Particle-number Projection Method Research Articles

A Time-Dependent-Density-Functional-Theory Study of Charge Transfer Processes of Li2+ Colliding with Ar in the MeV Region

We study charge transfer of a multi-electron collision system Li2+ + Ar using the time-dependent density functional theory non-adiabatically coupled to the molecular dynamics. By implementing the particle number projection method, the single- and double-charge transfer cross sections are extracted at MeV energies, which are in good agreement with the experimental data available. The analysis of charge transfer probabilities shows that for energies higher than 1.0 MeV, the single-charge transfer occurs for a broader range of impact parameters, while the double-charge transfer is dominated by close collisions. To gain the population of captured electrons on the projectile, we compute the orbital projection probabilities. It is found that the electrons of the Ar atom will most possibly transfer to the 2p orbitals of the Li2+, and only a small portion of captured electrons distribute on the s orbitals. This work verifies the capability of the present methodology in dealing with charge transfer in dressed ion collisions at MeV energies.

New and Practical Formulation for Overlaps of Bogoliubov Vacua.

In this Letter, we present a new expression for the overlaps of wave functions in Hartree-Fock-Bogoliubov based theories. Starting from the Pfaffian formula by Bertsch etal. [1], an exact and computationally stable formula for overlaps is derived. We illustrate the convenience of this new formulation with a numerical application in the context of the particle-number projection method. This new formula allows for substantially increased precision and versatility in chemical, atomic, and nuclear physics applications, particularly for methods dealing with superfluidity, symmetry restoration, and uses of nonorthogonal many-body basis states.

Microscopic studies of production cross sections in multinucleon transfer reaction Ni58+Sn124

Background: Multinucleon transfer reactions at low-energy collisions are considered to be promising for the production of new exotic nuclei, which are difficult to produce by other methods. Theoretical studies are required to provide reliable predictions for the experiments and to help understand the microscopic mechanism in multinucleon transfer reactions.Purpose: We provide a predictive approach for production cross sections and show how and to what extent the microscopic approach works well in multinucleon transfer reactions.Methods: We employ the $\text{TDHF}\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}\text{GEMINI}$ approach, which combines the microscopic time-dependent Hartree-Fock (TDHF) model with the state-of-art statistical model $\text{GEMINI}\phantom{\rule{0.16em}{0ex}}++$, to take into account both the multinucleon transfer dynamics and the secondary de-excitation process. The properties of primary products in multinucleon transfer process, such as transfer probabilities and primary cross sections, are extracted from TDHF dynamics using the particle-number projection method. Production cross sections for secondary products are evaluated using the statistical model $\text{GEMINI}\phantom{\rule{0.16em}{0ex}}++$.Results: We investigate the influence of colliding energies and deformation orientations of target and projectile nuclei on multinucleon transfer dynamics in the reaction $^{58}\mathrm{Ni}+^{124}\mathrm{Sn}$. More nucleons are observed to transfer in the tip collision than in the side collision. The production cross sections for secondary fragments with $\text{TDHF}\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}\text{GEMINI}$ calculations well reproduce the experimental measurements at energies close to the Coulomb barrier. At sub-barrier energy, the theoretical results gradually deviate from the experimental data with the increase of the number of transferred neutrons, showing the limitations of a single mean-field approximation in the TDHF approach. Possible origins for this discrepancy are discussed. The total cross sections integrated over all the neutron transfer channels are in good agreement with the experimental data for all the energies. We compare the production cross sections of $\text{TDHF}\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}\text{GEMINI}$ calculations with those from GRAZING model and find that our approach gives a description as quantitatively good as the semiclassical model, although there is no adjustable parameters for the reaction dynamics in the microscopic TDHF method.Conclusions: The microscopic $\text{TDHF}\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}\text{GEMINI}$ approach reasonably reproduces the experimental data at energies close to the Coulomb barrier and well accounts for the multinucleon transfer mechanism. The present studies clearly reveal the applicability of $\text{TDHF}\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}\text{GEMINI}$ method in multinucleon transfer reactions, which thus is a promising tool for predicting the properties of new reactions.

Production mechanism of neutron-rich nuclei around N = 126 in the multi-nucleon transfer reaction 132Sn + 208Pb**Supported by National Natural Science Foundation of China (11705118, 11475115, 11647026) and Natural Science Foundation of SZU (2016017)

The time-dependent Hartree-Fock approach in three dimensions is employed to study the multi-nucleon transfer reaction 132Sn + 208Pb at various incident energies above the Coulomb barrier. Probabilities for different transfer channels are calculated by using the particle-number projection method. The results indicate that neutron stripping (transfer from the projectile to the target) and proton pick-up (transfer from the target to the projectile) are favored. De-excitation of the primary fragments is treated by using the state-of-art statistical code GEMINI++. Primary and final production cross sections of the target-like fragments (with Z=77 to Z=87) are investigated. The results reveal that fission decay of heavy nuclei plays an important role in the de-excitation process of nuclei with Z > 82. It is also found that the final production cross sections of neutron-rich nuclei depend only slightly on the incident energy, while those of neutron-deficient nuclei depend strongly on the incident energy.

Time-dependent Hartree-Fock calculations for multinucleon transfer and quasifission processes in theNi64+U238reaction

Background: Multinucleon transfer (MNT) and quasifission (QF) processes are dominant processes in low-energy collisions of two heavy nuclei. They are expected to be useful to produce neutron-rich unstable nuclei. Nuclear dynamics leading to these processes depends sensitively on nuclear properties such as deformation and shell structure. Purpose: We elucidate reaction mechanisms of MNT and QF processes involving heavy deformed nuclei, making detailed comparisons between microscopic time-dependent Hartree-Fock (TDHF) calculations and measurements for the $^{64}$Ni+$^{238}$U reaction. Methods: Three-dimensional Skyrme-TDHF calculations are performed. Particle-number projection method is used to evaluate MNT cross sections from the TDHF wave function after collision. Results: Fragment masses, total kinetic energy (TKE), scattering angle, contact time, and MNT cross sections are investigated for the $^{64}$Ni+$^{238}$U reaction. They show reasonable agreements with measurements. At small impact parameters, collision dynamics depends sensitively on the orientation of deformed $^{238}$U. In tip (side) collisions, we find a larger (smaller) TKE and a shorter (longer) contact time. In tip collisions, we find a strong influence of quantum shells around $^{208}$Pb. Conclusions: It is confirmed that the TDHF calculations reasonably describe both MNT and QF processes in the $^{64}$Ni+$^{238}$U reaction. Analyses of this system indicates the significance of the nuclear structure effects such as deformation and quantum shells in nuclear reaction dynamics at low energies.

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.

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.

Particle-number conservation in thermodynamic systems within the Richardson model

To describe thermal pairing systems, an approach that combines the modified BCS method and the Lipkin-Nogami particle-number projection method is proposed. The expressions of the thermal pairing gap and internal energy are derived. The latter are numerically studied as a function of the temperature within the Richardson model.

Particle number projection in the macroscopic–microscopic approach

We perform nuclear ground-state pairing calculations with the monopole pairing interaction. The particle number fluctuations are taken into account by the particle number projection method, with variation after projection. The pairing-correction energies obtained in this approach are compared to the BCS-model results. We discuss extensively how to properly incorporate different pairing models in global macroscopic–microscopic nuclear mass calculations. A method to calculate the smoothly changing part of the particle number projected energy is developed based on the Strutinsky procedure, making it possible to extract a pairing-shell energy. The behavior of the different pairing models is investigated in detail in the nuclei 164Er and 165Tm. Calculations are then performed along the β-stability line and for several isotope and isotone chains from the proton drip-line to the neutron drip-line. The single-particle energy levels used are obtained from two different single-particle potentials: the folded-Yukawa and the modified-harmonic oscillator potentials. The pairing calculations in the two potentials differ slightly in the fine-structure but the overall results are very similar. When comparing the particle number projected model and the BCS model it is found that the pairing-shell energy is quite insensitive to which microscopic pairing model is used.

QUANTUM NUMBER PROJECTIONS OF NUCLEAR DEFORMED STATES

We combine the exact particle-number projection method with the method of Peierls-Yoccoz in order to build the simultaneous eigen-functions of the particle number and the angular momentum operators. In the axial symmetry case, the general expression of the system energy resulting from this double projection is derived. In order to overcome the complexity of the method, the calculations are performed within the Gaussian overlap approximation. It turns out that, on the one hand, the double projection introduces a non–negligible correction of the energy of the system, and on the other hand, this correction is sensitive to the deformation. Future calculations have to therefore include an evaluation of the equilibrium deformation.

Microscopic HFB Fission Barriers for r-Process Nucleosynthesis Calculations

New systematic calculations of fission barriers are performed in the 3-dimensional deformation space using the constrained Skyrme-Hartree-Fock-Bogoliubov (SHFB) plus Particle-Number Projection method. The BSk8 Skyrme force, fitted to the full set of more than 2000 experimentally known masses, is used. The energy surfaces are analysed using the flooding method, a necessary tool if more than two dimensions are spanned. The SHFB outer fission barriers are compared with experimental data. The reflection asymmetry is shown to play a significant role in the Z ≈ 94 open-shell region. Extrapolations in the neutron-rich region can differ with the previous compilations of fission barriers.

Mean-field based approaches to pairing correlations in atomic nuclei

The evolution of the pairing correlations from closed shell to midshell nuclei is analyzed in the Sn isotopes with the Finite Range Density Dependent Gogny force. As theoretical approaches we use the Hartree–Fock–Bogoliubov, the Lipkin–Nogami, their particle number projected counterparts and the full variation after particle number projection method. We find that whereas all approaches succeed rather well in the description of the total energy they differ significantly in the pairing correlation content of the wave functions. The description of the evolution from the weak to the strong pairing regime is also approach dependent, specially at shell closure.

Particle number projection with effective forces

The particle-number projection method is formulated for density-dependent forces and in particular for the finite-range Gogny force. Detailed formula for the projected energy and its gradient are provided. The problems arising from the neglection of any exchange term, which may lead to divergences, are thoroughly discussed and the possible inaccuracies estimated. Numerical results for the projection after variation method are shown for the nucleus 164Er and for the projection before variation approach for the nuclei 48,50Cr. We also confirm the Coulomb antipairing effect found in mean-field theories.

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.

Approximate particle number projection for rotating nuclei.

Pairing correlations in rotating nuclei are discussed within the Lipkin-Nogami method. The accuracy of the method is tested for the Krumlinde-Szyma\ifmmode \acute{n}\else \'{n}\fi{}ski R(5) model. The results of calculations are compared with those obtained from the standard mean field theory and particle-number projection method, and with exact solutions.