Three-dimensional Time-dependent Hartree-Fock Research Articles

Asymmetry in high-order harmonic generation of polar molecule CO

Compared with nonpolar molecules, owing to the inherent asymmetry, polar molecules exhibit rich and very complex electronic dynamics under the interaction with strong laser fields. In this work, high-order harmonic generation (HHG) of polar molecules CO is investigated by using the three-dimensional time-dependent Hartree-Fock (3D-TDHF) theory, with all electrons active. Through the high harmonic spectra and time-frequency analyses, it is found that when the laser field polarizes along the molecular axis, the ionized electrons from the two sides (C side and O side) contribute differently to the harmonic radiation. On the one hand, the harmonic intensity from the C side is greater than that from the O side, which is caused by the ionization rate. On the other hand, for the lower-order (7<sup>th</sup>–17<sup>th</sup> order) harmonics of plateau region, only the electrons from the C side participate in the HHG. However, for its higher part (18<sup>th</sup>–36<sup>th</sup> order), the electrons from both C side and O side contribute to high harmonics simultaneously. Moreover, the difference between contributions from two sides is related to the alignment angle <i>θ</i> between the laser polarization and the molecular axis, and it reaches a maximum value around <i>θ</i> = 0º and a minimum value around <i>θ</i> = 90º. There are two strong resonances around harmonic order H12.6 (19.5 eV) and H18 (27.9 eV) in the harmonic spectra when <i>θ</i> = 0º. The first resonance around H12.6 reveals that part of electrons ionized from the C side recombine to the vicinity of the further O nucleus. Near the second resonance around H18, there appears a shape resonance. Nevertheless, the shape resonances from the C and O sides are disparate. Based on the strong-field approximation theory, the ratio between photoionization cross sections from C and O sides around the shape resonance is calculated. The ratio is about 5.5 from 3D-TDHF, which is greater than the result of 3 simulated by ePloyScat, where only HOMO is considered. This discrepancy reveals that multi-electron effects enhance the asymmetry of polar molecules. This work provides an in-depth insight into the asymmetry in HHG of polar molecules, which benefits the generation of isolated attosecond pulse . It also promotes the application of high harmonic spectra in tracking the ultrafast dynamics of electrons.

Quantal diffusion approach for multinucleon transfer processes in theNi58,64+Pb208reactions: Toward the production of unknown neutron-rich nuclei

Background: In recent years, substantial efforts have been made for the study of multinucleon transfer reactions at energies around the Coulomb barrier both experimentally and theoretically, aiming at the production of unknown neutron-rich heavy nuclei. It is crucial to provide reliable theoretical predictions based on microscopic theories with sufficient predictive power.Purpose: This paper aims to clarify the applicability of the quantal diffusion approach based on the stochastic mean-field (SMF) theory for multinucleon transfer processes. Isotope production cross sections are evaluated for the reactions of $^{64}\mathrm{Ni}+^{208}\mathrm{Pb}$ at ${E}_{\mathrm{c}.\mathrm{m}.}$ $=$ 268 MeV and $^{58}\mathrm{Ni}+^{208}\mathrm{Pb}$ at ${E}_{\mathrm{c}.\mathrm{m}.}$ $=$ 270 MeV and are compared with available experimental data.Methods: Three-dimensional time-dependent Hartree-Fock (TDHF) calculations are carried out for a range of initial orbital angular momenta with Skyrme SLy4d functional. Quantal diffusion equations, derived based on the SMF theory, for variances and covariance of neutron and proton numbers of reaction products are solved, with microscopic drift and diffusion coefficients obtained from time evolution of occupied single-particle orbitals in TDHF. Secondary de-excitation processes, both particle evaporation and fission, are simulated by a statistical compound-nucleus de-excitation model, gemini++.Results: Dynamics of a fast isospin equilibration process followed by a slow drift toward the mass symmetry are commonly observed, as expected. Various reaction outcomes are evaluated, including average mass and charge numbers of reaction products, total kinetic energy loss (TKEL), scattering angle, contact time, and production cross sections for primary and secondary products. By comparing with the experimental data, we find that SMF and TDHF quantitatively reproduce experimental data for few-nucleon-transfer channels around the average values. In contrast, for many-nucleon-transfer channels, we find that the SMF approach provides much better description of the experimentally measured isotopic distributions. The results underline the importance of beyond-mean-field effects, especially one-body (mean-field) fluctuations and correlations, in describing multinucleon transfer processes. Moreover, through a combined analysis of SMF with a statistical model, gemini++, we find a significant contribution of transfer-induced fission, which is consistent with the experimental observation. In some cases, the SMF approach overestimates the isotopic width, requiring further improvements of the theoretical description. Possible ways to improve the description are discussed.Conclusions: The SMF approach is designed to describe the quantum many-body problem according to an ensemble of mean-field trajectories, taking into account part of many-body correlations in the description. As it requires feasible computational costs comparable to the ordinary TDHF approach, together with further model improvements, it will be a promising tool in the search for optimal reaction conditions to produce yet-unknown neutron-rich heavy nuclei through the multinucleon transfer reaction.

Predictions of New Neutron-Rich Isotopes at N = 126 in the Multinucleon Transfer Reaction 136Xe + 194Ir

Aiming to produce new neutron-rich nuclei around $N=126$, the multi-nucleon transfer reaction $^{136}$Xe+$^{194}$Ir is investigated by using the GRAZING model and the three-dimensional time-dependent Hartree-Fock (TDHF) approach. De-excitation processes of the primary fragments are taken into account in both models. The results of GRAZING indicate that this system is a promising candidate for producing new neutron-rich isotones with $N=126$. In contrast, no such nuclei are observed in TDHF. Nucleon transfer process in TDHF collisions is studied and the relation to final isotopic production cross sections is discussed.

Multiconfiguration time-dependent Hartree-Fock treatment of electron correlation in strong-field ionization of H2 molecules

Electron correlation plays an important role in the multielectron interactions of many physical and chemical processes.The investigation of correlation effects in the non-perturbative electronic dynamics (e.g.non-sequential double ionization) when atoms and molecules are subjected to strong laser fields requires non-perturbative theoretical treatments. The direct numerical integration of the time-dependent Schrödinger equation successfully explains many experimental results,but it is computationally prohibitive for systems with more than two electrons.There is clearly a need for a theory which can treat correlation dynamics self-consistently in strong time-dependent electric fields.In this paper we develop a three-dimensional multiconfiguration time-dependent Hartree-Fock method,which can be applied to the non-perturbative electronic dynamics for diatomic molecules,and it can also investigate the effect of electron correlation in strong-field ionization of H2 molecules.This method adopts the prolate spheroidal coordinates (which can treat the two-center Coulomb potential accurately) and the finite-element method together with discrete-variable representation (which lowers the calculation burden from two-electron integrations).For the temporal propagation,we use the efficient short iterative Lanczos algorithm for the equation which governs the configuration expansion coefficients,while an eight-order Runge-Kutta (RK) method and an Bulirsch-Stoer (BS) extrapolation method,both with adaptive precision controls,are implemented to solve the nonlinear orbital equation.While both methods yield correct results,the BS method displays a better stability in the realtime propagation,while the RK method demands less computation.The alignment-dependent ionization probabilities of H2 molecules in intense extreme ultraviolet pulses are calculated.Comparisons between multi-configuration and single-configuration results show that electron correlation has little effect on the single ionization process,but it plays an important role in double ionization,leading to the decrease in the ionization probability.The double ionization probability from the single-configuration space 1σ is about three times larger that from 4σ1π.The ionization probability increases monotonically when the alignment angle increases from 0° to 90°, yielding a ratio of 2.6(1.5) between 90° and 0° for the double (single) ionization process.This method presents the basis for the future study of electron correlation in strong-field processes.

Configuration transition effect in heavy-ion fusion reactions with deformed nuclei

Fusion reactions of 16O + 24Mg are investigated with the three-dimensional time-dependent Hartree-Fock (TDHF) method. The resultant fusion excitation functions show some novel features that are beyond the current understandings and the predictions of various theoretical models. We will demonstrate that these phenomena cannot be explained by a density rearrangement effect but by the configuration transition mechanism in the dynamical process.

Microscopic study of noncentral effects in heavy-ion fusion reactions with spherical nuclei

Fusion reactions of $^{16}\mathrm{O}+^{16}\mathrm{O},^{40}\mathrm{Ca}+^{40}\mathrm{Ca}$, and $^{40}\mathrm{Ca}+^{90}\mathrm{Zr}$ have been investigated by using the three-dimensional (3D) time-dependent Hartree-Fock approach with and without density constraint. The interaction potentials and effective mass parameters are studied. It is found that both the nucleus-nucleus interaction potentials and the effective mass parameters of these systems depend on the impact parameter. Such noncentral effects on fusion cross sections are also investigated. It is shown that the noncentral effect can reduce the fusion cross sections significantly when the incident energy is just above the barrier, while it becomes weaker with the increasing incident energy and can be totally neglected when the energy is up to twice as much as the barrier height in these spherical colliding systems.

Dissipation dynamics and spin-orbit force in time-dependent Hartree-Fock theory

We investigate the one-body dissipation dynamics in heavy-ion collisions of $^{16}\mathrm{O}+^{16}\mathrm{O}$ using a fully three-dimensional time-dependent Hartree-Fock (TDHF) theory with the modern Skyrme energy functional and without any symmetry restrictions. The energy dissipation is revealed to decrease in deep-inelastic collisions of the light systems as the bombarding energy increases owing to the competition between collective motion and single-particle degrees of freedom. The role of spin-orbit force is given particular emphasis in deep-inelastic collisions. The spin-orbit force causes a significant enhancement of the dissipation. The time-even coupling of spin-orbit force plays a dominant role at low energies, while the influence of time-odd terms is notable at high energies. About 40--65% of the total dissipation depending on the different parameter sets is predicted to arise from the spin-orbit force. The theoretical fusion cross section has a reasonably good agreement with the experimental data, considering that no free parameters are adjusted to reaction dynamics in the TDHF approach.

High-spin torus isomers and their precession motions

We systematically investigate the existence of exotic torus isomers and their precession motions for a series of $N=Z$ even-even nuclei from $^{28}$Si to $^{56}$Ni. We analyze the microscopic shell structure of the torus isomer and discuss why the torus shape is generated beyond the limit of large oblate deformation. We use the cranked three-dimensional Hartree-Fock (HF) method with various Skyrme interactions in a systematic search for high-spin torus isomers. We use the three-dimensional time-dependent Hartree-Fock (TDHF) method for describing the precession motion of the torus isomer. We obtain high-spin torus isomers in $^{36}$Ar, $^{40}$Ca, $^{44}$Ti, $^{48}$Cr, and $^{52}$Fe. The emergence of the torus isomers is associated with the alignments of single-particle angular momenta, which is the same mechanism as found in $^{40}$Ca. It is found that all the obtained torus isomers execute the precession motion at least two rotational periods. The moment of inertia about a perpendicular axis, which characterizes the precession motion, is found to be close to the classical rigid-body value. The high-spin torus isomer of $^{40}$Ca is not an exceptional case. Similar torus isomers exist widely in nuclei from $^{36}$Ar to $^{52}$Fe and they execute the precession motion. The torus shape is generated beyond the limit of large oblate deformation by eliminating the $0s$ components from all the deformed single-particle wave functions to maximize their mutual overlaps.

Effect of tensor force on dissipation dynamics in time-dependent Hartree-Fock theory

The role of tensor force on the collision dynamics of 16O+16O is investigated in the framework of a fully three-dimensional time-dependent Hartree-Fock theory. The calculations are performed with modern Skyrme energy functional plus tensor terms. Particular attention is given on the analysis of dissipation dynamics in heavy-ion collisions. The energy dissipation is found to decrease as an initial bombarding energy increases in deep-inelastic collisions for all the Skyrme parameter sets studied here because of the competition between the collective motion and the single-particle degrees of freedom. We reveal that the tensor forces may either enhance or reduce the energy dissipation depending on the different parameter sets. The fusion cross section without tensor force overestimates the experimental value by about 25%, while the calculation with tensor force T11 has good agreement with experimental cross section.

Pure collective precession motion of a high-spin torus isomer

We investigate the precession motion of the exotic torus configuration in high-spin excited states of ${}^{40}$Ca. For this aim, we use the three-dimensional time-dependent Hartree-Fock (TDHF) method. Although the high-spin torus isomer is a unique quantum object characterized by the alignment of angular momenta of independent single-particle motions, we find that the obtained moment of inertia for rotations about an axis perpendicular to the symmetry axis is close to the rigid-body value. We also analyze the microscopic structure of the precession motion using the random-phase approximation (RPA) method for high-spin states. In the RPA calculation, the precession motion of the torus isomer is generated by coherent superposition of many one-particle--one-hole excitations across the sloping Fermi surface that strongly violates the time-reversal symmetry. By comparing results of the TDHF and the RPA calculations, we find that the precession motion obtained by the TDHF calculation is a pure collective motion well decoupled from other collective modes.

Dynamic Core Polarization in Strong-Field Ionization of CO Molecules

The orientation-dependent strong-field ionization of CO molecules is investigated using the fully propagated three-dimensional time-dependent Hartree-Fock theory. The full ionization results are in good agreement with recent experiments. The comparisons between the full method and the single active orbital method show that although the core electrons are generally more tightly bound and contribute little to the total ionization yields, their dynamics cannot be ignored, which effectively modifies the behavior of electrons in the highest occupied molecular orbital. By incorporating it into the single active orbital method, we identify that the dynamic core polarization plays an important role in the strong-field tunneling ionization of CO molecules, which is helpful for the future development of the tunneling ionization theory of molecules beyond the single active electron approximation.

Microscopic study of16O+16Ofusion

We perform a study of $^{16}$O+$^{16}$O fusion at above and below the interaction barrier energies using three-dimensional time-dependent Hartree-Fock (TDHF) calculations at above barrier energies and density-constrained TDHF calculations for the entire energy range. We discuss the variations of the experimental data at above the barrier energies. Calculations reasonably reproduce the observed energy-dependent broad oscillations in the fusion excitation functions. These oscillations result from overcoming $L$-dependent fusion barriers. The role of the coupling to low-lying octupole states is also discussed.

Actinide collisions for QED and superheavy elements with the time-dependent Hartree-Fock theory and the Balian-Vénéroni variational principle

Collisions of actinide nuclei form, during very short times of few zs ($10^{-21}$ s), the heaviest ensembles of interacting nucleons available on Earth. Such collisions are used to produce super-strong electric fields by the huge number of interacting protons to test spontaneous positron-electron pair emission (vacuum decay) predicted by the quantum electrodynamics (QED) theory. Multi-nucleon transfer in actinide collisions could also be used as an alternative way to fusion in order to produce neutron-rich heavy and superheavy elements thanks to inverse quasifission mechanisms. Actinide collisions are studied in a dynamical quantum microscopic approach. The three-dimensional time-dependent Hartree-Fock (TDHF) code {\textsc{tdhf3d}} is used with a full Skyrme energy density functional to investigate the time evolution of expectation values of one-body operators, such as fragment position and particle number. This code is also used to compute the dispersion of the particle numbers (e.g., widths of fragment mass and charge distributions) from TDHF transfer probabilities, on the one hand, and using the Balian-Veneroni variational principle, on the other hand. A first application to test QED is discussed. Collision times in $^{238}$U+$^{238}$U are computed to determine the optimum energy for the observation of the vacuum decay. It is shown that the initial orientation strongly affects the collision times and reaction mechanism. The highest collision times predicted by TDHF in this reaction are of the order of $\sim4$ zs at a center of mass energy of 1200 MeV. According to modern calculations based on the Dirac equation, the collision times at $E_{cm}>1$ GeV are sufficient to allow spontaneous electron-positron pair emission from QED vacuum decay, in case of bare uranium ion collision. A second application of actinide collisions to produce neutron-rich transfermiums is discussed. A new inverse quasifission mechanism associated to a specific orientation of the nuclei is proposed to produce transfermium nuclei ($Z>100$) in the collision of prolate deformed actinides such as $^{232}$Th+$^{250}$Cf. The collision of the tip of one nucleus with the side of the other results in a nucleon flux toward the latter. The probability distributions for transfermium production in such a collision are computed. The produced nuclei are more neutron-rich than those formed in fusion reactions, thus, leading to more stable isotopes closer to the predicted superheavy island of stability. In addition to mass and charge dispersion, the Balian-Veneroni variational principle is used to compute correlations between $Z$ and $N$ distributions, which are zero in standard TDHF calculations.

Two different modes associated with the composite nucleus- the charge and mass distributions for exotic nuclear synthesis -

The mass-dependent structure of the composite nucleus is shown based on three-dimensional timedependent Hartree-Fock calculations with Skyrme interactions (SLy4d and SkM*). One remarkable result is that the isovector monopole excitation dominantly appears for collisions of heavy nuclei, and the isovector dipole excitation for those of light ones. Such a difference found in the dynamical structure of composite nucleus plays a role in the equilibration of charge.

Erratum to: Synthesis of exotic nuclei in heavy-ion collisions at higher energies

The charge equilibration upper energy-limit formula is proposed. It shows not only the basic mechanism of charge equilibration but also does imply the possibility of controlling the exotic nuclear synthesis. To understand the mechanism of charge equilibration, charge and mass distributions of final fragments of the heavy-ion collisions are studied based on three-dimensional time-dependent Hartree-Fock calculations with a Skyrme interaction.

Mass dispersions from giant dipole resonances using the Balian–Vénéroni variational approach

The Balian–Vénéroni variational approach has been implemented using a three-dimensional time-dependent Hartree–Fock (TDHF) code with realistic Skyrme interactions and used to investigate the mass dispersions from giant dipole resonances in 32S and 132Sn decaying through particle emission. The fluctuations obtained are shown to be quantitatively larger than the standard TDHF results.

Three-dimensional time-dependent Hartree-Fock approach for arbitrarily oriented molecular hydrogen in strong electromagnetic fields

We present a theoretical framework for the electronic dynamics of arbitrarily oriented molecular hydrogen in strong and short electromagnetic fields. The ground state of ${\mathrm{H}}_{2}$ is obtained by propagating the time-dependent Schr\"odinger equation in imaginary time by assuming the Hartree-Fock ansatz for the interaction between the electrons. The interaction of ${\mathrm{H}}_{2}$ with the radiation field is considered in the single-active-electron approximation, with the continuum electron subject to Hartree-Fock radial potentials. We propagate the wave function by a split-operator scheme projected on a spherical harmonics basis. Alignment-dependent yields and angular distributions for one- and two-photon ionization induced by an external femtosecond light source are presented and compared with available theoretical results.

Boost-invariant mean field approximation and the nuclear Landau-Zener effect

We investigate the relation between time-dependent Hartree-Fock (TDHF) states and the adiabatic eigenstates by constructing a boost-invariant single-particle Hamiltonian. The method is numerically realized within a full three-dimensional TDHF which includes all the terms of the Skyrme energy functional and without any symmetry restrictions. The study of free translational motion of a nucleus demonstrates the validity of the concept of boost-invariant and adiabatic TDHF states. The interpretation is further corroborated by the test case of fusion of 16 O + 16 O. As a first application, we present a study of the nuclear Landau-Zener effect on a collision of 4 He + 16 O.

Direct inelastic scattering ofN14+C12in a three-dimensional time-dependent Hartree-Fock scheme

Momentum space three-dimensional time-dependent Hartree-Fock (TDHF) calculations are performed for $^{12}\mathrm{C}$ + $^{14}\mathrm{N}$ and $^{16}\mathrm{O}$ + $^{16}\mathrm{O}$ collisions. The potential is the simplified Skyrme interaction which gives the single-particle potential ${v}_{N}(r)=\ensuremath{-}a\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+b{\ensuremath{\rho}}^{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, and includes the direct part of the Coulomb interaction. We use a predictor corrector method to step in time and a fast Fourier transform (FFT) method to compute the kinetic energy in momentum space. Three types of events are obtained: vibrational instability scattering at low impact parameters $b$, near orbiting at medium $b$ values, and rotational instability scattering beyond that. The discussion points out that further improvements will require a fusion window and quantal corrections for angular and energy uncertainties which go beyond the TDHF method.NUCLEAR REACTIONS $^{14}\mathrm{N}$(112MeV)+$^{12}\mathrm{C}$, $^{16}\mathrm{O}$(128 MeV)+$^{16}\mathrm{O}$, calculated deflection functions using time-dependent Hartree-Fock approximation.

Comparisons of two- and three-dimensional time-dependent Hartree-Fock calculations of the reactionsO16+O16andCa40+Ca40

Two-dimensional and three-dimensional time-dependent Hartree-Fock calculations have been performed over a wide range of angular momenta for $^{16}\mathrm{O}$ + $^{16}\mathrm{O}$ at ${E}_{\mathrm{lab}}=105$ MeV and for $^{40}\mathrm{Ca}$ + $^{40}\mathrm{Ca}$ at ${E}_{\mathrm{lab}}=192$ MeV. In all of the two-dimensional calculations it is assumed that the nuclear system is axially-symmetric about the line joining the mass centers of the colliding ions. Two very different two-dimensional, axially-symmetric models are considered. (a) In the first case, it is assumed that after the two ions interpenetrate the moment of inertia of the system attains the rigid-body value. (b) In the second model, each single-particle wave function is assumed to be multiplied by an extra phase factor which depends upon the azimuthal angle. This model yields an irrotational fluid flow. The results of time-dependent Hartree-Fock (TDHF) calculations with both of these models are compared with each other and with three-dimensional TDHF results. It is concluded that the two-dimensional calculations reproduce reasonably well the three-dimensional results for values of the angular momentum both below and above the three-dimensional fusion window. As the laboratory bombarding energy is decreased, there is better agreement between the two- and three-dimensional calculations, including cases in which the angular momentum is within the fusion region.NUCLEAR REACTIONS $^{16}\mathrm{O}(^{16}\mathrm{O}, x)$ and $^{40}\mathrm{Ca}(^{40}\mathrm{Ca}, x)$ in 2- and 3-dimensional time-dependent Hartree-Fock calculations. Comparisions between 2- and 3- dimensional results. Fusion and strongly damped collisions.