclei close to neutron drip li…
We discuss pairing correlations in weakly bound neutron rich nuclei, by using the coordinate-space Hartree-Fock-Bogolyubov approach which allows to take properly into account the coupling to particle continuum. We show that the additional pairing binding energy acts against a development of an infinite rms radius, even in situations when an l=0 single-particle orbital becomes unbound.
With the help of a unified description of the nuclear structure and of the direct reaction mechanism we show that a recent 1H(11Li,9Li)3H experiment provides, for the first time in nuclear physics, direct evidence of phonon mediated pairing.
We present a simple scheme for taking into account the resonant continuum coupling in the Relativistic Mean Field- BCS (RMF-BCS) calculations. In this scheme, applied before in non-relativistic calculations, the effect of the resonant continuum on pairing correlations is introduced through the scattering wave functions located in the region of the resonant states. These states are found by solving the relativistic mean field equations with scattering-type boundary conditions for continuum spectrum. The calculations are done for the neutron-rich Zr isotopes. It is shown that the sudden increase…
Using a Glauber model with our relativistic fully microscopic structure model input, we give a full description of the halo nature of 31Ne that includes a self-consistent use of pairing and continuum contributions that makes predictions consistent with reaction cross section measurements. Our predictions of total reaction and one-neutron removal cross sections of 31Ne on a Carbon target were significantly enhanced compared with those of neighboring Neon isotopes, agreeing well with measurements at 240 MeV/nucleon and consistent with a single neutron halo. Furthermore, our calculations of the i…
We explore the evolution of the structure of the ground state of a nucleus with two valence nucleons as the system approaches the two particle threshold. We use a three-body model of $core+n+n$ where the core is deformed and allowed to excite. We find that both NN correlations and correlations due to deformation/excitation of the core inhibit the formation of halos. Our results suggest that it is unlikely to find halo nuclei on the dripline of deformed nuclei.
The Relativistic Hartree Bogoliubov (RHB) model is applied in the self-consistent mean-field approximation to the description of the neutron halo in the mass region above the s-d shell. Pairing correlations and the coupling to particle continuum states are described by finite range two-body forces. Finite element methods are used in the coordinate space discretization of the coupled system of Dirac-Hartree-Bogoliubov integro-differential eigenvalue equations, and Klein-Gordon equations for the meson fields. Calculations are performed for the isotopic chains of Ne and C nuclei. We find evidence…
We study the roles of the tensor and pairing correlations on the halo formation in 11Li with an extended 9Li+n+n model. We first solve the ground state of 9Li in the shell model basis by taking 2p-2h states using the Gaussian functions with variational size parameters to take into account the tensor correlation fully. In 11Li, the tensor and pairing correlations in 9Li are Pauli-blocked by additional two neutrons, which work coherently to make the configurations containing the 0p1/2 state pushed up and close to those containing the 1s1/2 state. Hence, the pairing interaction works efficiently …
Important features of low-frequency collective vibrational excitations in neutron drip line nuclei are studied. We emphasize that pairing anti-halo effect in the Hartree-Fock-Bogoliubov (HFB) theory play crucial roles to realize collective motions in loosely bound nuclei. We study the spatial properties of one particle - one hole (1p-1h) states with/without selfconsistent pairing correlations by solving simplified HF(B) equations in coordinate space. Next, by performing Skyrme-HFB plus selfconsistent quasiparticle random phase approximation (QRPA) we investigate the first 2+ states in neutron …
Matter radii of the $^{34-40}$Mg nuclei are investigated by self-consistent Hartree-Fock-Bogolyubov calculations assuming the axial symmetry. With the semi-realistic M3Y-P6 interaction, the $N$-dependence of the matter radii observed in the experiments is reproduced excellently. Both the pairing and the deformation play significant roles in an intertwined manner. The $^{35}$Mg nucleus has a smaller radius than the neighboring even-$N$ nuclei, which is attributed to its smaller deformation. In contrast, a neutron halo is obtained in $^{37}$Mg. We point out that, in contrast to the pairing anti-…
Background: A newly identified dripline nucleus $^{31}$F offers a unique opportunity to study the two-neutron ($2n$) correlation at the east shore of the island of inversion where the $N = 28$ shell closure is lost. Purpose: We aim to present the first three-body theoretical results for the radius and total reaction cross sections of $^{31}$F. This will further help to investigate how the pairing and breakdown of the $N = 28$ shell closure influence the formation of the $2n$-halo structure and the anti-halo effect in this mass region. Methods: A $^{29}$F$+n+n$ three-body system is described by…
An unified shell model scheme to evaluate simultaneously the contribution of antibound states and Gamow resonances to the nuclear halos is presented. The calculations, performed in the complex energy plane, are applied to the case of $^{11}$Li. It is found that $^{11}$Li may develop a resonant state excitation near the breakup threshold.
The influence of pairing correlations on the neutron root mean square (rms) radius of nuclei is investigated in the framework of self-consistent Skyrme Hartree-Fock-Bogoliubov calculations. The continuum is treated appropriately by the Green's function techniques. As an example the nucleus $^{124}$Zr is treated for a varying strength of pairing correlations. We find that, as the pairing strength increases, the neutron rms radius first shrinks, reaches a minimum and beyond this point it expands again. The shrinkage is due to the the so-called `pairing anti-halo effect', i. e. due to the decreas…
We take a new look at $^{6}\textrm{He}$ halo nucleus and set up a halo effective field theory at low energies to calculate the charge form factor of ${}^{6}\textrm{He}$ system with resonant P-wave interaction. P-wave Lagrangian has been introduced and the charge form factor of ${}^{6}\textrm{He}$ halo nucleus has been obtained at Leading-Order. In this study, the mean-square charge radius of ${}^{6}\textrm{He}$ nucleus relative to ${}^{4}\textrm{He}$ core and the root-mean-square (r.m.s) charge radius of ${}^{6}\textrm{He}$ nucleus have been estimated as $\langle r_{E}^{2}\rangle=1.408\; \text…
We investigate the spatial extension of weakly bound Ne and C isotopes by taking into account the pairing correlation with the Hartree-Fock-Bogoliubov (HFB) method and a 3-body model, respectively. We show that the odd-even staggering in the reaction cross sections of $^{30,31,32}$Ne and $^{14,15,16}$C are successfully reproduced, and thus the staggering can be attributed to the pairing anti-halo effect. A correlation between a one-neutron separation energy and the anti-halo effect is demonstrated for $s$- and p-waves using the HFB wave functions.
Continuum effects in the weakly bound nuclei close to the drip-line are investigated using the analytically soluble Poschl-Teller-Ginocchio potential. Pairing correlations are studied within the Hartree-Fock-Bogoliubov method. We show that both resonant and non-resonant continuum phase space is active in creating the pairing field. The influence of positive-energy phase space is quantified in terms of localizations of states within the nuclear volume.
The pairing Hamiltonian constitutes an important approximation in many- body systems, it is exactly soluble and quantum integrable. On the other hand, the continuum single particle level density (CSPLD) contains information about the continuum energy spectrum. The question whether one can use the Hamiltonian with constant pairing strength for correlations in the continuum is still unanswered. In this paper we generalize the Richardson exact solution for the pairing Hamiltonian including correlations in the continuum. The resonant and non-resonant continuum are included through the CSPLD. The r…
A method for incorporating the effect of the resonant continuum into Hartree-Fock+BCS equations is proposed. The method is applied for the case of a neutron-rich nucleus calculated with a Skyrme-type force plus a zero-range pairing interaction and the results are compared with Hartree-Fock-Bogoliubov calculations. It is shown that the widths of resonant states have an important effect on the pairing properties of nuclei close to the drip line.
The pairing anti-halo effect is a phenomenon that a pairing correlation suppresses a divergence of nuclear radius, which happens for single-particle states with orbital angular momenta of $l$ = 0 and 1 in the limit of vanishing binding energy. While this effect has mainly been discussed in terms of Hartree-Fock-Bogoliubov (HFB) theory, we here use a three-body model and provide its new intuitive concept as a localized wave packet for a quasi-particle, that is, a coherent superposition of a weakly bound and continuum wave functions due to a pairing interaction. We show that the one-particle den…
We derive the approximate solution for the pairing Hamiltonian in the Berggren ensemble of single particle states including bound, resonance and non-resonant scattering states. We show that this solution is reliable in the limit of a weak pairing interaction.
A correlated fermion system is considered surrounding a finite cavity with virtual levels. The pairing properties are calculated and the influence of the cavity is demonstrated. To this end the Gell-Mann and Goldberger formula is generalized to many-body systems. We find a possible enhancement of pairing temperature if the Fermi momentum times the cavity radius fulfills a certain resonance condition which suggests an experimental realization.
The Hartree-Fock-Bogoliubov (HFB) equations in coordinate representation are solved exactly, i.e., with correct asymptotic boundary conditions for the continuous spectrum. The calculations are preformed with effective Skyrme interactions. The exact HFB solutions are compared with HFB calculations based on box boundary conditions and with resonant continuum Hartree-Fock-BCS (HF-BCS) results. The comparison is done for the neutron-rich Ni isotopes. It is shown that close to the drip line the amount of pairing correlations depends on how the continuum coupling is treated. On the other hand, the r…
The influence of pairing correlations on the nuclear size and in particular on the formation of nuclear halos is studied in the framework of relativistic continuum Hartree-Bogoliubov (RCHB) theory. It turns out that the contributions from the weakly-bound orbits with low orbital angular momenta $l$ play an important role. As an example, we investigate the neutron-rich Mg isotopes as a function of the pairing strength in situations, where the neuron Fermi surface are below, between and above two weakly-bound $2p$ levels. We find that the size of the pairing correlations has a two-fold influence…
Nuclear many-body theory is used to study nuclear matter and finite nuclei at extreme isospin. In-medium interactions in asymmetric nuclear matter are obtained from (Dirac-) Brueckner theory. Neutron skin formation in Ni and Sn isotopes is investigated by relativistic mean-field calculations in DDRH theory with density dependent meson-nucleon vertices. Applications to light nuclei are discussed with special emphasis on pairing and core polarization in weakly bound nuclei. Approaches accounting for continuum coupling in dripline pairing and core polarization are presented. Calculations for the …
We perform a detailed analysis for the structure of 31F, which is a candidate of the halo nucleus. We calculate the radius and reaction cross-section using a three-body model of 29F+n+n and discuss how the competition between the neutron-pairing and the single-particle energy induces structural changes of 31F. The present analysis further clarifies a new aspect of the anti-halo efect that suppresses the halo structure.
A study of the effect of deformation and pairing on the development of halo nuclei is presented. Exploratory three-body $core+n+n$ calculations show that both the NN interaction and the deformation/excitation of the core hinder the formation of the halo. Preliminary self-consistent mean-field calculations are used to search for regions in the nuclear chart where halos could potentially develop. These are also briefly discussed.
We study weakly-bound deformed nuclei based on the coordinate-space Skyrme Hartree-Fock-Bogoliubov approach, in which a large box is employed for treating the continuum and surface diffuseness. Approaching the limit of core-halo deformation decoupling, calculations found an exotic "egg"-like structure consisting of a spherical core plus a prolate halo in $^{38}$Ne, in which the resonant continuum plays an essential role. Generally the halo probability and the decoupling effect in heavy nuclei are reduced compared to light nuclei, due to denser level densities around Fermi surfaces. However, de…
Finite-temperature Hartree-Fock-Bogoliubov theory using Skyrme interactions and Relativistic Hartree-Fock effective Lagrangians, predicts $^{48}$Ni as being a possible candidate for the finite temperature pairing re-entrance phenomenon. For this proton-drip-line nucleus, proton resonant states are expected to contribute substantially to pairing correlations and the two predicted critical temperatures are $T_{c1}\sim 0.08-0.2$~MeV and $T_{c2}\sim 0.7-0.9$~MeV. It is also shown that pairing re-entrance modifies the proton single particle energies around the Fermi level, as well as occupation num…
Relativistic Hartree-Bogoliubov (RHB) theory in coordinate space is used to describe the chain of even-even Zirconium isotopes reaching from Zr116 to the drip line nucleus Zr140. Pairing correlations are taken into account by a density dependent force of zero range. For neutron numbers larger than the magic number N=82 a giant neutron halo outside the core of Zr122 is observed. It is formed by up to six neutrons.
The giant halos predicted in neutron-rich Zr isotopes with $A=124-138$ are investigated by using the self-consistent continuum Skyrme Hartree-Fock-Bogoliubov approach, in which the asymptotic behavior of continuum quasiparticle states is properly treated by the Green's function method. We study in detail the neutron pair correlation involved in the giant halo by analyzing the asymptotic exponential tail of the neutron pair condensate (pair density) in addition to that of the neutron particle density. The neutron quasiparticle spectra associated with these giant halo nuclei are examined. It is …
We review the properties of nuclear halos and nuclear skins in drip line nuclei in the framework of the spherical Hartree-Fock-Bogoliubov theory with continuum effects and projection on good particle number with the Gogny force. We first establish the position of the un-projected HFB drip lines for the two most employed parametrizations of the Gogny force and show that the use of finite-range interactions leads almost always to small-sized halos, even in the least bound nuclei, which is in agreement with most mean-field predictions. We also discuss the size of the neutron skin at the drip line…
With the help of the static and dynamic mean field spectroscopic amplitudes, taking into account successive and simultaneous transfer channels properly corrected because of non-orthogonality effects, as well as describing the associated elastic channels in terms of experimentally determined optical potentials, one obtains absolute, two-particle transfer differential cross sections which provide an overall account of the data within experimental errors. One of the first results connected with such quantitative studies of pairing correlations in nuclei is the observation of phonon mediated pairi…
We make a systematic study of Li isotopes (A=9,10,11) in the tensor optimized shell model for 9Li and treat the additional valence neutrons in the cluster model approach by taking into account the Pauli-blocking effect caused by the tensor and pairing correlations. We describe the tensor correlations in 9Li fully in the tensor-optimized shell model, where the variation of the size parameters of the single particle orbits is essential for getting strong tensor correlations. We have shown in our previous study that in $^{10,11}$Li the tensor and pairing correlations in 9Li are Pauli-blocked by a…
Role of pairing correlation for emergence of low-frequency vibrational excitations in neutron drip line nuclei is discussed paying special attention to neutrons with small orbital angular momentum l. Self-consistent pairing correlation in the Hartree-Fock-Bogoliubov (HFB) theory causes the change of the spatial structure of the quasiparticle wave functions; "the pairing anti-halo effect" in the lower component and "the broadening effect" in the upper component. The resultant spatial distribution of the two-quasiparticle states among low-l neutrons, "the broad localization", brings about qualit…
The Hartree-Fock-Bogolyubov (HFB) problem for the cutoff local energy-density functional is solved numerically by using the Gor'kov formalism with an exact treatment of the particle continuum. The contributions from the resonant and "gas" continuum to the spectral density of the HFB eigenstates as well as the shifting and broadening of the discrete HF hole orbitals are clearly demonstrated with the illustrative example of the drip-line nucleus ^{70}Ca. The structure of the neutron density distribution in the localized ground state is analyzed, and the formation of its extended tail ("halo") is…
Models for the description of breakup reactions used to study the structure of exotic cluster structures like halos are reviewed. The sensitivity of these models to the projectile description is presented. Calculations are sensitive to the projectile ground state mostly through its asymptotic normalisation coefficient (ANC). They also probe the continuum of the projectile. This enables studying not only the bound states of the projectile but also its continuum, both resonant and non-resonant. This opens the possibility to study correlations between both halo neutrons in two-neutron halo nuclei…
We present a microscopic study of halo nuclei, starting from the Paris and Bonn potentials and employing a two-frequency shell model approach. It is found that the core-polarization effect is dramatically suppressed in such nuclei. Consequently the effective interaction for halo nucleons is almost entirely given by the bare G-matrix alone, which presently can be evaluated with a high degree of accuracy. The experimental pairing energies between the two halo neutrons in $^6$He and $^{11}$Li nuclei are satisfactorily reproduced by our calculation. It is suggested that the fundamental nucleon-nuc…
The BCS and HFB theories which can accommodate the pairing correlations in the ground states of atomic nuclei are presented. As an application of the pairing theories, we investigate the spatial extension of weakly bound Ne and C isotopes by taking into account the pairing correlation with the Hartree-Fock-Bogoliubov (HFB) method and a 3-body model, respectively. We show that the odd-even staggering in the reaction cross sections of $^{30,31,32}$Ne and $^{14,15,16}$C are successfully reproduced, and thus the staggering can be attributed to the unique role of pairing correlations in nuclei far …
Paring correlations in weakly bound nuclei on the edge of neutron drip line is studied by using a three-body model. A density-dependent contact interaction is employed to calculate the ground state of halo nuclei $^{6}$He and $^{11}$Li, as well as a skin nucleus $^{24}$O. Dipole excitations in these nuclei are also studied within the same model. We point out that the di-neutron type correlation plays a dominant role in the halo nuclei $^{6}$He and $^{11}$Li having the coupled spin of the two neutrons $S$=0, while the correlation similar to the BCS type is important in $^{24}$O. Contributions o…
Recent data resulting from studies of two-nucleon transfer reaction on 11Li, analyzed through a unified nuclear-structure-direct-reaction theory have provided strong direct as well as indirect confirmation, through the population of the first excited state of 9Li and of the observation of a strongly quenched ground state transition, of the prediction that phonon mediated pairing interaction is the main mechanism binding the neutron halo of the 8.5 ms-lived 11Li nucleus. In other words, the ground state of 11Li can be viewed as a neutron Cooper pair bound to the 9Li core, mainly through the exc…
Making use of the fact that the collective modes associated with the spontaneous (static and dynamic) violation of gauge invariance in atomic nuclei (pairing rotations and pairing vibrations) are amenable to a simple, quite accurate nuclear structure description (BCS and QRPA respectively), it is possible to quantitatively test the reaction mechanism which is at the basis of two-nucleon transfer reactions, specific probe of pairing in nuclei. With the help of the static and dynamic mean field spectroscopic amplitudes, taking into account successive and simultaneous transfer channels properly c…
The role of different continuum components in the weakly-bound nucleus $^6$He is studied by coupling unbound spd-waves of $^5$He by means of simple pairing contact-delta interaction. The results of our previous investigations in a model space containing only p-waves, showed the collective nature of the ground state and allowed the calculation of the electric quadrupole transitions. We extend this simple model by including also sd-continuum neutron states and we investigate the electric monopole, dipole and octupole response of the system for transitions to the continuum, discussing the contrib…
Using a simple model based on the knowledge of spherical and deformed Woods-Saxon potentials, it is shown that the recent observation of halo phenomena in $^{29}$F can be interpreted as an evidence for the prolate deformation of the ground state of $^{29}$F. The prolate deformation is the result of the shell structure, which is unique in one-neutron resonant levels, in particular near degeneracy of the neutron 1$f_{7/2}$ and 2$p_{3/2}$ resonant levels, together with the strong preference of prolate shape by the proton number $Z$ = 9. On the other hand, in oxygen isotopes spherical shape is so …
Role of different continuum components in the weakly bound nucleus 6He is studied by coupling unbound spd-waves of 5He by using simple pairing contact-delta interaction. The results show that 6He ground state $0^+$ displays collective nature by taking contribution from five different oscillating continuum states that sum up to give an exponentially decaying bound wavefunction emerging from five different possible configurations i.e. $(s_{1/2})^2$, $(p_{1/2})^2$, $(p_{3/2})^2$, $(d_{3/2})^2$ and $(d_{5/2})^2$. The ground state properties of 6He has been calculated.
We study the two-neutron correlations in the ground state of the weakly-bound two-neutron halo nucleus $^{22}$C sitting at the edge of the neutron-drip line and also in the unbound nucleus $^{26}$O sitting beyond the neutron dripline. For the present study, we employ a three-body (core$+n+n$) structure model designed for describing the two-neutron halo system by explicit inclusion of unbound continuum states of the subsystem (core$+n$). We use either a density-independent or a density-dependent contact-delta interaction to describe the neutron-neutron interaction and its strength is varied to …
We show how the quasiparticle picture of quarks changes near but above the critical temperature T_c of the color-superconducting phase transition in the heated quark matter. We demonstrate that a non-Fermi liquid behavior of the matter develops drastically when the diquark coupling constant is increased owing to the coupling of the quark with the pairing soft mode: We clarify that the depression and eventually the appearance of a gap structure in the spectral function as well as the anomalous quark dispersion relation of the quark can be understood in terms of the resonant scattering between t…
The breakup cross section (BUX) of 22C by 12C at 250 MeV/nucleon is evaluated by the continuum-discretized coupled-channels method incorporating the cluster-orbital shell model (COSM) wave functions. Contributions of the low-lying 0+_2 and 2+_1 resonances predicted by COSM to the BUX are investigated. The 2+_1 resonance gives a narrow peak in the BUX, as in usual resonant reactions, whereas the 0+_2 resonant cross section has a peculiar shape due to the coupling to the nonresonant continuum, i.e., the Fano effect. By changing the scattering angle of 22C after the breakup, a variety of shapes o…
We develop a perturbative model to treat the off-diagonal components in the Hartree-Fock-Bogoliubov (HFB) transformation matrix, which are neglected in the BCS approximation. Applying the perturbative model to a weakly bound nucleus $^{84}$Ni, it is shown that the perturbative approach reproduces well the solutions of the HFB method both for the quasi-particle energies and the radial dependence of quasi-particle wave functions. We find that the non-resonant part of the continuum single-particle state can acquire an appreciable occupation probability when there exists a weakly bound state close…
Particles in the classically forbidden regions are studied based on the deformed relativistic Hartree-Bogoliubov theory in continuum with PC-PK1 for magnesium isotopes. By analyzing the neutron and proton radii, it is found that the largest deviations from the empirical values appear at the predicted neutron halo nuclei $^{42}$Mg and $^{44}$Mg. Consistently, notable increases at $^{42}$Mg and $^{44}$Mg are found in the total number of neutrons in the classically forbidden regions that includes the number of neutrons in continuum. It is shown that the deformation effect, in general, increases t…
We study the pairing dynamics of polar states in a single species p-wave superfluid Fermi gas following a sudden change of the interaction strength. The anisotropy of pair interaction together with the presence of the centrifugal barrier results in profoundly different pairing dynamics compared to the s-wave case. Depending on the direction of quenches, quench to the BCS regime results in a large oscillatory depletion of momentum occupation inside the Fermi sea or a large filling of momentum occupation. We elucidate a crucial role of the resonant state supported by the centrifugal barrier in t…
We discuss the role of pairing anti-halo effect in the observed odd-even staggering in reaction cross sections for $^{30,31,32}$Ne and $^{36,37,38}$Mg isotopes by taking into account the ground state deformation of these nuclei. To this end, we construct the ground state density for the $^{30,31}$Ne and $^{36,37}$Mg nuclei based on a deformed Woods-Saxon potential, while for the $^{32}$Ne and $^{38}$Mg nuclei we also take into account the pairing correlation using the Hartree-Fock-Bogoliubov method. We demonstrate that, when the one-neutron separation energy is small for the odd-mass nuclei, a…
