Particle Number Projection Research Articles

Evaluation of theβ+-decay logftvalue with inclusion of the neutron-proton pairing and particle-number projection

The neutron-proton isovector pairing effect on the beta-plus decay log ft values is studied in typical mirror N≃Z nuclei. The log ft values are calculated by including or not the isovector pairing before and after a particle-number projection using the Sharp-Bardeen-Cooper-Schrieffer (SBCS) method. It is shown that the values obtained after projection in the isovector pairing case are the closest ones to experimental data. The effect of the deformation of the mother and daughter nuclei on the log ft is also studied.

Persistence of Vibrational Collectivity in Nuclear Level Densities

By applying the particle-number projection to the finite-temperature BCS theory to the state densities in the rare-earth nuclei and comparing its results to the SMMC ones, we investigate effects of the particle-number conservation on the collective enhancement factor for the state densities. Once we restore the particle-number conservation, the rapid decay of vibrational enhancement disappears. This suggests that the vibrational collectivity may survive up to higher energy and the relevant enhancement factor is insensitive to the excitation energy.

Particle-number conservation in quasiparticle representation in the isovector neutron–proton pairing case

Until now, the Sharp-Bardeen–Cooper–Schrieffer (SBCS) particle-number projection method, in the isovector neutron–proton pairing case, has been developed in the particle representation. However, this formalism is sometimes complicated and cumbersome. In this work, the formalism is developed in the quasiparticle representation. An expression of the projected ground state wave function is proposed. Expressions of the energy as well as the expectation values of the total particle-number operator and its square are deduced. It is shown that these expressions are formally similar to their homologues in the pairing between like-particles case. They are easier to handle than the ones obtained using the particle representation and are more adapted to numerical calculations. The method is then numerically tested within the schematic one-level model, which allows comparisons with exact results, as well as in the case of even–even nuclei within the Woods–Saxon model. In each case, it is shown that the particle-number fluctuations that are inherent to the BCS method are completely eliminated by the projection. In the framework of the one-level model, the values of the projected energy are clearly closer to the exact values than the BCS ones. In realistic cases, the relative discrepancies between projected and unprojected values of the energy are small. However, the absolute deviations may reach several MeV.

Beyond relativistic mean-field approach for nuclear octupole excitations

We report the first beyond-mean-field study of low-lying parity-doublet states in 224Ra by extending the multireference relativistic energy density functional method to include dynamical correlations related to symmetry restoration and quadrupole-octupole shape fluctuation with a generator coordinate method combined with parity, particle-number, and angular-momentum projections. We clarify full microscopically that the origin of spin-dependent parity splitting in low-spin states is related to the octupole shape stabilization of positive-parity states, the dominated shapes of which drift gradually to that of negative-parity ones.

Microscopic study of low-lying spectra ofΛhypernuclei based on a beyond-mean-field approach with a covariant energy density functional

We present a detailed formalism of the microscopic particle-rotor model for hypernuclear low-lying states based on a covariant density functional theory. In this method, the hypernuclear states are constructed by coupling a hyperon to low-lying states of the core nucleus, which are described by the generator coordinate method (GCM) with the particle number and angular momentum projections. We apply this method to study in detail the low-lying spectrum of $_{\mathrm{\ensuremath{\Lambda}}}^{13}\mathrm{C}$ and $_{\mathrm{\ensuremath{\Lambda}}}^{21}\mathrm{Ne}$ hypernuclei. We also briefly discuss the structure of $_{\mathrm{\ensuremath{\Lambda}}}^{155}\mathrm{Sm}$ as an example of heavy deformed hypernuclei. It is shown that the low-lying excitation spectra with positive-parity states of the hypernuclei, which are dominated by $\mathrm{\ensuremath{\Lambda}}$ hyperon in the $s$ orbital coupled to the core states, are similar to that for the corresponding core states, while the electric quadrupole transition strength, $B(E2)$, from the ${2}_{1}^{+}$ state to the ground state is reduced according to the mass number of the hypernuclei. Our study indicates that the energy splitting between the first $1/{2}^{\ensuremath{-}}$ and $3/{2}^{\ensuremath{-}}$ hypernuclear states is generally small for all the hypernuclei which we study. However, their configurations depend much on the properties of a core nucleus, in particular on the sign of deformation parameter. That is, the first $1/{2}^{\ensuremath{-}}$ and $3/{2}^{\ensuremath{-}}$ states in ${}_{\mathrm{\ensuremath{\Lambda}}}^{13}\mathrm{C}$ are dominated by a single configuration with $\mathrm{\ensuremath{\Lambda}}$ particle in the $p$-wave orbits and thus provide good candidates for a study of the $\mathrm{\ensuremath{\Lambda}}$ spin-orbit splitting. On the other hand, those states in the other hypernuclei exhibit a large configuration mixing and thus their energy difference cannot be interpreted as the spin-orbit splitting for the $p$ orbits.

Shape and energy spectra ofΛhypernuclei from a Skyrme-Hartree-Fock model and a beyond-mean-field calculation

A triaxial shape-constrained Skyrme-Hartree-Fock mean-field model, employing a density-dependent pairing interaction and the $\mathrm{NSC}89 N\ensuremath{\Lambda}$ interaction, is used to investigate the potential energy surfaces of the $\ensuremath{\Lambda}$ hypernuclei $_{\ensuremath{\Lambda}}^{13}\mathrm{C},\phantom{\rule{0.16em}{0ex}}_{\ensuremath{\Lambda}}^{21}\mathrm{Ne},\phantom{\rule{0.16em}{0ex}}_{\ensuremath{\Lambda}}^{25,27}\mathrm{Mg}$, and $_{\ensuremath{\Lambda}}^{27,29}\mathrm{Si}$. Angular momentum projection and particle number projection techniques are applied to the axially deformed mean-field states to restore the symmetries lost in the mean-field calculation, and the energy spectra and $E2$ transition probabilities of several (hyper)nuclei are studied, demonstrating the shrinkage effect of the $\ensuremath{\Lambda}$, which is analyzed in detail.

Nuclear pairing correlations within and beyond HFB-BCS models

Pairing correlations in nuclear systems play a crucial role in several aspects, i.e. binding energies and odd-even effects, superfluid phenomena and pair transfer mechanisms, just to quote few of them. On the theoretical side, the standard description of these features is done by using BCS or HFB models which allow to describe in a simple way pairing effects. However, due to the explicit breaking of the particle number, these theories present some limitations which can be cured by using particle number projection techniques. In this paper, we will show the merits of these techniques and their accuracy in treating pairing correlations. First, a beyond BCS-HFB method is introduced where the effect of four quasi-particle states is included perturbatively and subsequently the particle number is restored. We will then show the need of restoring the good particle number also for excited states that are essential in the pair transfer process between superfluid systems. Applications to the Richardson model are shown and discussed.

Time-dependent Hartree-Fock calculations for multi-nucleon transfer processes: Effects of particle evaporation on production cross sections

We present a microscopic calculation of multi-nucleon transfer reactions employing the time-dependent Hartree-Fock (TDHF) theory. In our previous publication [Phys. Rev. C 88, 014614 (2013)], we reported our analysis for the multi-nucleon transfer processes for several systems. Here we discuss effects of particle evaporation processes on the production cross sections. Since particle evaporation processes may not be described adequately by the TDHF calculations, we evaluate them using a statistical model. As an input of the statistical model, excitation energies of the final fragments are necessary. We evaluate them from the TDHF wave function after collisions, extending the particle number projection technique. From the calculation, the particle evaporation effects are found to improve descriptions of the production cross sections. However, the production cross sections are still underestimated for processes where a number of protons are transferred. Possible origins of the discrepancy are discussed.

Particle-number projection method in time-dependent Hartree-Fock theory: Properties of reaction products

Background: The time-dependent Hartree-Fock (TDHF) theory has been successful in describing low-energy heavy ion collisions. Recently, we have shown that multinucleon transfer processes can be reasonably described in the TDHF theory combined with the particle-number projection technique. Purpose: In this work, we propose a theoretical framework to analyze properties of reaction products in TDHF calculations. Methods: TDHF calculation in three-dimensional Cartesian grid representation combined with particle number projection method. Results: We develop a theoretical framework to calculate expectation values of operators in the TDHF wave function after collision with the particle-number projection. To show how our method works in practice, the method is applied to $^{24}$O+$^{16}$O collisions for two quantities, angular momentum and excitation energy. The analyses revealed following features of the reaction: The nucleon removal proceeds gently, leaving small values of angular momentum and excitation energy in nucleon removed nuclei. Contrarily, nuclei receiving nucleons show expectation values of angular momentum and excitation energy which increase as the incident energy increases. Conclusions: We have developed a formalism to analyze properties of fragment nuclei in the TDHF theory combined with the particle-number projection technique. The method will be useful for microscopic investigations of reaction mechanisms in low-energy heavy ion collisions as well as for evaluating effects of particle evaporation on multinucleon transfer cross sections.

Microscopic particle-rotor model for the low-lying spectrum ofΛhypernuclei

We propose a novel method for low-lying states of hypernuclei based on the particle-rotor model, in which hypernuclear states are constructed by coupling the hyperon to low-lying states of the core nucleus. In contrast to the conventional particle-rotor model, we employ a microscopic approach for the core states, that is, the generator coordinate method (GCM) with the particle number and angular momentum projections. We apply this microscopic particle-rotor model to $^9_\Lambda $Be employing a point-coupling version of the relativistic mean-field Lagrangian. A reasonable agreement with the experimental data for the low-spin spectrum is achieved using the $\Lambda N$ coupling strengths determined to reproduce the binding energy of the $\Lambda$ particle.

Low-velocity transient-field technique with radioactive ion beams:gfactor of the first excited 2+state in72Zn

The $g$ factor of the first excited ${2}^{+}$ state in ${}^{72}\mathrm{Zn}$ has been measured using the transient-field (TF) technique in combination with Coulomb excitation in inverse kinematics. This experiment presents only the third successful application of the TF method to a short-lived radioactive beam in 10 y, highlighting the intricacies of applying this technique to present and future isotope separator on-line facilities. The significance of the experimental result, $g({2}_{1}^{+})=+0.47(14)$, for establishing the structure of the Zn isotopes near $N=40$ is discussed on the basis of shell-model and beyond-mean-field calculations, the latter accounting for the triaxial degree of freedom, configuration mixing, and particle number and angular momentum projections.

Particle-number projected electric quadrupole moment of even-even proton-rich nuclei in the isovector pairing case

The effect of the particle-number projection on the electric quadrupole moment (Q 2 ) of even-even proton-rich nuclei is studied in the isovector neutron-proton (np) pairing case. As a first step, an expression of the electric quadrupole moment, which takes into account the isovector np pairing effect and which conserves the particle-number, is established within the Sharp-BCS (SBCS) method. This expression does generalize the one used in the pairing between like-particles case. As a second step, Q 2 is calculated for even-even proton-rich nuclei using the single-particle energies of a Woods-Saxon mean-field. The obtained results are compared with the results obtained in the pairing between like-particles case. It is shown that the np pairing effect, as well as the projection one, is maximal when N=Z.

Pair condensation in a finite trapped Fermi gas

Superfluidity in the cold atomic two-species Fermi gas system in the unitary limit of infinite scattering length remains incompletely understood. In particular, a pseudogap phase has been proposed to exist above the superfluid critical temperature. Here we apply the auxiliary-field quantum Monte Carlo method to perform the first ab initio calculations of the temperature dependence of three quantities -- the energy-staggering pairing gap, the condensate fraction and the heat capacity -- in a trapped finite-size cold atom system. As the calculations of the energy-staggering pairing gap require the use of the canonical ensemble, we employ a novel algorithm for the stabilization of particle-number projection that is essential for reaching convergence in the size of the model space. We observe clear signatures of the superfluid phase transition in all three quantities, including a signature of the recently measured lambda peak in the heat capacity, but find no evidence of a pseudogap effect in the energy-staggering pairing gap.

Laboratory versus intrinsic description of nonaxial nuclei above doubly magic78Ni

We study the development of collectivity in neutron rich nuclei in the close vicinity of ${}^{78}$Ni. We report on the large scale shell model calculations in the $N=52--54$ even-even nuclei with $Z=30--36$. We predict maximum of triaxiality in ${}^{86}$Ge and explain this phenomenon on the basis of a pseudo-SU(3) symmetry interpretation. For the cases where signs for nonaxial shapes appear, we perform the triaxial Gogny calculations with particle number and angular momentum projections. The comparison of results obtained in the laboratory and intrinsic frames provides a comprehensive and complete picture of nuclear deformation in this region.

MICROSCOPIC ANALYSIS OF SHAPE EVOLUTION IN Mo ISOTOPES

The shape evolution of even–even Mo isotopes from the line of stability up to the two-neutron drip-line is investigated within the self-consistent Hartree–Fock–Bogoliubov theory in both the axial and triaxial symmetries. The Skyrme energy density functional SLy4 has been considered in the particle-hole channel, while the zero range delta-interaction has been employed in the particle–particle channel. In order to correctly treat the pairing correlations, a particle-number projection was carried out by the Lipkin–Nogami (LN) method. The two-neutron separation energies and root-mean-square (rms) charge radii are investigated and compared with available experimental data. The evolution of the potential energy surfaces in the (β, γ) deformation plane is presented and discussed. In addition, the obtained ground state deformation parameters are compared to those obtained by other models.

Masses and rotational energies of the heaviest nuclei

The ground-state masses of even–even 226Ra–278Cn isotopes are obtained within the macroscopic–microscopic model using the Lublin Strasbourg drop mass formula for the macroscopic part and the Yukawa-folded single-particle potential to evaluate the microscopic energy corrections. The pairing correlations are included within the Bardeen–Cooper–Schrieffer formalism with an approximated particle number projection. The coupling of the rotation with the pairing field is taken into account when evaluating the energies of the ground-state rotational band.

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.

Effects of Thermodynamic Pairing on Nuclear Level Density

Thermodynamic properties of some selected even-even nuclei such as $^{56}$Fe, $^{60}$Ni, $^{98}$Mo, and $^{116}$Sn are studied within the Bardeen-Cooper-Schrieffer theory at finite temperature (FTBCS) taking into account pairing correlations. The theory also incorporates the particle-number projection within the Lipkin-Nogami method (FTLN). The level densities are derived based on the statistical theory of the grand-canonical ensemble (GCE). The results obtained are compared with the recent experimental data by Oslo (Norway) group. It is found that pairing correlations have significant effects on nuclear level density, especially at low and intermediate excitation energies.

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.

NEUTRON–PROTON PAIRING EFFECT ON ONE-PROTON AND TWO-PROTON SEPARATION ENERGIES IN RARE-EARTH PROTON-RICH NUCLEI

The one-proton and two-proton separation energies are studied for "ordinary" and rare-earth proton-rich nuclei by including the isovector neutron–proton (np) pairing correlations using the BCS approximation. Even–even as well as odd nuclei are considered. In the latter case, the wave function is defined using the blocked-level technique. The single-particle energies used are those of a deformed Woods–Saxon mean field. It is shown that the np isovector pairing effects on the one-proton and two-proton separation energies are non-negligible. However, the only isovector BCS approximation seems to be inadequate for a good description of these quantities when including the np pairing effects: either a particle-number projection or the inclusion of the isoscalar pairing effect seems to be necessary. Another possible improvement would be a more realistic choice of the pairing strengths.