HFB Equations Research Articles

Weakly bound nuclei: Exotic ground state properties and collective excitations

Weakly bound nuclei are open quantum many-body systems and can have exotic structures and dynamics in contrast to our knowledges of stable nuclei. Currently the related studies are among the central topics in nuclear physics and are important for further improvements of nuclear theory and understandings of element enrichment processes in astrophysical environments. Since the improvements of intensity and quality of radioactive nuclear beams are very difficult, theoretical studies will be very helpful for planning experiments and the interpretation of results. Here we review recent developments of continuum density functional theory and studies of weakly bound nuclei. The theoretical challenge is to self-consistently describe the weak-binding effects, exotic shapes, continuum effects and dynamics together. To this end, our main work is to solve the Hartree-Fock-Bogoliubov equation in non-spherical coordinate- spaces and then to describe the ground state properties as well as collective excitations of weakly bound nuclei. In the 2D lattice, we solved the HFB equation with the B-spline techniques. More recently, we solved the HFB equation in the 3D coordinate-space with the adaptive multi-resolution multi-wavelet techniques, which is novel and much more efficient. Based on our calculations, we discussed the exotic egg-like deformed halo structures, the core-halo deformation decoupling, and the role of continuum contributions. We also see pairing density distributions have remarkable deformed halo structures. The deformed cores are not so favorable for the formation of 2 n -halos, in particular heavy 2 n -halos. The non-resonant continuum rather than the resonant continuum is mainly responsible for halo features. With the quasiparticle blocking calculations, we also studied the deformed 1 n -halo nuclei, in which the pairing density is unusual due to the spin polarization. In addition, we developed the deformed continuum FAM-QRPA calculations in coordinate spaces and studied the emergent soft monopole collective excitations. The soft collective modes are unique in weakly bound nuclei and are useful for studies of nuclear properties in extreme conditions. Compared to the conventional QRPA diagonalization method, the FAM-QRPA solves the QRPA equation iteratively and is more efficient. Otherwise the deformed QRPA calculations with continuum would be very challenging. We see the pairing density halos and non-resonant continuum play an important role in these soft collective excitations. Our studies involved large coordinate-space calculations in Tianhe supercomputers to take into account shape deformations, large spatial extensions and continuum spectra. Otherwise, calculations within small coordinate spaces would produce a coarse continuum discretization and even false resonance peaks. Finally we give some persectives on studies of weakly bound nuclei and our plans. Since weakly bound nuclei exhibit few-body physics in many-body systems, the conventional shell-structure picture breaks down in this respect. Theorefore new many-body techniques, time-dependent 3D dynamics and improved effective interactions are very desirable. The studies of weakly bound nuclei as open quantum systems that involve threshold effects and continuum contributions can have broad interdisciplinary interests.

On the non-unitarity of the Bogoliubov transformation due to the quasiparticle space truncation

We show that due to the energy cutoff in the Hartree-Fock-Bogoliubov quasiparticle space, the Bogoliubov transformation becomes non-unitary. We propose a method of restoring the unitarity by introducing a truncated single-particle Hilbert space, in which the HFB equations are to be solved.

Canonical-basis solution of the Hartree-Fock-Bogoliubov equation on a three-dimensional Cartesian mesh

A method is presented to obtain the canonical-form solutions of the HFB equation for atomic nuclei with zero-range interactions like the Skyrme force. It is appropriate to describe pairing correlations in the continuum in coordinate-space representations. An improved gradient method is used for faster convergences under constraint of orthogonality between orbitals. To prevent high-lying orbitals to shrink into a spatial point, a repulsive momentum dependent force is introduced, which turns out to unveil the nature of high-lying canonical-basis orbitals. The asymptotic properties at large radius and the relation with quasiparticle states are discussed for the obtained canonical basis.

Pairing correlations and particle-number projection methods

A systematic study of the pairing-correlations derived from various particle-number projection methods is performed in an exactly soluble cranked-deformed shell model Hamiltonian. It is shown that most of the approximate particle-number projection methods including the method of Lipkin-Nogami, which is used quite extensively in nuclear structure studies, breakdown in the weak pairing limit. The results obtained from the recently formulated number-projected Hartree-Fock-Bogoliubov (PHFB) equations, on the other hand, are in complete agreement with the exact solutions of the model Hamiltonian. The pairing-energy calculated from PHFB method is shown to be finite in all the studied limits. More importantly, the numerical work involved in the solution of the PHFB equations appears to be comparable to the solution of the bare HFB equations.

Hartree–Fock–Bogoliubov calculation of charge radii of Sn, Ba, Yb and Pb isotopes

Charge radii of Sn, Ba, Yb and Pb isotopes are calculated within Hartree–Fock–Bogoliubov theory with a Skyrme force and a density-dependent delta-force pairing. We investigate mean-field effects of the pairing upon odd–even staggering of isotope shifts. HFB equations are solved in the canonical basis. Odd nuclei are treated in the blocking approximation.

Statistical orientation fluctuations in 188Os

Abstract The role of the nuclear orientation in hot rotating nuclei is explored microscopically. We calculate statistical fluctuations in the orientation of the nucleus 188 Os relative to the spin vector by solving the finite-temperature HFB equation with three-dimensional cranking and the pairingplus-quadrupole interaction. The nuclear principal axis frame is defined self-consistently for each orientation by constraining the quadrupole mass tensor to be diagonal. We study the thermal and rotational dependence of the orientation probability distribution within the constant average spin constraint. Orientation fluctuations are also computed with the macroscopic Landau theory of hot rotating nuclei, and the results compare favorably with those of the microscopic HFB calculations. The equilibrium nuclear orientation is calculated as a function of temperature and spin to investigate the possibility of stable tilted rotation for this nucleus. For all rotational configurations considered, the nuclear free energy is found to be a minimum when the spin axis coincides with a principal axis of the quadrupole shape.

Two-neutrino double-beta decay in coupled QRPA with neutron-proton pairing

Two-neutrino double-beta decay rates of 46 20Ca 26, 48 20Ca 28, 76 32Ge 44 and 82 34Se 48 are calculated with inclusion of neutron-proton (np) pairing correlations in describing the ground states of initial and final nuclei. Inclusion of np pairing correlations yields stronger bound nuclei as solutions of the HFB equation. Our results show that even in medium and medium-heavy nuclei, np pairing correlations reduce the Gamow-Teller matrix elements compared to the results without np pairing. Further, the dependence of these matrix elements on the particle-particle strength parameter g pp is reduced remarkably.

Microscopic properties of the superdeformed rotational states in light rare-earth nuclei 132Ce and 134, 136Nd

The particle number- and angular-momentum-constrained HFB equation is solved with the monopole- and quadropole-pairing plus the quadrupole-quadrupole interactions. The numerical results predict that the superdeformed bands become yrast from spin 32ħ in 132Ce, 26ħ in 134Nd and 32ħ in 136Nd. The superdeformed states in 132Ce and 134Nd are characterized by vanishing static pairing gaps, while in 136Nd the neutron pairing gap remains finite even at spin 60ħ. This is due to the neutron spin alignment which is large enough to suppress the dissociation of the other neutron pairs.

Pairing properties of nuclear matter from the gogny force

Gogny's effective interaction has proved to give among many other things very reasonable pairing properties for finite nuclei. It is therefore interesting to study the pairing properties of this force also for infinite nuclear matter in the perspective of setting a benchmark for other recent and more microscopic forces. The solution of the HFB equations yields a relatively high value of Δ nm ⋍0.7 MeV for nuclear matter at saturation. The result is compatible with experimental data of heavy nuclei but in contradiction with most microscopic nuclear matter calculations which give a negligible value of Δ nm.

Multipole pair fields in the actinides

The method recently developed to solve the HFB equations in the canonical representation is generalized to the case of odd-number parity. It is then applied to the nuclei 235U, 236U, 232U and 237Np. The influence of higher multipole pairing is studied with respect to yrast lines, moments of inertia, g-factors and quasiparticle energies.

Hartree-Fock-Bogolyubov description of nuclei near the neutron-drip line

We consider the Hartree-Fock-Bogolyubov theory of nuclei in the coordinate representation and derive and solve the HFB equation for the Skyrme effective interaction. Ground-state wave functions and energies of the tin isotopes with 100 ⩽ A ⩽ 176 have been determined and the results have been compared with the predictions of the HF+BCS and macroscopic-microscopic models. The lightest tin isotope which is unstable with respect to a neutron emission is predicted by the HFB method to be 153Sn. In the region of nuclei where experimental data are not available the macroscopic-microscopic and self-consistent approximations give substantially different results.

Different treatments of pairing correlations for nuclear shapes at high spin

Strutinsky calculations are used to investigate the shape and fission stability of rotating nuclei. A cranked Woods-Saxon single-particle Hamiltonian is provided to describe triaxial and dumb-bell shapes. Since the cranking term violates time-reversal symmetry the convenient BCS method cannot be applied. On the other hand, the consideration of pairing by the solution of HFB equations is possible for a few selected nuclei only. As demonstrated recently, pairing influences the shape of fission barriers up to very high angular momenta and should not be neglected. Thus, modifications of the BCS equations are desirable. Such methods are discussed and their results are compared with the HFB results.

The mixing of ground-state rotational and gamma and beta vibrational bands in the regionA ≥ 228

The mixing of beta, gamma and ground-state bands has been investigated through the experimental determination of mixing parametersZ γ andZ βγ . TheseZ γ values have been compared with the theoretical calculations of this parameter from the solutions of time-dependent HFB equations on the adiabatic and nonadiabatic assumptions. The experimental values are in better agreement with the results obtained under the nonadiabatic assumption, valid for small deviations from the spherical symmetry.

Symmetry conserving generalisation of Hartree Fock Bogoliubov theory

We derive a generalisation of the HFB equations which conserve particle number. This is achieved in using the equation of motion method or alternatively the Green's function technique. The price we have to pay is that there is not only one mean field for the particle numberN but a set of coupled mean field equations for the whole bandN, N±2,N±4... Nevertheless we think that our theory is a quite interesting variant in comparison with the conventional projection technique. We apply our theory to simple models and find that the results are excellent.

Theory of the cranked temperature-dependent Hartree-Fock-Bogoliubov approximation and parity projected statistics

Projected statistics is proposed to select states of specified space parity and number parity from the Fock space. In the framework of the self-consistent mean field method the temperature-dependent (or thermal) HFB (THFB) equation with the constraints of angular momentum as well as particle number can be derived from the variation after projections of space parity and number parity. By considering the behavior of the quasiparticle distribution function in the limit of zero temperature, it is shown that the self-consistently blocked HFB equation follows from the THFB equation in this limit. An improved level density formula which is free from the problem of enhancement at low temperatures is derived.

Prediction of a new phase in deformed nuclei

A possible first-order phase transition across the “zero-rotation” line in the low-spin and high-temperature region (kT ≳ 0.5MeV) is indicated by the solution of the cranked temperature-dependent HFB equation for a deformed nucleus.

Current distributions for rotating nuclei

Current distributions for collective rotations have been obtained from self-consistent solutions of the cranked HF and HFB equations for the Skyrme force for a rare-earth nucleus. Currents are shown both for the ground-state band as well as for very high-spin states. The effects of the pairing correlations and those of a rotational alignment of single nucleons are discussed. A decomposition of the currents into irrotational and rigid components is performed.

Nuclear rotational anomaly and the HFB equations

We show that the S-shape of the plot of moment of inertia versus square of the angular frequency observed experimentally for the rotational spectrum of rare-earth nuclei follows naturally from the solution of the Hartree-Fock-Bogoliubov equations. We consider the case of eight particles in the 1 h 11 2 level outside a closed shell and use the Sussex interaction for the nucleon-nucleon potential. Some properties of the upper, middle and lower branches are looked into and it is established that the transition indeed occurs due to reduction in the strength of the pairing part of the interaction due to the Coriolis force. Methods for calculating the moment of inertia and other nuclear properties in the transitional region, i.e. the middle branch, are suggested and it is found that the properties there are intermediate between the upper and lower branch. We also suggest the importance of higher (> 0) particle-particle channel coupled angular momentum of the nucleon force for the rotational spectrum of deformed nuclei at high angular momentum. A comment is also made on misleading results that can be obtained for e.g. moment of inertia, by projecting only even integral values of angular momentum in Hartree-Fock type solutions and thereby ignoring finer details.