The application of the time-dependent Hartree-Fock method to the calculation of excited state-excited state transition densities of atoms /molecules/
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Csanak, G.↗
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Microscopic predictions for the production of neutron-rich nuclei in the reaction Yb 176 + Yb 176
Background: Production of neutron-rich nuclei is of vital importance to both understanding nuclear structure far from stability and to informing astrophysical models of the rapid neutron capture process (r-process). Multinucleon transfer (MNT) in heavy-ion collisions offers a possibility to produce neutron-rich nuclei far from stability. Purpose: The 176 Yb + 176 Yb reaction has been suggested as a potential candidate to explore the neutron-rich region surrounding the principal fragments. The current study has been conducted with the goal of providing guidance for future experiments wishing to study this (or similar) system. Methods: Time-dependent Hartree-Fock (TDHF) and its time-dependent random-phase approximation (TDRPA) extension are used to examine both scattering and MNT characteristics in 176 Yb + 176 Yb. TDRPA calculations are performed to compute fluctuations and correlations of the neutron and proton numbers, allowing for estimates of primary fragment production probabilities. Results: Both scattering results from TDHF and transfer results from the TDRPA are presented for different energies, orvientations, and impact parameters. In addition to fragment composition, scattering angles and total kinetic energies, as well as correlations between these observables are presented. Conclusions: 176 Yb + 176 Yb appears to be an interesting probe for the midmass neutron-rich region of the chart of nuclides. Lastly, the predictions of both TDHF and TDRPA are speculative, and will benefit from future experimental results to test the validity of this approach to studying MNT in heavy, symmetric collisions.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Laser pulse induced second- and third-harmonic generation of gold nanorods with real-time time-dependent density functional tight binding (RT-TDDFTB) method
In this paper, we investigate second- and third-harmonic generation processes in Au nanorod systems using the real-time time-dependent density functional tight binding method. Our study focuses on the computation of nonlinear signals based on the time dependent dipole response induced by linearly polarized laser pulses interacting with nanoparticles. We systematically explore the influence of various laser parameters, including pump intensity, duration, frequency, and polarization directions, on harmonic generation. We demonstrate all the results using Au nanorod dimer systems arranged in end-to-end configurations, and disrupting the spatial symmetry of regular single nanorod systems is crucial for second-harmonic generation processes. Furthermore, we study the impact of nanorod lengths, which lead to variable plasmon energies, on harmonic generation, and estimates of polarizabilities and hyper-polarizabilities are provided.
77 NANOSCIENCE AND NANOTECHNOLOGY↗
Scalable learning of potentials to predict time-dependent Hartree–Fock dynamics
We propose a framework to learn the time-dependent Hartree–Fock (TDHF) inter-electronic potential of a molecule from its electron density dynamics. Although the entire TDHF Hamiltonian, including the inter-electronic potential, can be computed from first principles, we use this problem as a testbed to develop strategies that can be applied to learn a priori unknown terms that arise in other methods/approaches to quantum dynamics, e.g., emerging problems such as learning exchange–correlation potentials for time-dependent density functional theory. We develop, train, and test three models of the TDHF inter-electronic potential, each parameterized by a four-index tensor of size up to 60 × 60 × 60 × 60. Two of the models preserve Hermitian symmetry, while one model preserves an eight-fold permutation symmetry that implies Hermitian symmetry. Across seven different molecular systems, we find that accounting for the deeper eight-fold symmetry leads to the best-performing model across three metrics: training efficiency, test set predictive power, and direct comparison of true and learned inter-electronic potentials. All three models, when trained on ensembles of field-free trajectories, generate accurate electron dynamics predictions even in a field-on regime that lies outside the training set. To enable our models to scale to large molecular systems, we derive expressions for Jacobian-vector products that enable iterative, matrix-free training.
97 MATHEMATICS AND COMPUTING↗
Pauli energy contribution to nucleus-nucleus interaction
The investigation delves into understanding how the Pauli exclusion principle influences the bare potential between atomic nuclei through the application of advanced theoretical methodologies. Specifically, the application of the novel Frozen-Hartree-Fock (DCFHF) technique is employed. The resulting potentials demonstrate a noticeable repulsion at short distances, attributed to the effects of the Pauli exclusion principle. To account for dynamic phenomena, such as nucleon transfer processes, the density-constrained time-dependent Hartree-Fock (DC-TDHF) method is utilized. This approach integrates isovector contributions into the potential, shedding light on their influence on fusion reactions. Notably, the inclusion of isovector effects leads to a reduction or enhancement in the inner part of the potential, suggesting a nuanced role of transfer in the fusion process.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Quantal diffusion approach for multinucleon transfer processes in the 58,64 Ni + 208 Pb reactions: Toward the production of unknown neutron-rich nuclei
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 article 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 Ni + 208 Pb at E c.m. = 268 MeV and 58 Ni + 208 Pb at E c.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++. 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. 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.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Pauli energy contribution to the nucleus-nucleus interaction
Background: The Pauli exclusion principle plays a crucial role as a building block of many-body quantal systems comprised of fermions. It also induces a “Pauli repulsion” in the interaction between di-nuclear systems. It has been shown in [Phys. Rev. C 95, 031601(R) (2017)] that the Pauli repulsion widens the nucleus-nucleus potential barrier, thus hindering sub-barrier fusion. Purpose: To investigate the proton and neutron contributions to the Pauli repulsion, both in the bare potential neglecting shape polarization and transfer between the reactants, as well as in the dynamical potential obtained by accounting for such dynamical rearrangements. Methods: As the basis of our study we utilize the Pauli kinetic energy (PKE) obtained by studying the nuclear localization function (NLF). Recently this approach has been generalized to incorporate all of the dynamical and time-odd terms present in the nuclear energy density functional. This approach is employed in the density constrained frozen Hartree-Fock (DCFHF) and in the density constrained time-dependent Hartree-Fock (DC- TDHF) microscopic methods. Results: The PKE spatial distribution shows that a repulsion occurs in the neck between the nuclei when they first touch. Inside the barrier, neutrons can contribute significantly more to the Pauli repulsion in neutron-rich systems. Dynamical effects tend to lower the Pauli repulsion near the barrier. Proton and neutron dynamical contributions to the PKE significantly differ inside the barrier for asymmetric collisions, which is interpreted as an effect of multinucleon transfer. Conclusions: The PKE is shown to make a significant contribution to nuclear interaction potentials. Protons and neutrons can play very different roles in both the bare potential and in the dynamical rearrangement. Further microscopic studies are required to better understand the role of transfer and to investigate the effect of pairing and deformation.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Role of isospin composition in low-energy nuclear fusion
We employ a microscopic approach that examines the impact of isospin dynamics on the process of low- energy nuclear fusion along an isotope chain and dependence on deformation. Our method utilizes the density constrained time-dependent Hartree-Fock theory (DC-TDHF), where isoscalar and isovector characteristics of the energy density functional (EDF) are examined in turn. This approach is applied to a series of fusion interactions of 176 Yb with increasingly neutron rich isotopes of calcium. By evaluating the contributions from the isoscalar and isovector components of the EDF, we look to quantify the influence of isospin composition on the conditions under which fusion is most likely to take place. Furthermore, our findings reveal that, in nonsymmetric systems, the isovector dynamics plays a significant role. Its typical effect is a reduction in the potential barrier, which turns into enhancement for neutron-rich systems.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Microscopic description of the torque acting on fission fragments
When two fragments are created in a fission decay, any torque due to nuclear and Coulomb interaction can change the fragment's angular momentum. This paper explores the character and magnitude of the angular momentum as a function of the initial conditions around the scission point using the time-dependent Hartree-Fock theory. To understand the torque acting on the fragments, the Frozen Hartree-Fock method is also used to determine the collective potential at scission. Two 240 Pu fission channels ( 132 Sn + 108 Ru and 144 Ba + 96 Sr) are studied. Further, these two channels cover different shapes (spherical, quadrupole, and octupole deformation) of the fragments. It is found that the angular momentum generated by the Coulomb interaction after fission is mainly collective, while this is not the case for the angular momentum generated at scission. The competition between rotational modes (bending, wriggling, and twisting) is discussed and shows that the angular momentum is generated mainly perpendicular to the fission axis.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Microscopic study of the fusion reactions 40,48 Ca+ 78 Ni and the effect of the tensor force
In this work, we provide a microscopic description of the fusion reactions between 40,48 Ca and 78 Ni. The internuclear potentials are obtained using the density-constrained (DC) time-dependent Hartree-Fock (TDHF) approach and fusion cross sections are calculated via the incoming wave boundary condition method. By performing DC-TDHF calculations at several selected incident energies, the internuclear potentials for both systems are obtained and the energy-dependence of fusion barrier are revealed. The influence of tensor force on internuclear potentials of 48 Ca+ 78 Ni is more obvious than those of 40 Ca+ 78 Ni. By comparing the calculated fusion cross sections between 40 Ca+ 78 Ni and 48 Ca+ 78 Ni, an interesting enhancement of subbarrier fusion cross sections for the former system is found, which can be explained by the narrow width of internuclear potential for 40 Ca+ 78 Ni while the barrier heights and positions are very close to each other. The tensor force suppresses the subbarrier fusion cross sections of both two systems.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Nonlinear Optimal Control of Electron Dynamics Within Hartree-Fock Theory
Consider the problem of determining the optimal applied electric field to drive a molecule from an initial state to a desired target state. For even moderately sized molecules, solving this problem directly using the exact equations of motion—the time-dependent Schrödinger equation (TDSE)—is numerically intractable. Here, we present a solution of this problem within time-dependent Hartree-Fock (TDHF) theory, a mean field approximation of the TDSE. Optimality is defined in terms of minimizing the total control effort while maximizing the overlap between desired and achieved target states. We frame this problem as an optimization problem constrained by the nonlinear TDHF equations; we solve it using trust region optimization with gradients computed via a custom-built adjoint state method. For three molecular systems, we show that with very small neural network parametrizations of the control, our method yields solutions that achieve desired targets within acceptable constraints and tolerances.
97 MATHEMATICS AND COMPUTING↗
Description of the multinucleon transfer mechanism for Ca 48 + Pu 244 and Kr 86 + Pt 198 reactions in a quantal transport approach
Multinucleon transfer (MNT) reactions involving heavy projectile and target combinations stand as a promising method for synthesizing new neutron-rich exotic nuclei, which may not be possible using hot or cold fusion reactions or fragmentation. Exploring the mechanisms behind MNT reactions is essential and it requires a comprehensive theoretical framework that can explain the physical observables in these reactions. This work aims to show that the quantal diffusion approach based on the stochastic mean-field (SMF) theory is capable of explaining the reaction dynamics observed in MNT reactions. Primary product mass distributions in 48 Ca + 244 Pu reaction at E c.m. = 203.2 MeV and 86 Kr + 198 Pt reaction at E c.m. = 324.2 MeV are calculated and compared with the available experimental data. In this work, we utilize the time-dependent Hartree-Fock (TDHF) calculations to analyze the mean-field reaction dynamics computationally in the reactions 48 Ca + 244 Pu and 86 Kr + 198 Pt for a broad range of initial angular momenta. Quantal transport description based on the SMF approach is used to calculate quantal diffusion coefficients and mass variances in 48 Ca + 244 Pu and 86 Kr + 198 Pt systems. The primary products arising from quasifission reactions are described by joint probability distribution in the SMF approach and those arising from fusion-fission are estimated by using the statistical deexcitation code gemini + +. Mean values of charge and mass numbers, scattering angles of the primary reaction products, and the total kinetic energies after the collision are calculated within the TDHF framework for a broad range of initial angular momenta. Throughout all the collisions, drift toward the mass symmetry and large mass dispersion associated with this drift are observed. Here, the calculated primary fragment and mass distributions using the SMF approach successfully explain experimental observations for the 48 Ca + 244 Pu and 86 Kr + 198 Pt systems. The primary mass distributions, mean values of binary products, and mass dispersions are determined and results are compared with the available experimental data. The observed agreement between the experimental data and SMF results highlights the effectiveness of the quantal diffusion mechanism based on the SMF approach, which does not include any adjustable parameters other than standard parameters of Skyrme energy density functional.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Quantal diffusion description of isotope production via the multinucleon transfer mechanism in 48 Ca+ 238 U collisions
As an extension of previous work, we calculate the production cross section of heavy neutron-rich isotopes by employing the quantal diffusion description to 48 Ca+ 238 U collisions. The quantal diffusion is deduced from stochastic mean-field approach, and transport properties are determined in terms of time-dependent single-particle wave functions of the time-dependent Hartree-Fock theory. As a result, the approach allows for prediction of production cross sections without any adjustable parameters. Furthermore, the secondary cross sections by particle emission are calculated with the help of the statistical gemini++ code.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Search for beyond-mean-field signatures in heavy-ion fusion reactions
Examination of high-resolution, experimental fusion excitation functions for 16,17,18 O + 12 C reveals a remarkable irregular behavior that is rooted in the structure of both the colliding nuclei and the quasi-molecular composite system. The impact of the l-dependent fusion barriers is assessed using a time-dependent Hartree-Fock model, viewed as a mean-field reference. Barrier penetrabilities, taken directly from a density-constrained calculation, provide a significantly improved description of the experimental data as compared to the standard Hill-Wheeler approach. Furthermore, the remaining deviations between the parameter-free theoretical mean-field predictions and experimental fusion cross sections are exposed and discussed.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Merging of transport theory with the time-dependent Hartree-Fock approach: Multinucleon transfer in U + U collisions
Multinucleon transfer mechanism in the collision of 238 U + 238 U system is investigated at E c.m. = 833 MeV in the framework of the quantal diffusion description based on the stochastic mean-field approach. Double cross sections σ (N, Z ) as a function of the neutron and proton numbers, the cross sections σ (Z ) and σ (A) as a function of the atomic numbers and the mass numbers are calculated for production of the primary fragments. The calculation indicates the 238 U + 238 U system may be located at an unstable equilibrium state at the potential energy surface with a slightly negative curvature along the β stability line on the (N, Z ) plane. Here, this behavior may lead to rather large diffusion along the β stability direction.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Kinetic-energy dissipation and fluctuations in strongly damped heavy-ion collisions within the stochastic mean-field approach
Microscopic mean-field approaches have been successful in describing the most probable reaction outcomes in low-energy heavy-ion reactions. However, those approaches are known to severely underestimate dispersions of observables around the average values that has limited their applicability. Recently it has been shown that a quantal transport approach based on the stochastic mean-field (SMF) theory significantly improves the description, while its application has been limited so far to fragment mass and charge dispersions. In this work, we extend the quantal transport approach based on the SMF theory for relative kinetic energy dissipation and angular momentum transfer in low-energy heavy-ion reactions. Based on the SMF concept, analytical expressions are derived for the radial and tangential friction and associated diffusion coefficients. Those quantal transport coefficients are calculated microscopically in terms of single-particle orbitals within the time-dependent Hartree-Fock (TDHF) approach. As the first application of the proposed formalism, we consider the radial linear momentum dispersion, neglecting the coupling between radial and angular momenta. We analyze the total kinetic energy (TKE) distribution of binary reaction products in the 136 Xe + 208 Pb reaction at E c.m. =526 MeV and compare with experimental data. From time evolution of single-particle orbitals in TDHF, the radial diffusion coefficient is computed on a microscopic basis, while a phenomenological treatment is introduced for the radial friction coefficient. By solving the quantal diffusion equation for the radial linear momentum, the dispersion of the radial linear momentum is obtained, from which one can construct the TKE distribution. We find that the calculations provide a good description of the TKE distribution for strongly damped events with large energy losses, TKEL ≳ 150 MeV. However, the calculations underestimate the TKE distribution for smaller energy losses. Further studies are needed to improve the technical details of calculations.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Proton and neutron density distributions at supranormal density in low- and medium-energy heavy-ion collisions. II. Central Pb + Pb collisions
This paper represents a continuation of our investigation into the limitations on total, proton, and neutron particle number densities, as well as the asymmetry of proton and neutron density distributions achievable in central heavy-ion collisions. We explore these aspects at low and medium energies within the framework of the Boltzmann-Uhlenbeck-Uehling (pBUU) transport and time-dependent Hartree-Fock (TDHF) models. Previous studies, focusing on symmetric and asymmetric collisions of Ca and Sn nuclei [1], and initial results on Pb-nuclei collisions [2], emphasized the role of the Coulomb interaction in these events. Our findings indicated that: (i) the highest total densities predicted at 𝐸 beam = 800 MeV/nucleon were on the order of ≈ 2.5𝜌 0 (𝜌 0 = 0.16 fm −3 ), (ii) the proton-neutron asymmetry for maximal densities, 𝛿=(𝜌$^{max}_{n}$− 𝜌$^{max}_{p}$)/(𝜌$^{max}_{n}$+𝜌$^{max}_{p}$), did not generally exceed the asymmetry in the initial state of the collision at all beam energies and tended to decrease during the reaction, and (iii) a significant portion of this asymmetry had its microscopic origin in Coulomb forces, masking the pure nuclear contribution. These new findings, particularly relevant in the astrophysical context, are further examined in this work, focusing on the heaviest target-projectile combination 212,208 Pb accessible in an experiment. We introduce the SkT3 Skyrme force model, not previously used for the Pb system, and compare it to the SV-bas and SV-sym34 models to explore the symmetry-energy dependence of the results. Contour plots of nucleonic densities are presented, contrasting the time evolution of the density distributions in low (TDHF) and high (pBUU) models. We also present the evolution of normalized maximal proton, neutron, and total nucleon number density with increasing beam energy in the full pBUU model and the Vlasov approximation, aiming to explore the impact of correlations in the reaction. The time evolution of the proton and neutron density distributions in the plane transverse to the beam direction is illustrated at both low and high beam energy. In conclusion, our detailed examination of the Pb system in this work provides further essential evidence that the aforementioned findings (i)–(iii) are only weakly dependent on system size and a symmetry-energy model, and thus, they are of more general importance.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗