Angular momentum dependence of quasifission dynamics in the reaction <Superscript>48</Superscript>Ca+<Superscript>244</Superscript>Pu

The quasifission dynamics in the reaction ⁴⁸Ca+²⁴⁴Pu is investigated in the framework of time-dependent Hartree-Fock (TDHF) theory. The calculations are performed in three-dimensional Cartesian coordinate without any symmetry restrictions. The full Skyrme energy functional is incorporated in our TDHF implementation. The quasifission dynamics is quite sensitive to the angular momentum of colliding system. The contact time of quasifission decreases as a function of angular momentum and then forms a plateau with small oscillations. The quasifission process is accompanied by an important multi-nucleon transfer. The quantum shell effect plays a crucial role in the mass and charge of quasifission fragments. The mass-angle distribution of the fragments is calculated, which can be compared directly with future experiments.     

A generator co-ordinate approach to the breathing mode

In the harmonic approximation of the generator co-ordinate method (GCM) with the Skyrme interaction we show that the frequency of the breathing mode is essentially the same as the semiclassical result. The accuracy of the harmonic approximation is tested in ¹⁶O. Also, a comparison is made between the adiabatic time-dependent Hartree-Fock (ATDHF) theory and an anharmonic version of the GCM.     

Evaluation of the optimal path in ATDHF theory

The formal resemblance between the adiabatic time-dependent Hartree-Fock (ATDHF) theory of Villars and of Baranger and Vénéroni (BV) has been demonstrated. While the BV approach mixes the zeroth- and second-order quantities. Villars neglects the second-order equation and suggests a self-consistent method of solving the zeroth- and first-order equations. It is shown here that in order to obtain the time evolution of the TDHF solution one has to include the second-order equation along with zeroth- and first-order equations. It is then shown by an analytic method that the consistent solutions of the three equations follow the bottom of the valley of the collective potential.     

张量力在16O+16O熔合动力学中的效应（英文）

利用时间相关Hartree-Fock 理论和完整Skyrme 有效相互作用研究了16O+16O 碰撞在库仑位垒附近的熔合动力学。数值计算是在没有任何对称性约束的三维笛卡尔基下完成。将时间相关Hartree-Fock 理论和冻结密度近似下的能量密度泛函方法给出的库仑位垒与实验结果进行了比较,发现同位旋标量的张量项能降低自旋饱和体系16O+16O的库仑位垒,而库仑位垒高度随着同位旋矢量的张量项的耦合常数减小而降低。并计算了包含和不包含张量力的16O+16O熔合截面,发现张量力对16O+16O碰撞在库仑位垒附近的熔合动力学影响较小。The fusion dynamics of 16O+16O around Coulomb barrier has been studied in the timedependent Hartree-Fock (TDHF) theory with the full Skyrme effective interaction. The calculations have been carried out in three-dimensional Cartesian basis without any symmetry restrictions. We have included the full tensor force and all the time-odd terms in Skyrme energy density functional (EDF). The Coulomb barrier obtained from the dynamical TDHF calculations and EDF with frozen density approximation has been compared with the available experimental data. The isoscalar tensor terms and the rearrangement of other terms are found to decrease the barrier height in the spin-saturated system 16O+16O, while the energy of Coulomb barrier tends to decrease as the isovector coupling constant decreases. The fusion cross section for 16O+16O collision has been calculated with and without the tensor force. We found that the tensor force has minor effect on the fusion dynamics of 16O+16O at the energies around Coulomb barrier.     

Diabatic aspects of single-particle motion within the time-dependent Hartree-Fock theory

Time-dependent Hartree-Fock (TDHF) calculations for central collisions of⁸⁶Kr on¹⁶⁶Er are investigated with respect to quasi-stationary properties for various energies in the center-of-mass system. A quasi-stationary potential as well as quasi-stationary single-particle levels are extracted as functions of the relative distance between the two nuclei. A quantitative comparison with the diabatic two-center shell-model yields surprising agreement with the results from TDHF-calculations for both the nucleus-nucleus potential and the single-particle levels. From this we conclude that the approach phase of a nucleus-nucleus collision can be well described within a diabatic approximation for the single-particle motion.     

Low energy fusion of 12C + 12C and 16O + 16O systems

The adiabatic time-dependent Hartree-Fock method (ATDHF) is applied to the calculation of low energy fusion of ¹²C + ¹²C and ¹⁶O + ¹⁶O systems. The energy dependence of the results is in good agreement with experiment, while the order of magnitude is not correct. It is shown that the dynamical effects included in ATDHF are very important and cannot be neglected at the energies of astrophysical interest.     

Nuclear giant resonances and linear response

We search for nonlinear effects in nuclear giant resonances (GRs), in particular the isovector dipole and the isoscalar quadrupole modes. To that end, we employ a spectral analysis of time-dependent Hartree-Fock (TDHF) dynamics using Skyrme forces. Based on TDHF calculations over a wide range of excitation amplitudes, we explore the collectivity and degree of harmonic motion in these modes. Both GR modes turn out to be highly harmonic in heavy nuclei from A = 100 on. There is no trace of a transition to irregular motion and multiple resonances are predicted. Slight anharmonicities are seen for light nuclei, particularly for ¹⁶O. These are mainly caused by the spin-orbit splitting.     

A treatment of renormalization,potential energy of intermediate states and re-arrangement effects in brueckner calculations of closed shell nuclei

In the Brueckner-Hartree-Fock theory of finite nuclei, the problems of renormalization, potential energy of intermediate states and re-arrangement effects are examined. For practical calculations it is shown that satisfactory solutions to them can be obtained if the parametrization chosen for the single-particle potential in finite nuclei is linked closely to nuclear matter results. We make the link with similar problems in the density dependent Hartree-Fock theory and emphasize the possibility of such a parametrization by recalling the existence of a formal solution for the Hartree-Fock single-particle potential if the effective interaction is of the δ-function type. A method of solution of the Bethe-Goldstone equation is then presented which separates the intermediate states into those of an “open shell” and of a “continuum”. Finally results of model calculations of ¹⁶O and ⁴⁰Ca with harmonic oscillator functions are presented in which the parametrization chosen for the BHF single-particle potential is taken from the Skyrme-Vautherin δ-function force. A self-consistent determination of certain parameters of this form of force leads to values in close agreement with the empirical estimate made by Vautherin and Brink in ¹⁶O, with the exception of the spin-orbit splittings. Limitations and possible improvements of this type of approach are discussed for ⁴⁰Ca.     

Entrance-channel dynamics in the reaction ~(40)Ca+~(208)Pb

The entrance-channel dynamics including capture, fusion, and quasifission processes for the reaction ~(40)Ca+~(208)Pb is investigated in the fully microscopic time-dependent Hartree-Fock(TDHF) theory. The calculations are performed in three-dimensional Cartesian coordinate without any symmetry restrictions, in which the full Skyrme energy functional SLy4d and SLy5 are adopted.We study the energy dependence of capture cross sections, and find that the experimental data are well reproduced by the TDHF calculations. Both fusion and quasifission events are observed in the reaction ~(40)Ca+~(208)Pb. The contact time, mass and charge of quasifission fragments show a wide distribution in SLy4d compared with SLy5, implying that more nucleons are transferred in the SLy4d calculations. We find that the total kinetic energy of quasifission fragments in the TDHF calculations is distributed around Viola systematics, indicating that most of the relative kinetic energy is dissipated in quasifission dynamics.     

DWIA calculations of proton momentum distributions in 12C and 16O(e,e'p) reactions

The DWIA analysis of the quasi-free (e, e'p) process on ¹²C and ¹⁶O has been performed within the framework of the single-particle shell model. The spherical Hartree-Fock wave functions for the effective Skyrme forces are used in describing the ground-state properties of these nuclei. The competition between the structure effects and the final proton-nucleus interaction in forming the momentum distributions of outgoing protons is studied in a wide range of recoil momenta (up to 3 fm^?1). The distorting optical potential has been calculated on the basis of the Watson multiple scattering theory. The results of the analysis are compared with the new data obtained at Saclay.     

Phonon–particle coupling effects in odd–even mass differences of semi-magic nuclei

A method to evaluate the particle–phonon coupling (PC) corrections to the single-particle energies in semi-magic nuclei, based on a direct solving the Dyson equation with PC corrected mass operator, is used for finding the odd–even mass difference between 18 even Pb isotopes and their odd-proton neighbors. The Fayans energy density functional (EDF) DF3-a is used which gives rather high accuracy of the predictions for these mass differences already on the mean-field level, with the average deviation from the existing experimental data equal to 0.389 MeV. It is only a bit worse than the corresponding value of 0.333 MeV for the Skyrme EDF HFB-17, which belongs to a family of Skyrme EDFs with the highest overall accuracy in describing the nuclear masses. Account for the PC corrections induced by the low-laying phonons 2 ₁ ⁺ and 3 ₁ ^- significantly diminishes the deviation of the theory from the data till 0.218 MeV.     

Self-consistent theory of finite Fermi systems and Skyrme–Hartree–Fock method

Recent results obtained on the basis of the self-consistent theory of finite Fermi systems by employing the energy density functional proposed by Fayans and his coauthors are surveyed. These results are compared with the predictions of Skyrme–Hartree–Fock theory involving several popular versions of the Skyrme energy density functional. Spherical nuclei are predominantly considered. The charge radii of even and odd nuclei and features of low-lying 2⁺ excitations in semimagic nuclei are discussed briefly. The single-particle energies ofmagic nuclei are examined inmore detail with allowance for corrections to mean-field theory that are induced by particle coupling to low-lying collective surface excitations (phonons). The importance of taking into account, in this problem, nonpole (tadpole) diagrams, which are usually disregarded, is emphasized. The spectroscopic factors of magic and semimagic nuclei are also considered. In this problem, only the surface term stemming from the energy dependence induced in the mass operator by the exchange of surface phonons is usually taken into account. The volume contribution associated with the energy dependence initially present in the mass operator within the self-consistent theory of finite Fermi systems because of the exchange of high-lying particle–hole excitations is also included in the spectroscopic factor. The results of the first studies that employed the Fayans energy density functional for deformed nuclei are also presented.

     

Angular momentum dependence of quasifission dynamics in the reaction ~(48)Ca+~(244)Pu

The quasifission dynamics in the reaction ~(48)Ca+~(244)Pu is investigated in the framework of time-dependent Hartree-Fock(TDHF)theory. The calculations are performed in three-dimensional Cartesian coordinate without any symmetry restrictions. The full Skyrme energy functional is incorporated in our TDHF implementation. The quasifission dynamics is quite sensitive to the angular momentum of colliding system. The contact time of quasifission decreases as a function of angular momentum and then forms a plateau with small oscillations. The quasifission process is accompanied by an important multi-nucleon transfer. The quantum shell effect plays a crucial role in the mass and charge of quasifission fragments. The mass-angle distribution of the fragments is calculated, which can be compared directly with future experiments.     

A mean-field description of (γ, pi) cross sections at medium energies

Differential (γ, p_i) cross sections on ¹⁶O and ⁴⁰Ca have been calculated within a single-particle direct knock-out model, for photon energies between 40 and 400 MeV. It is shown that generating the bound and scattered states in a Skyrme Hartree-Fock (HF) mean-field, one can account for the global features of the data. Hereby, the role of the velocity-dependent part of the nucleon-nucleon interaction is emphasized.     

Density as a constraint and the separation of internal excitation energy in TDHF

We present a fast and efficient constrained Hartree-Fock iteration scheme which constraints the complete density distribution to remain constant. The scheme is particularly suited to a coordinate- or momentum-space representation. The technique is applied to separate the collective and the internal energy in a propagating TDHF state. We study the behavior of these two energies in an¹⁶O+¹⁶O collision.     

Role of the nuclear surface in a unified description of the damping of single-particle states and giant resonances

The damping widths of single-particle states and of giant resonances are estimated in spherical nuclei, based on the excitation of surface modes.A Skyrme III interaction with an effective mass consistent with that resulting from infinite nuclear matter calculations with “realistic” forces $\text{(m}{}^1\text{/m = 0.76)}$, was utilized. The single-particle basis needed to construct the unperturbed nuclear response function for each multipolarity was obtained, treating this force in the Hartree-Fock approximation. Diagonalizing a schematic interaction in this basis, the surface modes were calculated. They are used to dress the single-particle and single-hole states and to renormalize the vertex interaction, taking into account the proper energy dependence of the couplings.The essential new feature of the present calculation as compared to the calculations reported in ref.¹) is that the energy dependence of the real and imaginary part of the self-energy is taken into account. This is done utilizing a strength function model.About 70 % of the damping widths arise from the coupling to specific intermediate states containing one low-lying collective surface vibration. The rest, from the coupling to many nonspecific states.Qualitative agreement is found with the experimental data for spherical nuclei throughout the mass table for both the single-particle states and the giant resonances. The model seems however to predict widths which are smaller than those experimentally observed.     

Properties of the quasiparticle excitations in 207Pb and 209Pb from an extrapolation of the optical-model potential

A previously developed dispersion relation approach is used to calculate the shell-model potential in the case of neutrons in ²⁰⁸Pb, in the energy domain (-50 MeV, 0). This potential contains a dispersive contribution besides a Hartree-Fock type component, and thereby includes correlation and polarization effects. The shell-model and the Hartree-Fock type potentials are assumed to have Woods-Saxon shapes with diffuseness a_v = 0.70 fm; the energy dependence of their depths and radii is calculated. The energy dependence of the shell-model potential is characterized by the effective mass, whose dependence upon radial distance and neutron energy is determined. The effective mass is a sensitive function of energy, in contrast to its Hartree-Fock type component which is nearly independent of energy. Attention is drawn to the fact that the effective mass in nuclear matter cannot be straightforwardly identified with the effective mass at the nuclear centre. The effective mass presents a sharp peak at the nuclear surface near the Fermi energy and a dip at the surface for energies 10 to 20 MeV away from the Fermi energy. The spectroscopic factors of single-particle excitations in ²⁰⁷Pb and ²⁰⁹Pb are calculated from the difference between the effective mass and its Hartree-Fock type component. The predicted values of the valence single-particle wave functions at large radial distances are in fair agreement with experimental values deduced from analyses of sub-Coulomb pickup reactions. It is shown that the dispersive contribution increases the level density parameter by about 25%, in agreement with previous microscopic or semi-phenomenological models; the calculated level density parameter is in good agreement with the empirical value.     

The generator-coordinate-method with conjugate parameters and the unification of microscopic theories for large amplitude collective motion

A dynamic theory of large amplitude collective motion of many particle systems is presented which is relevant, for example, to nuclear fission. The theory is microscopic and makes use of a collective path, i.e. a suitably constructed set of distorted nonequilibrium Slater determinants. The approach is a generalization of the generator coordinate method (GCM) and improves its dynamic aspects by extending it to a pair of conjugate generator parameters q and p (DGCM). The problems connected with redundancy and superfluous degrees of freedom are solved by prediagonalizing the local oscillations about each point of the dynamic collective basis | q, p ～. For adiabatic large amplitude collective motion a Schrödinger equation is derived which appears to be nearly identical to the one obtained by a consistent quantization of semiclassical approaches as e.g. the adiabatic time dependent Hartree-Fock theory (ATDHF). In turn a collective path constructed by ATDHF proves to be particularly suited for being used in the present DGCM formalism. Altogether the formalism unifies two classes of microscopic approaches to collective motion, viz. the quantum mechanical GCM and the classical theories like cranking and ATDHF.     

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.