Approximate Particle Number Projection Research Articles

Rotational excitations in rare-earth nuclei: A comparative study within three cranking models with different mean fields and treatments of pairing correlations

High-spin rotational bands in rare-earth Er ($Z=68$), Tm ($Z=69$) and Yb ($Z=70$) isotopes are investigated by three different nuclear models. These are (i) the cranked relativistic Hartree-Bogoliubov (CRHB) approach with approximate particle number projection by means of the Lipkin-Nogami (LN) method, (ii) the cranking covariant density functional theory (CDFT) with pairing correlations treated by a shell-model-like approach (SLAP) or the so called particle-number conserving (PNC) method, and (iii) cranked shell model (CSM) based on the Nilsson potential with pairing correlations treated by the PNC method. A detailed comparison between these three models in the description of the ground state rotational bands of even-even Er and Yb isotopes is performed. The similarities and differences between these models in the description of the moments of inertia, the features of band crossings, equilibrium deformations and pairing energies of even-even nuclei under study are discussed. These quantities are considered as a function of rotational frequency and proton and neutron numbers. The changes in the properties of the first band crossings with increasing neutron number in this mass region are investigated. On average, a comparable accuracy of the description of available experimental data is achieved in these models. However, the differences between model predictions become larger above the first band crossings. Because of time-consuming nature of numerical calculations in the CDFT-based models, a systematic study of the rotational properties of both ground state and excited state bands in odd-mass Tm nuclei is carried out only by the PNC-SCM. With few exceptions, the rotational properties of experimental 1-quasiparticle and 3-quasiparticle bands in $^{165,167,169,171}$Tm are reproduced reasonably well.

Benchmarking mean-field approximations to level densities

We assess the accuracy of finite-temperature mean-field theory using as a standard the Hamiltonian and model space of the shell model Monte Carlo calculations. Two examples are considered: the nucleus $^{162}$Dy, representing a heavy deformed nucleus, and $^{148}$Sm, representing a nearby heavy spherical nucleus with strong pairing correlations. The errors inherent in the finite-temperature Hartree-Fock and Hartree-Fock-Bogoliubov approximations are analyzed by comparing the entropies of the grand canonical and canonical ensembles, as well as the level density at the neutron resonance threshold, with shell model Monte Carlo (SMMC) calculations, which are accurate up to well-controlled statistical errors. The main weak points in the mean-field treatments are seen to be: (i) the extraction of number-projected densities from the grand canonical ensembles, and (ii) the symmetry breaking by deformation or by the pairing condensate. In the absence of a pairing condensate, we confirm that the usual saddle-point approximation to extract the number-projected densities is not a significant source of error compared to other errors inherent to the mean-field theory. We also present an alternative formulation of the saddle-point approximation that makes direct use of an approximate particle-number projection and avoids computing the usual three-dimensional Jacobian of the saddle-point integration. We find that the pairing condensate is less amenable to approximate particle-number projection methods due to the explicit violation of particle-number conservation in the pairing condensate. Nevertheless, the Hartree-Fock-Bogoliubov theory is accurate to less than one unit of entropy for $^{148}$Sm at the neutron threshold energy, which is above the pairing phase transition.

Potential energy landscapes of Th isotopes within the Lublin Strasbourg drop + Yukawa-folded model

Advanced calculations of potential energy surfaces in a four-dimensional deformation space were performed for even–even Th isotopes using the macroscopic–microscopic method. The macroscopic part of the total energy was derived from the Lublin Strasbourg drop model. The Yukawa-folded potential was used to generate the single-particle spectra. The asymptotic Strutinsky shell-correction method and the Bardeen–Cooper–Schrieffer theory with approximate particle number projection were used to evaluate the microscopic part of the nuclear energy. Predictions for the ground-state masses and fission-barrier heights agree well with the experimental data. An accurate analysis of the computed multi-dimensional energy maps is in progress. Our preliminary results indicate that the present approach is capable of addressing the evolution of fission modality in the actinide region.

Microscopic nuclear mass table with high-performance computing

We overview the methodology behind the large-scale mass table calculations based on the nuclear density functional theory (DFT) with Skyrme energy density functionals (EDFs). The calculations employing massively parallel computers are done in a large configuration space of an axially deformed harmonic oscillator. Nuclear mass tables tabulating global nuclear properties such as binding energies, radii, shape deformations, and pairing gaps are obtained for several Skyrme EDFs augmented by a mixed-type pairing functional, using an approximate particle number projection before variation. Specialty visualization tools have been developed to analyze the results. As an illustrative example of our current capabilities, we show some results for a wide range of even-even nuclei with Z ≤ 120 and N ≤ 300.

PAIRING AS A COLLECTIVE MODE

The collective pairing hamiltonian is obtained in the framework of the generator coordinate method using the Gaussian overlap approximation. A slightly modified BCS wave function is used as a generator function. The pairing gap parameter Δ and the gauge transformation angle ϕ are chosen as generator coordinates. This model gives not only an approximate particle-number projection but also the form of the ground state collective wave function in the (Δ, ϕ) plane. It was found that the most probable Δ does not corresponds to the minimum (Δ eq ) of the energy function E BCS (Δ) but it is systematically shifted towards smaller values than Δ eq . This effect is due to the strong dependence of the mass parameters on the pairing gap. The influence of the dynamics in the pairing degrees of freedom on the fission life-times as well as the quadrupole collective excitations is examined.

Superheavy nuclei: a relativistic mean field outlook

The analysis of quasi-particle spectra in the heaviest A∼250 nuclei with spectroscopic data provides an additional constraint for the choice of effective interaction for the description of superheavy nuclei. It strongly suggests that only the parametrizations which predict Z=120 and N=172 as shell closures are reliable for superheavy nuclei within the relativistic mean field theory. The influence of the central depression in the density distribution of spherical superheavy nuclei on the shell structure is studied. A large central depression produces large shell gaps at Z=120 and N=172. The shell gaps at Z=126 and N=184 are favoured by a flat density distribution in the central part of the nucleus. It is shown that approximate particle number projection (PNP) by means of the Lipkin–Nogami (LN) method removes pairing collapse seen at these gaps in the calculations without PNP.

Deformation effects on the structures of [formula omitted] halo nuclei

The structures of N = 7 isotones have been investigated using the deformed Skyrme–Hartree–Fock model. The approximate particle number projection by means of the Lipkin–Nogami method was used with the blocking effect of the odd neutron taken into account. For nuclei near the drip lines, a volume–surface mixing pairing interaction was employed, which can well reproduce odd–even differences for the long chain of carbon isotopes up to drip lines. The calculated potential-energy curve shows that the first 1 2 + state of the 11Be nucleus has a very large prolate deformation. Such a large deformation can result in a good agreement between the calculated and experimental density distribution of the nucleus. The first 1 2 + state in 13C is also calculated to have a large deformation, and the 1 2 + state in 9He could be a spherical halo attached to a deformed core. The possible halo structure of the 1 2 + neutron orbit in the N = 7 isotones is discussed.

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.

Rotational properties of 252, 253, 254No: influence of pairing correlations

Rotational bands of 252,253,254No and the fission barriers of 254No at spin 0 ℏ and 20 ℏ are calculated with the Hartree–Fock–Bogolyubov theory and the Lipkin–Nogami approximate particle number projection. The SLy4 Skyrme force is used in the particle–hole channel. A zero-range force with and without density-dependence is used in the particle–particle channel. The experimental ground state deformation ( Q 20=32.8 b) is reproduced as well as the increase of the dynamical moment of inertia with frequency both for 252No and 254No. The rotational band of 253No is also calculated. Fission barriers of 254No at spin 0 ℏ and 20 ℏ show the robustness of shell-corrections against rotation in these heavy nuclei.

Moments of inertia of nuclei in the rare earth region: A relativistic versus nonrelativistic investigation

A parameter-free investigation of the moments of inertia of ground-state rotational bands in well-deformed rare-earth nuclei is carried out using cranked relativistic Hartree-Bogoliubov (CRHB) and nonrelativistic cranked Hartree-Fock-Bogoliubov (CHFB) theories. In CRHB theory, the relativistic fields are determined by the nonlinear Lagrangian with the NL1 force and the pairing interaction by the central part of a finite-range Gogny D1S force. In CHFB theory, the properties in particle-hole and particle-particle channels are defined solely by Gogny D1S forces. Using an approximate particle number projection before variation by means of the Lipkin-Nogami method improves the agreement with the experimental data, especially in CRHB theory. The effect of the particle number projection on the moments of inertia and pairing energies is larger in relativistic than in nonrelativistic theory.

Cranked relativistic Hartree–Bogoliubov theory: formalism and application to the superdeformed bands in the A∼190 region

Cranked relativistic Hartree–Bogoliubov theory without and with approximate particle number projection by means of the Lipkin–Nogami method is presented in detail as an extension of relativistic mean field theory with pairing correlations to the rotating frame. Pairing correlations are taken into account by a finite range two-body force of Gogny type. The applicability of this theory to the description of rotating nuclei is studied in detail on the example of superdeformed bands in even–even nuclei of the A∼190 mass region. Different aspects such as the importance of pairing and particle number projection, the dependence of the results on the parametrization of the RMF Lagrangian and Gogny force, etc., are investigated in detail. It is shown that without any adjustment of new parameters the best description of experimental data is obtained by using the well established parameter sets NL1 for the Lagrangian and D1S for the pairing force. Contrary to previous studies at spin zero it is found that the increase of the strength of the Gogny force is not necessary in the framework of relativistic Hartree–Bogoliubov theory provided that particle number projection is performed.

Implementation of a variational approach to approximate particle number projection with effective forces

Kamlah's second-order method for approximate particle number projection is applied for the first time to variational calculations with effective forces. High spin states of normal and superdeformed nuclei have been calculated with the finite range density dependent Gogny force for several nuclei. Advantages and drawbacks of the Kamlah second-order method as compared to the Lipkin–Nogami recipe are thoroughly discussed. We find that the Lipkin–Nogami prescription occasionally may fail to find the right energy minimum in the strong pairing regime and that Kamlah's second-order approach though providing better results than the LN one may break down in some limiting situations.

Approximate particle number projection with density dependent forces: superdeformed bands in the A=150 and A=190 regions

We derive the equations for approximate particle number projection based on mean field wave functions with finite range density dependent forces. As an application ground bands of even- A superdeformed nuclei in the A=150 and A=190 regions are calculated with the Gogny force. We discuss nuclear properties such as quadrupole moments, moments of inertia and quasiparticle spectra, among others, as a function of the angular momentum. We obtain a good overall description.

Cranked relativistic Hartree-Bogoliubov theory: Superdeformed bands in theA∼190region

Cranked Relativistic Hartree-Bogoliubov (CRHB) theory is presented as an extension of Relativistic Mean Field theory with pairing correlations to the rotating frame. Pairing correlations are taken into account by a finite range two-body force of Gogny type and approximate particle number projection is performed by Lipkin-Nogami method. This theory is applied to the description of yrast superdeformed rotational bands observed in even-even nuclei of the $A\sim 190$ mass region. Using the well established parameter sets NL1 for the Lagrangian and D1S for the pairing force one obtains a very successful description of data such as kinematic ($J^{(1)}$) and dynamic ($J^{(2)}$) moments of inertia without any adjustment of new parameters. Within the present experimental accuracy the calculated transition quadrupole moments $Q_t$ agree reasonably well with the observed data.

Collective quadrupole excitations in the 50 < Z, N < 82 nuclei with the general Bohr Hamiltonian

The general Bohr Hamiltonian is applied to a description of low-lying collective excitations in even-even isotopes of Te, Xe, Ba, Ce, Nd and Sm. The collective potential and inertial functions are determined by means of the Strutinsky method and the cranking model, respectively. A shell-dependent parametrization of the Nilsson potential is used. An approximate particle-number projection is performed in the treatment of pairing correlations. The effect of coupling with the pairing vibrations is taken into account approximately when determining the inertial functions. The calculation does not contain any free parameter.

Self-consistent description of the excited superdeformed bands in 191Hg

Abstract The superdeformed ground and excited bands of the nucleus 191 Hg are calculated self-consistently within the Hartree-Fock-Bogoliubov theory with approximate particle number projection. In the numerical calculations we use the finite range density dependent Gogny force. We describe quadrupole moments, excitation energies and moments of inertia the experimentally observed four superdeformed bands. Former band assignments based in a more phenomenological approximation are confirmed. The agreement with the available experimental results is very satisfactory.

Approximate number projection at finite temperature

The approximate particle number projection method based on an idea of the famous Lipkin — Nogami approximation is developed in the framework of the thermo field dynamics.

Lipkin-Nogami pairing scheme in self-consistent nuclear structure calculations.

We investigate the pairing treatment with approximate particle number projection, known as the Lipkin-Nogami scheme, in connection with nuclear Hartree-Fock models. A prescription for the pairing strength is developed. It is based on the average gap method and employs a Thomas-Fermi model to estimate the average level density at the Fermi surface. The second-order variation of the Skyrme energy functional gives feedback to the self-consistent density and thus to the mean field. This effect turns out to be small but non-negligible. The Lipkin-Nogami scheme is compared with the conventional treatment of pairing based on the BCS approximation. The bulk observables, especially those which are related to surface properties, depend on the pairing scheme employed, with the differences most noticeable for light nuclei. Pronounced effects on deformation energy surfaces were found, leading to sizable differences in the deformation energies. {copyright} {ital 1996 The American Physical Society.}

Approximate particle number projection for finite range density dependent forces.

The Lipkin-Nogami method is generalized to deal with finite range density dependent forces. New expressions are derived and realistic calculations with the Gogny force are performed for the nuclei $^{164}\mathrm{Er}$ and $^{168}\mathrm{Er}$. The sharp phase transition predicted by the mean field approximation is washed out by the Lipkin-Nogami approach. A much better agreement with the experimental data is reached with the new approach than with the Hartree-Fock-Bogoliubov one, especially at high spins. \textcopyright{} 1996 The American Physical Society.

Superdeformed rotational bands in the mercury region. A cranked Skyrme-Hartree-Fock-Bogoliubov study

A study of rotational properties of the ground superdeformed bands in \Hg{0}, \Hg{2}, \Hg{4}, and \Pb{4} is presented. We use the cranked Hartree-Fock-Bogoliubov method with the {\skm} parametrization of the Skyrme force in the particle-hole channel and a seniority interaction in the pairing channel. An approximate particle number projection is performed by means of the Lipkin-Nogami prescription. We analyze the proton and neutron quasiparticle routhians in connection with the present information on about thirty presently observed superdeformed bands in nuclei close neighbours of \Hg{2}.