New neutron-rich isotope production in $^{154}$Sm+$^{160}$Gd

Deep inelastic scattering in $^{154}$Sm+$^{160}$Gd at energies above the Bass barrier is for the first time investigated with two different microscopic dynamics approaches: improved quantum molecular dynamics (ImQMD) model and time dependent Hartree-Fock (TDHF) theory. No fusion is observed from both models. The capture pocket disappears for this reaction due to strong Coulomb repulsion and the contact time of the di-nuclear system formed in head-on collisions is about 700 fm/c at an incident energy of 440 MeV. The isotope distribution of fragments in the deep inelastic scattering process is predicted with the simulations of the latest ImQMD-v2.2 model together with a statistical code (GEMINI) for describing the secondary decay of fragments. More than 40 extremely neutron-rich unmeasured nuclei with $58 \le Z\le 76$ are observed and the production cross sections are at the order of ${\rm \mu b}$ to mb. The multi-nucleon transfer reaction of Sm+Gd could be an alternative way to synthesize new neutron-rich lanthanides which are difficult to be produced with traditional fusion reactions or fission of actinides.


I. INTRODUCTION
The heavy-ion reaction at energies around the Coulomb barrier is an important way not only for the study of the nuclear structures, but also for the synthesis of unstable or even exotic (neutron-rich, neutron-deficient, superheavy) nuclei for which no experimental data exist [1][2][3][4][5][6][7][8]. For light and intermediate fusion systems, the fusion process is usually described by the penetration of the fusion barriers. The fusion (capture) cross sections can be accurately predicted by using the fusion coupled channel calculations or empirical barrier distribution approaches [9][10][11][12][13][14]. For fusion systems leading to the synthesis of super-heavy elements, the quasi-fission and fusion-fission process significantly complicates the description of fusion process. The very shallow capture pockets in such kind of reaction systems may cause some difficulties in the applications of the barrier-penetration approaches. Although macroscopic dynamics models [15][16][17][18][19] met with some success for describing the residual evaporation cross sections of measured super-heavy systems, the uncertainty of the predicted fusion probability from these different models for unmeasured systems is still large due to the uncertainty of model parameters [20,21] and ambiguity of reaction mechanism.
For example, with the fusion-by-diffusion model, Choudhury and Gupta [22] investigated symmetric heavy-ion reaction of 154 Sm+ 154 Sm and obtained measurable evaporation residue cross sections (∼ 0.6 pb). However, Cap et al. [23] investigated the same reaction and found the cross sections are extremely small (about 10 −13 pb ) and probably never reachable. The contradictory predictions imply some key model parameters such as the injection point distance and the dynamical nucleus-nucleus potential are far from clear for this reaction. It is therefore necessary to investigate the dynamics process and fusion probability in this kind of reactions with self-consistent microscopic dynamics models.
In addition to the formation of superheavy nuclei, the synthesis of extremely neutron-rich heavy nuclides through multi-fragmentation, deep inelastic scattering and quasi-fission are of exceptional importance to advance our understanding of nuclear structure at the extreme isospin limit of the nuclear landscape [24][25][26][27][28]. Neutron-rich lanthanides, such as 182 70 Yb 112 with "false magic numbers", are of importance for understanding the strength of spin-orbit interaction which influences the positions of the island of stability for super-heavy nuclei.
Unfortunately, if one glances at the chart of nuclides (see the positions of known nuclei in AME2012 [29]), one notes that the number of observed neutron-rich nuclides is very limited at mass region A > 160, due to that neither traditional fusion reactions with stable beams nor fission of actinides easily produce new neutron-rich heavy nuclei in this region. Recently, some heavy neutron-rich nuclei with 70 ≤ Z ≤ 79 were produced in projectile fragmentation of 197 Au primary beams bombarding on thick 9 Be target at GSI [30]. In addition to the fragmentation of heavy nuclei, multi-nucleon transfer process might be helpful to produce neutron-rich heavy nuclei [31][32][33][34]. Zychor et al. have performed a systematic study on the productions of Hafnium and Lutetium isotopes with the reactions induced on a thick tungsten target by 40 Ar, 84 Kr, and 136 Xe, respectively. The study indicated that the absolute production cross sections of neutron-rich heavy isotopes increase with increasing projectile mass [35]. It is necessary and important to study the multi-nucleon transfer between two nuclei in the rare-earth region for producing new neutron-rich lanthanides, considering the fission barriers of lanthanides are relatively high to prevent fission of heavy fragments in the secondary decay process. The investigation of deep inelastic scattering in 154 Sm+ 160 Gd at energies above the Coulomb barrier is therefore interesting, not only for the study of the production probability of super-heavy nuclei, but also for the synthesis of unmeasured lanthanides.
To understand the dynamical process in fusion and deep inelastic scattering reactions, some microscopical dynamics models, such as the time-dependent Hartree-Fock (TDHF) [36][37][38][39][40][41][42] and some different extended versions of quantum molecular dynamics (QMD) model [43] including IQMD [44,45], CoMD [46][47][48], ImQMD [49][50][51][52], EQMD [53,54], etc. have been developed. TDHF theory has many successful applications in the description of nuclear large amplitude collective motions, for example, heavy-ion collisions, giant resonance, fission dynamics, and nuclear molecular resonance; for a recent review see Ref. [41]. TDHF in a nuclear context means a time-dependent mean-field theory derived from an effective energy functional. The most widely used is the Skyrme energy density functional (EDF) which leads to an accurate description of selected static properties in nuclei. Static and dynamical mean-field theories, by considering directly single-particle degrees of freedom interacting, was a major breakthrough in nuclear physics to describe static and dynamical nuclear properties [55]. Comparing with the semi-classic molecular dynamics simulations, the TDHF calculations can describe better the structure effects of nuclear system such as the shell effects and nuclear shapes in heavy-ion reaction at low incident energies.
On the other hand, at the very early stage of the application of TDHF, it was already realized that the independent particle picture used in the mean-field theory leads to severe limitations [55]. It is known the one-body microscopic dynamics models based on the mean-field theory are difficult to describe a multi-fragmentation process, due to the fact that the correlations treated in the one-body approach are not able to describe the large fluctuations [46]. This difficulty can be solved by adopting more suitable treatments of the N-body problem like molecular dynamics. In the improved quantum molecular dynamics (ImQMD) model, the standard Skyrme force with the omission of spin-orbit term is adopted for describing not only the bulk properties but also the surface properties of nuclei. Simultaneously, the Fermi constraint is used to maintain the fermionic feature of the nuclear system.
In the Fermi constraint which was previously proposed by Papa et al. in the CoMD model [46] and improved very recently in Refs. [56,57], the phase space occupation probabilityf i of the i-th particle is checked during the propagation of nucleons. Iff i > 1, i.e. violation of the Pauli principle, the momentum of the particle i is randomly changed by a series of two-body "elastic scattering" and "inelastic scattering" between this particle and its neighboring particles, together with Pauli blocking condition being checked after the momentum re-distribution. In other words, both the self-consistently generated mean-field and the momentum re-distribution in the Fermi constraint which introduces additional fluctuations and two-body dissipation affect the movements of nucleons in the simulations. The ImQMD model allows to investigate the formation of fragments during a heavy-ion collision in a consistent N-body treatment, through event-by-event simulations, with which the charge and isotope distributions of fragments can be obtained.
Considering the advantage of the TDHF theory in the description of nuclear structure effects and that of the ImQMD model in the description of fluctuations and fragment formation, it is therefore necessary to investigate the same reaction system with these two different microscopic dynamics models, for exploring the dynamical mechanism and improving the reliability of model predictions for unmeasured reaction systems.
The structure of this paper is as follows: In sec. II, the frameworks of TDHF and ImQMD will be introduced. In sec. III, the deep inelastic scattering process of 154 Sm+ 160 Gd at an incident energy of E c.m. = 440 MeV will be investigated with the two models. In Sec. IV, the isotope distribution, angular distribution and production cross sections of some neutronrich nuclei with unmeasured masses will be studied with the ImQMD model. Finally a brief summary is given in Sec. V.

II. Theoretical Frameworks
In the TDHF theory, the complicated many-body problem is replaced by an independent particle problem, i.e., the many-body wave functions are approximated as the antisymmetrized independent particle states to assure an exact treatment of Pauli principle during time evolution. In the nuclear context, the basic ingredient of TDHF is the energy functional composed by the various one-body densities. Here, we adopt the full Skyrme EDF with the parameter set SLy5 [59]. The Skyrme parameters have been fitted with the ground state properties of the selected nuclei. For the heavy-ion collisions, there is no adjustable free parameters in TDHF. The dynamical evolution of the mean-field is expressed by TDHF with the single-particle Hamiltonian h[ρ] and the one-body densityρ. Taking the nuclear ground state as an initial state of the dynamical evolution, TDHF time evolution is determined by the dynamical unitary propagator. Earlier TDHF calculations imposed the various approximations on the effective interaction and geometric symmetry. The development of computational power allows a fully three-dimensional (3D) TDHF calculation with the modern effective interaction and without symmetry restrictions, which significantly improves the physical scenario in heavy-ion collisions [60]. In this work, the set of nonlinear TDHF equation is solved on a three-dimensional Cartesian coordinate-space without any symmetry restrictions. We use the fast Fourier transformation method to calculate the derivatives. The conservation of total energy and particle number is assured during the time evolution by choosing the parameters of grid spacing as 1 fm and time step ∆t = 0.2 fm/c.
In the ImQMD simulations, each nucleon is represented by a coherent state of a Gaussian wave packet where r i and p i are the centers of the i-th wave packet in the coordinate and momentum space, respectively. σ r represents the spatial spread of the wave packet. The total N-body wave function is assumed to be the direct product of these coherent states.
The anti-symmetrization effects are additionally simulated by introducing the Fermi constraint mentioned previously (In the traditional QMD calculations, the Pauli potential [61] or momentum-dependent two-body repulsion [62,63] and the collision term [64] are usually used to simulate the effects). Through a Wigner transformation, the one-body phase space distribution function and the density distribution function ρ of a system are obtained. The propagation of nucleons is governed by the self-consistently generated  configuration. We note that the compound nuclei are not formed in the two different dynamics simulations, even the incident energy is obviously higher than the Bass barrier. At t = 1000 fm/c, the neck of the di-nuclear system (DNS) becomes narrow and the system tend to split up. The contact-times of the DNS in this reaction are about 700 fm/c, which is much shorter than the typical contact-times of quasi-fission (usually greater than 1500 fm/c but much shorter than typical fusion-fission times) [24,25]. Here, we would like to emphasize that the density distributions from the ImQMD simulations in Fig. 1 represent the average value over a large number simulation events. Simultaneously, we investigate the relative motion of 154 Sm+ 160 Gd. Fig. 2 shows the time evolution of the relative distance between two nuclei. The squares and the circles denote the results of TDHF at different orientations for the deformed reaction partners, which are comparable with those of ImQMD (solid curve). From Fig. 2, one sees that the results of TDHF are slightly different from those of ImQMD at touching configuration, which is due to the deformation effects of re- there is about 1% of simulation events at central collisions in which the ternary breakup rather than traditional binary scattering is observed in the ImQMD simulations.
To understand the behavior of deep inelastic scattering at energies above the Bass barrier, it is necessary to investigate the nucleus-nucleus potential between these two nuclei. In heavy-ion fusion reactions, some static nucleus-nucleus potential are successfully proposed for describing the fusion barrier, such as the Bass potential [66] and the Woods-Saxon (WS) parametrization of the nuclear potential given by Broglia and Winther from a knowledge of the densities of the colliding nuclei and an effective two-body force [72,73]. In addition, the Skyrme EDF together with extended Thomas-Fermi (ETF2) approach is also frequently used to investigate the Coulomb barrier based on the sudden approximation for the densities of the reaction partners [11][12][13]. Considering the uncertainty of these static/empirical potential at short distances, it is of importance to investigate the dynamical nucleus-nucleus potential.
According to the energy conservation, we have [51] E c.m.
where E c.m. is the incident center-of-mass energy, T is the relative motion kinetic energy of two colliding nuclei, E * is the excitation energy, and T oth is other collective kinetic energy, such as vibrational energy of neck and rotational energy. Before the neck of DNS being well formed in head-on collisions, E * and T oth could be negligible, the nucleus-nucleus potential is approximately expressed as At the closest distance R min , i.e. the smallest value in Fig. 2, T (R min ) = 0. Therefore, the nucleus-nucleus potential at R min can be roughly estimated by the corresponding incident energy, V (R min ) ≃ E c.m. , in the elastic and inelastic scattering collisions.
In Fig. 3, we show the static nucleus-nucleus potential of 154 Sm+ 160 Gd. The dot-dashed curve, the crosses and the solid curve denote the Bass potential, the Woods-Saxon potential of Broglia and Winther and the potential based on ETF2 approach, respectively. At the regions where two nuclei begin to touch each other (R < 14 fm), Bass potential is flat, whereas the other two potentials suggest a strong repulsion between the reaction partners.
The circles and squares denote the upper limit of the dynamical potential according to Eq. (6) together with calculated closest distance R min from ImQMD and TDHF, respectively. Both the calculations of these static models and the results of ImQMD and TDHF indicate the capture pocket of this reaction system disappears in generally, which explains the deep

IV. Production of neutron-rich isotopes in 154 Sm+ 160 Gd
Although it is almost impossible to produce super-heavy nuclei in 154 Sm+ 160 Gd considering the disappearance of the capture pocket and the rapid increase of the potential with decreasing of the relative distance, it might produce new neutron-rich nuclide during the deep inelastic scattering process. Here, we study the isotope distribution of fragments in 154 Sm+ 160 Gd from central to peripheral collisions with the ImQMD-v2.2 model [57]. Before investigating the isotope distribution of fragments in Sm+Gd, we have already tested the ImQMD-v2.2 model for description of the isotope distribution in the multi-nucleon transfer of 86 Kr+ 64 Ni at an incident energy of 25 MeV/nucleon [74]. The measured isotope distribution of products can be reasonably well reproduced by using the ImQMD model together In Table 1, we list the production cross sections of some neutron-rich heavy nuclei with unknown masses. Here, we only list the nuclei with cross sections larger than 20µb. Through multi-nucleon transfer in the deep inelastic scattering reaction of 154 Sm+ 160 Gd, more than 40 neutron-rich nuclei with unknown masses can be produced, which implies that the deep inelastic scattering between two lanthanides is an efficient way to synthesize new neutron-rich heavy nuclei. The production cross sections decrease exponentially with further increasing of neutrons in an isotope chain. The production cross section of 182 Yb is about 46µb. In the table, the predicted mass excesses of these nuclei from a macroscopic-microscopic mass model, Weizsäcker-Skyrme (WS4) model [67] are also presented.
Simultaneously, the angular distribution of these neutron-rich nuclei produced in the deep inelastic scattering process are analyzed. Fig. 5 shows the calculated angular distribution of heavy fragments with Z ≥ 62 from the ImQMD simulations. The different curves denote the results at different impact parameter. Most heavy fragments are emitted from forward angles. The double-peak structure of the angular distribution at the semi-central collisions can be clearly observed. We note that the sum of the two emission angles of the projectilelike and target-like nuclei is about 96 • (the corresponding value is 90 • in elastic scattering between two identical particles), which represents the behavior of elastic and inelastic scat- description of the deep inelastic scattering of 154 Sm+ 160 Gd at energies above the Bass barrier. The fusion process is neither observed from the improved quantum molecular dynamics (ImQMD) simulations nor from the time dependent Hartree-Fock (TDHF) calculations. The contact time of the di-nuclear system formed in head-on collisions is about 700 fm/c at an incident energy of 440 MeV, which is much shorter than the typical contact-times of quasifission. The time evolutions of the relative distance between the reaction partners at this energy from the two models are in good agreement with each other. Through investigating the nucleus-nucleus potential, we find that the capture pocket in 154 Sm+ 160 Gd generally disappears, which leads to that the deep inelastic scattering process is a dominant process at central collisions. The isotope distribution of fragments in the deep inelastic scattering process is calculated with the ImQMD-v2.2 model together with the statistical decay model (GEMINI) for describing the secondary decay of fragments. More than 40 extremely neutron-rich nuclei with unknown masses are observed and the production cross sections are at the order of µb to mb. The multi-nucleon transfer in the deep inelastic scattering reaction of Sm+Gd seems to be an efficient way to produce new neutron-rich lanthanides.
By analyzing the angular distribution of the produced heavy fragments, we suggest that 20 • < Θ lab < 60 • might be a suitable angular range to detect these extremely neutron-rich heavy nuclei.