Search

Background: 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. Purpose: 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. Results: 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 $E_\mathrm{c.m.}=526$ 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 large values of energy losses, TKEL $\gtrsim$ 150 MeV. However, the calculations underestimate the TKE distribution for smaller energy losses. Further studies are needed to improve the technical details of calculations. (Shortened due to the word limit), Comment: 17 pages, 6 figures, 2 tables (plus 5 figures in Appendix); v2 - Version accepted for publication in Physical Review C, with additional analysis on DD-TDHF in Appendix C

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 $^{64}$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 the time evolution of occupied single-particle orbitals in TDHF. Secondary deexcitation processes, both particle evaporation and fission, are simulated by a statistical model, GEMINI++. Results: We find that the quantal diffusion approach reproduces the measured cross sections for abundant reaction products, including both projectile-like and target-like fragments, as well as fragments of transfer-induced fission. The results underline the importance of beyond mean-field effects in describing multinucleon transfer processes. We emphasize that the quantal diffusion description does not involve any adjustable parameters, once an energy density functional is given. Possible future directions are discussed. (Shortened due to the word limit), Comment: 26 pages, 12 figures (plus 4 figures, 2 tables in Appendix); v2 - Version accepted for publication in Physical Review C; Supplemental movies can be found at https://www.youtube.com/playlist?list=PLKTmRVRArzsQUKpMiOXAzliVSgzjD0ZG2

We analyze here in details the probability to find a given number of particles in a finite volume inside a normal or superfluid finite system. This probability, also known as counting statistics, is obtained using projection operator techniques directly linked to the characteristic function of the probability distribution. The method is illustrated in atomic nuclei. The nature of the particle number fluctuations from small to large volumes compared to the system size are carefully analyzed in three cases: normal systems, superfluid systems and superfluid systems with total particle number restoration. The transition from Poissonian distribution in the small volume limit to Gaussian fluctuations as the number of particles participating to the fluctuations increases, is analyzed both in the interior and at the surface of the system. While the restoration of total number of particles is not necessary for small volume, we show that it affects the counting statistics as soon as more than very few particles are involved.

In this note we provide answers to criticisms made the appendix D of the article [Phys. Rev. C100, 034615 (2019)] against to the theory developed recently and known a "Stochastic Mean-Field" (SMF) approach and its applications. Briefly, we list our replies and leave the judgment to the readers., Comment: 3 pages, no figures

The multi-nucleon exchange mechanism in ${}^{58} \text{Ni}+{}^{60} \text{Ni}$ and ${}^{60} \text{Ni}+{}^{60}\text{Ni}$ collisions is analyzed in the framework of the stochastic mean-field approach. The results of calculations are compared with the TDRPA calculations and the recent data of ${}^{58} \text{Ni}+{}^{60} \text{Ni}$. A good description of the data and a relatively good agreement with the TDRPA calculations are found., Comment: 6 pages, 6 figures

We show that the total kinetic energy (TKE) of nuclei after the spontaneous fission of $^{258}$Fm can be well reproduced using simple assumptions on the quantum collective phase-space explored by the nucleus after passing the fission barrier. Assuming energy conservation and phase-space exploration according to the stochastic mean-field approach, a set of initial densities is generated. Each density is then evolved in time using the nuclear time-dependent density-functional theory. This approach goes beyond mean-field by allowing spontaneous symmetry breaking as well as a wider dynamical phase-space exploration leading to larger fluctuations in collective space. The total kinetic energy and mass distributions are calculated. New information on the fission process: fluctuations in scission time, strong correlation between TKE and collective deformation of daughter nuclei as well as pre- and post-scission particle emission, are obtained., Comment: Accepted to PRL (Supplement material available upon demand). All calculations made with dynamical pairing

The stochastic mean-field (SMF) approach allows to treat correlations beyond mean-field using a set of independent mean-field trajectories with appropriate choice of fluctuating initial conditions. We show here, that this approach is equivalent to a simplified version of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy between one-, two-, ..., N-body degrees of freedom. In this simplified version, one-body degrees of freedom are coupled to fluctuations to all orders while retaining only specific terms of the general BBGKY hierarchy. The use of the simplified BBGKY is illustrated with the Lipkin-Meshkov-Glick (LMG) model. We show that a truncated version of this hierarchy can be useful, as an alternative to the SMF, especially in the weak coupling regime to get physical insight in the effect beyond mean-field. In particular, it leads to approximate analytical expressions for the quantum fluctuations both in the weak and strong coupling regime. In the strong coupling regime, it can only be used for short time evolution. In that case, it gives information on the evolution time-scale close to a saddle point associated to a quantum phase-transition. For long time evolution and strong coupling, we observed that the simplified BBGKY hierarchy cannot be truncated and only the full SMF with initial sampling leads to reasonable results., Comment: 10 pages, 4 figures

Mean-field approaches where a complex fermionic many-body problem is replaced by an ensemble of independent particles in a self-consistent mean-field can describe many static and dynamical aspects. It generally provides a rather good approximation for the average properties of one-body degrees of freedom. However, the mean-field approximation generally fails to produce quantum fluctuations of collective motion. To overcome this difficulty, noise can be added to the mean-field theory leading to a stochastic description of the many-body problem. In the present work, we summarize recent progress in this field and discuss approaches where fluctuations have been added either to the initial time, like in the Stochastic Mean-Field theory or continuously in time as in the Stochastic Time-Dependent Hartree-Fock. In some cases, the initial problem can even be re-formulated exactly by introducing Quantum Monte-Carlo methods in real-time. The possibility to describe superfluid systems is also invoked. Successes and shortcomings of the different beyond mean-field theories are discussed and illustrated., Comment: 34 pages, submitted to EPJA-Review section

Extending the stochastic mean-field model by including pairing, an approach is proposed for describing evolutions of complex many-body systems in terms of an ensemble of Time-Dependent Hartree-Fock Bogoliubov trajectories which is determined by incorporating fluctuations in the initial state. Non-linear evolution of the initial fluctuations provides an approximate description of quantal correlations and fluctuations of collective observables. Since the initial-state fluctuations break the particle-number symmetry, the dynamical description in which pairing correlations play a crucial role is greatly improved as compare to the mean-field evolution. The approach is illustrated for a system of particles governed by a pairing Hamiltonian., Comment: 5 pages, 2 figures, To appear in Phys. Rev. C (Rapid communication)

In the stochastic mean-field approach, an ensemble of initial conditions is considered to incorporate correlations beyond the mean-field. Then each starting pont is propagated separately using the Time-Dependent Hartree-Fock equation of motion. This approach provides a rather simple tool to better describe fluctuations compared to the standard TDHF. Several illustrations are presented showing that this theory can be rather effective to treat the dynamics close to a quantum phase transition. Applications to fusion and transfer reactions demonstrate the great improvement in the description of mass dispersion., Comment: Contribution to the International Conference "Nuclear Structure and Related Topics", July 2-July 7 (2012), Dubna, Russia

Dynamics of spontaneous symmetry breaking and fluctuations in the Lipkin-Meshkov-Glick model are investigated in a stochastic mean-field approach. Different from the standard mean-field, in the stochastic approach, initial state fluctuations, are incorporated. In weak coupling, the approach perfectly reproduces the exact quantal dynamics. On the other hand, for increasing coupling strength, above the symmetry breaking threshold, the approach provides description of gross properties (i.e. time averaged behavior) of the exact quantal evolution., Comment: 4 pages, 3 figures

Dissipation and fluctuations of one-body observables in heavy-ion reactions around the Coulomb barrier are investigated with a microscopic stochastic mean-field approach. By projecting the stochastic mean-field dynamics on a suitable collective path, transport coefficients associated with the relative distance between colliding nuclei and a fragment mass are extracted. Although microscopic mean-field approach is know to underestimate the variance of fragment mass distribution, the description of the variance is much improved by the stochastic mean-field method. While fluctuations are consistent with the empirical (semiclassical) analysis of the experimental data, concerning mean values of macroscopic variables the semiclassical description breaks down below the Coulomb barrier., Comment: 6 pages, 6 figures. A talk given at the FUSION11, May 2-6, 2011, Saint-Malo, France

Fusion reactions of heavy ions are investigated by employing a simple stochastic semi-classical model which includes the coupling between relative motion and low frequency collective surface modes of colliding ions similarly to the quantal coupled-channels description. The quantal effect enters into the calculation through the initial zero-point fluctuations of the surface vibrations. Good agreement with the result of coupled-channels calculations as well as data is obtained for the fusion cross sections of nickel isotopes. The internal excitations in non-fusing events as well as the fusion time are investigated., Comment: 8 pages, 8 figures, Published in Phys. Rev. C

The nucleus-nucleus potential and energy dissipation in fusion reactions are obtained from microscopic mean-field dynamics. The deduced potentials nicely reproduce the one extracted from experimental data. Energy dissipation shows a universal behaviour between different reactions. Also, the dispersion of mass distribution in transfer reaction is investigated in a stochastic mean-field dynamics. By including initial fluctuations in collective space, the description of the dispersion is much improved compared to that of mean field only. The result is consistent with the macroscopic phenomenological analysis of the experimental data., Comment: 4 pages, 4 figures. Proceedings of Second International Workshop on Compound Nuclear Reactions and Related Topics (CNR*09), October 5-8, 2009, Bordeaux, France

Nucleon transfer in symmetric heavy-ion reactions at energies below the Coulomb barrier is investigated in the framework of a microscopic stochastic mean-field theory. While mean-field alone is known to significantly underpredict the dispersion of the fragment mass distribution, a considerable enhancement of the dispersion is obtained in the stochastic mean-field theory. The variance of the fragment mass distribution deduced from the stochastic theory scales with the number of exchanged nucleon. Therefore, the new approach provides the first fully microscopic theory consistent with the phenomenological analysis of the experimental data., Comment: 4 pages, 2 figures, 1 table

By projecting the stochastic mean-field dynamics on a suitable collective path during the entrance channel of heavy-ion collisions, expressions for transport coefficients associated with relative distance are extracted. These transport coefficients, which have similar forms to those familiar from nucleon exchange model, are evaluated by carrying out TDHF simulations. The calculations provide an accurate description of the magnitude and form factor of transport coefficients associated with one-body dissipation and fluctuation mechanism., Comment: 9 pages, 5 figures

Information on dissipation in the entrance channel of heavy-ion collisions is extracted by macroscopic reduction procedure of Time-Dependent Hartree-Fock theory. The method gives access to a fully microscopic description of the friction coefficient associated with transfer of energy from the relative motion towards intrinsic degrees of freedom. The reduced friction coefficient exhibits a universal behavior, i.e. almost independent of systems investigated, whose order of magnitude is comparable with the calculations based on linear response theory. Similarly to nucleus-nucleus potential, especially close to the Coulomb barrier, there are sizable dynamical effects on the magnitude and form factor of friction coefficient., Comment: 7 pages, 10 figures

We propose a microscopic stochastic approach to improve description of nuclear dynamics beyond the mean-field approximation at low energies. It is shown that, for small amplitude fluctuations, the proposed model gives a result for the dispersion of a one-body observable that is identical to the result obtained previously through a variational approach. Furthermore, by projecting the proposed stochastic mean-field evolution on a collective path, a generalized Langevin equation is derived for collective variable, which incorporate one-body dissipation and one-body fluctuation mechanism in accordance with quantal fluctuation-dissipation relation., Comment: Paper is accepted for publication in Phys. Lett. B

We discuss quantum effects in the diffusion process which is used to describe the shape evolution from the touching configuration of fusing two nuclei to a compound nucleus. Applying the theory with quantum effects to the case where the potential field, the mass and friction parameters are adapted to realistic values of heavy-ion collisions, we show that the quantum effects play significant roles at low temperatures which are relevant to the synthesis of superheavy elements., Comment: 5 pages, 2 figures. Uses aipxfm.sty. A talk given at the FUSION06: International Conference on Reaction Mechanisms and Nuclear Structure at the Coulomb barrier, March 19-23, 2006, San Servolo, Venezia, Italy

We discuss a quantum effect in the diffusion process by developing a theory, which takes the finite curvature of the potential field into account. The transport coefficients of our theory satisfy the well-known fluctuation-dissipation theorem in the limit of Markovian approximation in the cases of diffusion in a flat potential and in a potential well. For the diffusion along a potential barrier, the diffusion coefficient can be related to the friction coefficient by an analytic continuation of the fluctuation-dissipation theorem for the case of diffusion along a potential well in the asymptotic time, but contains strong non-Markovian effects at short times. By applying our theory to the case of realistic values of the temperature, the barrier curvature, and the friction coefficient, we show that the quantum effects will play significant roles in describing the synthesis of superheavy elements, i.e., the evolution from the fusion barrier to the conditional saddle, in terms of a diffusion process. We especially point out the importance of the memory effect, which increases at lower temperatures. It makes the net quantum effects enhance the probability of crossing the conditional saddle., Comment: 12 pages, 3 figures, accepted for publication in Phys. Rev. C

Employing time-dependent projection formalism, a Fokker-Planck equation with non-Markovian transport coefficients is derived for large amplitude collective motion. Properties of transport coefficients for diffusion processes in a potential well and along a potential barrier are discussed, and their connection with the fluctuation-dissipation theorem is investigated. In the case of diffusion in a potential well, the formalism naturally leads to the generalization of the well known Einstein relation by including quantum fluctuations in addition to thermal fluctuations. Furthermore, it is shown that at low temperatures diffusion along potential barrier is significantly modified by quantum fluctuations. Explicit expressions for transport coefficients are presented for the case of the Caldeira-Leggett model., Comment: 19 pages, 5 figures

In order to incorporate the finite range effect into effective interactions, a modification of the Skyrme force by introducing a cut-off factor for high momentum transfers is proposed. The parameters of the cut-off factor are determined by fitting the microscopic in-medium cross-sections of Li and Machleidt over a wide range of energy and nuclear density. Results for the SkM* and SLy4 forces are presented., Comment: 13 pages including 3 figures

The giant monopole, dipole and quadrupole responses in $^{40}$Ca, $^{90}$Zr, $^{120}$Sn and $^{208}$Pb are investigated using linear response treatment based on a stochastic one-body transport theory. Effects of the coupling to low-lying surface modes (coherent mechanism) and the incoherent mechanism due to nucleon-nucleon collisions are included beyond the usual mean-field description. We emphasize the importance of both mechanism in the fragmentation and damping of giant resonance. Calculated spectra are compared with experiment in terms of percentage of Energy-Weighted Sum-Rules in various energy regions. We obtained reasonable agreement in all cases. A special attention as been given to the fragmentation of the Giant Quadrupole Resonance in calcium and lead. In particular, the equal splitting of the $2^{+}$ in $^{40}$Ca is correctly reproduced. In addition, the appearance of fine structure in the response $^{208}$Pb is partly described by the calculations in which the coherent mechanism play an important role., Comment: 44 pages, 8 figures, submitted to Physical Review C

Collisional damping widths of giant dipole excitations are calculated in Thomas-Fermi approximation by employing the microscopic in-medium cross-sections of Li and Machleidt and the phenomenological Gogny force. The results obtained in both calculations compare well, but account for about 25-35% of the observed widths in $^{120}Sn$ and $^{208}Pb$ at finite temperatures., Comment: Latex, 13 pages, 4 figures

A novel method is presented for implementation of the extended mean-field theory incorporating two-body collisions. At a given time, stochastic imaginary time propagation of occupied states are used to generate a convenient basis. The quantal collision terms, including memory effects, is then computed by a backward mean-field propagation of these single-particle states. The method is illustrated in an exactly solvable model. Whereas the usual TDHF fails to reproduce the long time evolution, a good agreement is found between the extended TDHF and the exact solution., Comment: 16 Latex pages including 3 figures (submitted to Nucl.Phys. A)

The nuclear collective response at finite temperature is investigated for the first time in the quantum framework of the small amplitude limit of the extended TDHF approach, including a non-Markovian collision term. It is shown that the collision width satisfies a secular equation. By employing a Skyrme force, the isoscalar monopole, isovector dipole and isoscalar quadrupole excitations in $^{40}Ca$ are calculated and important quantum features are pointed out. The collisional damping due to decay into incoherent 2p-2h states is small at low temperatures but increases rapidly at higher temperatures., Comment: 22 Latex pages including 9 figures. Phys. Rev. C (in press)