1 Introduction

Production of new heavy neutron-rich nuclei has attracted widespread interests in recent years [1-11]. The production of these exotic nuclei located far from the stability line is very important for exploring their structure and decay properties [12, 13], as well as a measurement of spins, neutron skin thickness [14], magnetic dipole, and electric quadrupole moments and charge radii, et al. In addition, it is also extremely important for understanding the r-process in nuclear astrophysics. The r-process path is located (and interrupted by fission) just in the region of unknown heavy nuclei with a large neutron excess. The neutron closed shell N = 126 is probably the last “waiting point” in the r-process of nucleosynthesis [15]. It is helpful for answering the origin of heavy elements from iron to uranium in the Universe.

Many new neutron-rich nuclei are produced by experiment with fragmentation method of projectile [16, 17]. For example, there are 60 new neutron-rich isotopes in the atomic number range of 60 ≤ Z ≤ 78 produced by using a 1 GeV/nucleon ²³⁸U beam impinged on a ⁹Be target by Kurcewicz et al. at GSI Darmstadt [16]. They found that the fragmentation plays a dominant role in the production of the isotopes of the elements above Z = 72. However, production cross sections of these nuclei with the fragmentation method are very small and are about a few ten pb orders of magnitude. It is difficult for further production of more neutron-rich nuclei with the fragmentation method. Recently, a multinucleon transfer (MNT) reaction experiment of ¹³⁶Xe+¹⁹⁸Pt was performed by Watanabe et al. at GANIL in Caen [9]. They found that the MNT approach has huge advantages for the production of very neutron-rich nuclei with Z⩽77 compared to the above fragmentation approach. It demonstrates that the MNT route is a more promising path to produce new neutron-rich isotopes around N=126 shell.

Radioactive ion beam facilities can provide very neutron-rich projectiles [18, 19] which could significantly improve the production cross section of heavy neutron-rich nuclei [20, 21]. Some facilities, such as the high intensity heavy ion accelerator facility (HIAF) in China [22, 23], have a plan to produce new heavy neutron-rich nuclei with intense secondary beams of exotic radioactive nuclei by the MNT reactions. Theoretical support for these very time-consuming and expensive experiments is vital for choosing the optimum target-projectile-energy combinations and for the estimation of cross sections. The dinuclear system (DNS) model is a semiclassical model which has shown reasonable success in predicting the evaporation residual cross section in fusion reactions [24-26] and multinucleon transfer reactions [27, 28]. In this work, we apply the DNS model to study the multinucleon transfer process with stable projectiles ¹³⁶Xe, ²³⁸U, and radioactive projectile ¹⁴⁸Xe impinged on a ²⁰⁸Pb target.

The structure of this paper is as follows. In Sect. 2, we briefly introduce the DNS model. In Sect. 3, the isotope production cross sections of ¹³⁶Xe+²⁰⁸Pb, ²³⁸U+²⁰⁸Pb, and ¹⁴⁸Xe+²⁰⁸Pb systems have been studied by using the DNS model. Finally, the conclusion is given in Sect. 4.

2 Theoretical descriptions

In the framework of the DNS model, the distribution probability can be obtained by solving a set of master equations numerically in the potential energy surface of the DNS. The probability distribution function P(Z₁,N₁,E₁,t) for fragment 1 with proton number Z₁, neutron number N₁ and excitation energy E₁ is described by the following master equation [27, 29]:

The sum is taken over all possible proton and neutron numbers that fragments $Z_1^{\text{'}}$ and $N_1^{\text{'}}$ may take, but only one nucleon transfer is considered in the model with the relation $Z_1^{\text{'}}=Z_1\pm 1$ and $N_1^{\text{'}}=N_1\pm 1$. $W_{Z_1\mathrm{,}{N}_1;Z_1^{\text{'}}\mathrm{,}{N}_1}$ denotes the mean transition probability from channel (Z₁,N₁,E₁) to ($z_1^{\prime }$,N₁, $E_1^{\prime }$), which can be read as

where, τ_mem and V denote memory time and single-particle potential, respectively. i denotes all remaining quantum numbers [30]. $d_{Z_1\mathrm{,}{N}_1}$ denotes the microscopic dimension corresponding to the macroscopic state (Z₁,N₁,E₁)

The n₁=g₁Δ ε and n₂=g₂Δ ε denote the numbers of valence states of fragments 1 and 2, respectively. g₁ and g₂ are mean single-particle level densities. $\Delta \epsilon =\sqrt{4{\epsilon }^{*}/g}$ is an energy range around Fermi energy, ε_F, of both fragments. m₁=n₁/2 and m₂=n₂/2 denote corresponding numbers of valence nucleons [30, 31]. The quasifission rate ${\Lambda }_{{}_{Z_1\mathrm{,}{N}_1}}^{qf}$ and the fission rate ${\Lambda }_{Z_1\mathrm{,}{N}_1}^{fis}$ of the heavy fragment are estimated with the one-dimensional Kramers formula [32]. $\Theta (t)=\sqrt{{\epsilon }^{*}/a}$ is the local temperature. a=Ai/12 is the level-density parameter. We assume that ε^* is divided between the two primary fragments according to their mass ratio. Hence the excitation energy of fragment 1 is expressed as E₁=A₁/(A₁+A₂)ε^*. A₁ and A₂ are mass numbers of fragment 1 and fragment 2, respectively.

The local excitation energy, ε_*, is defined as

Here E^diss is the energy dissipation from the relative kinetic energy loss, which can be seen in Ref. [33]. The dissipation of the relative motion and angular momentum of the DNS is described by the Fokker-Planck equation. The U(A₁,A₂) and U(AP,AT) are the driving potentials at the configuration of (A₁,A₂ and at the entrance point of DNS, which can be read as

Where, B(Z₁,N₁), B(Z₂,N₂), and B(Z,N) are the binding energies of the fragment 1, fragment 2, and the compound nucleus, respectively, in which the shell and the pairing corrections are included reasonably. $V_{rot}^{CN}$ is the rotation energy of the compound nucleus. The β₁ and β₂ are quadrupole deformation parameters of the two fragments at ground state which is static in the nucleon transfer process. θ₁ and θ₂ are angles between the collision orientations and the symmetry axes for fragment 1 and fragment 2, respectively. In Ref. [27], the calculations show that the tip-tip and side-side collisions give a similar result in the isotope distributions of ²³⁸U+²³⁸U collisions at incident energy near 1.1 VC. Hence in this work, only tip-tip cases are considered for calculations. The interaction potential between two fragments includes nuclear, Coulomb, and centrifugal parts

where μ is the reduced mass of system. The nuclear potential is calculated by using the double-folding method. The Coulomb potential is obtained by Wong’s formula [31].

The cross sections of the primary fragments (Z₁, N₁) are calculated as follows

where T(E_c.m.,J) is the capture probability. P(Z₁,N₁,τ_int) is the formation probability of primary fragments from the equilibrium state of DNS. The contribution of fission rate is ignored in this work. The $Y_{Z_1\mathrm{,}{N}_1}$ denotes the primary yields of fragments from the quasifission rate [32, 34], which is written as

The interaction time, τ_int, in the dissipation process of two colliding partners is dependent on the incident energy, E_c.m., in the center-of-mass frame and the angular momentum, J, which is calculated by using the deflection function method and has the value of a few 10^-20 s. For heavy systems with no potential pocket, the distance, R, between the centers of the two fragments is chosen to be the value at the touching point, in which the DNS is assumed to be formed. The maximal angular momentum is taken as the grazing collisions. More detailed descriptions of the DNS model can be found in Refs. [30, 31].

To obtain the final isotope cross section distributions, we apply the GEMINI to deal with the subsequent de-excitation process of primary fragments. The GEMINI is a Monte Carlo code which follows the decay of a compound nuclei by a series of sequential binary decays until the resulting products are unable to undergo any further decay. It allows not only light-particle emission and symmetric fission, but also the decaying nucleus to emit a fragment of any mass [35]. Nuclear level densities are taken as a Fermi-gas form with default parameters. In this work, we simulate 1,000 times the deexcitation processes for each primary fragment.

To test the DNS model for the description of multinucleon transfer reactions, the isotopic production cross sections for ¹³⁶Xe+²⁰⁸Pb at E_c.m.=450 MeV are calculated by using the DNS model. Figure 1 shows the isotopic production cross sections from Z=78 to 81. The dashed and solid lines denote the calculation results for primary and final fragments, respectively. The experimental data (solid circles) are from Ref. [2]. From Fig. 1, one can see that final yields shift to the left after the de-excitation process. The final isotopic cross sections are in good agreement with the experimental data. This indicates that the DNS model is applicable for the study of MNT reactions at near barrier energies.

Fig. 1Full-screen View

(Color online) The isotopic production cross sections from Z=78 to 81 for ¹³⁶Xe+²⁰⁸Pb at E_c.m.=450 MeV. The dashed and solid lines denote the calculation results for primary and final fragments, respectively. The experimental data (solid circles) are from Ref. [2].

3 Results and discussion

The radioactive neutron-rich nuclei have larger N/Z ratios. Such as ¹⁴⁸Xe is a known isotope of xenon with the largest neutron excess (N/Z=1.74). It can be as a radioactive beam in the future. First, we discuss the isotopic cross sections in different incident energies. Figure 2 shows the primary isotopic cross sections of Pt at incident energies E_c.m.=1.02, 1.06, and 1.10 V_C for an ¹⁴⁸Xe+²⁰⁸Pb system. V_C is the Coulomb barrier. The solid, dash-dotted and dotted lines denote the calculation results from incident energy E_c.m.=1.02, 1.06, and 1.10 V_C, respectively. One can see that isotopic cross sections increase with the increasing of incident energies both in neutron-rich and neutron-deficient sides. Larger incident energy improves the transfer probability of nucleons on the same potential energy surface. However, larger incident energy will result in higher excitation energy for primary fragments.

Fig. 2Full-screen View

(Color online) The primary isotopic cross section distributions of Pt at incident energy E_c.m.=1.02, 1.06 and 1.10 VC for ¹⁴⁸Xe+²⁰⁸Pb system. VC is the Coulomb barrier.

Figure 3 shows the final isotopic production cross sections from Z=75 to 80 for the reaction of ¹⁴⁸Xe+²⁰⁸Pb at different incident energies. One can see that in the neutron-deficient side, the isotopic production cross sections increase with increasing incident energies. It is very sensitive to incident energies. There are two reasons for this. One is that isotopic production cross sections in the neutron-deficient side at a larger incident energy are larger for primary fragments. The other one is that a larger incident energy leads to primary fragments obtaining more excitation energy. More neutrons evaporating causes a shift of final yields to the neutron-deficient side. While in the neutron-rich side, the final isotopic production cross sections are not sensitive to incident energies, especially for extremely neutron-rich nuclei. In addition, the values of peak for a few proton stripping channels change very slowly. Because of this, the driving potential of the system is reduced in the configuration of mass symmetry. The transfer of nucleons from target to projectile is dominant near the entrance point of DNS.

Fig. 3Full-screen View

(Color online) The final isotopic production cross section distributions from Z=75 to 80 for the reaction of ¹⁴⁸Xe+²⁰⁸Pb at different incident energies.

The N/Z ratio equilibration plays an important role between the projectile and target during the collision, which is one of the most important mechanisms to produce neutron-rich nuclei [36]. In Fig. 4, one shows the N=126 isotones distributions for the reactions of ¹³⁶Xe+²⁰⁸Pb and ¹⁴⁸Xe+²⁰⁸Pb at different incident energies. From Fig. 4(a), one can see that the cross sections of proton stripping channels are slightly lower than the pickup channels. The nucleus produced in the ²⁰⁸Pb+¹³⁶Xe reaction with the largest neutron excess is only ²⁰⁴Pt. It is because the N/Z value of ²⁰⁸Pb (1.54) is larger than that of ¹³⁶Xe (1.52). It is hard to transfer protons from ²⁰⁸Pb to ¹³⁶Xe. While in the Fig. 4(b), this phenomenon is opposite. The cross sections of proton stripping channels are obviously larger than the pickup channels. This reaction system can produce very neutron-rich nuclei (such as ¹⁹⁹Ta). Because the N/Z value of ¹⁴⁸Xe (1.74) is significantly larger than that of ²⁰⁸Pb (1.54), the ²⁰⁸Pb is easier to transfer protons to ¹⁴⁸Xe.

Fig. 4Full-screen View

(Color online) The N=126 isotones distributions (final fragments) for the reactions of ¹³⁶Xe+²⁰⁸Pb (up panel) and ¹⁴⁸Xe+²⁰⁸Pb (bottom panel) at different incident energies.

In Fig. 5, we show the isotope production cross sections of the ²⁰⁸Pb target bombarded with stable projectiles ¹³⁶Xe, ²³⁸U, and radioactive projectile, ¹⁴⁸Xe. The dashed, solid, and dotted lines denote the calculation results from ¹³⁶Xe+²⁰⁸Pb, ²³⁸U+²⁰⁸Pb, and ¹⁴⁸Xe+²⁰⁸Pb systems at incident energy E_c.m.=1.10 V_C. The open circles denote unknown neutron-rich nuclei. The N/Z values of ¹³⁶Xe, ²³⁸U, and ¹⁴⁸Xe are 1.52, 1.59 and 1.74, respectively. One can see that the isotope production cross sections in the neutron-rich side for each panel increase with the increasing N/Z value of projectile. Especially for larger proton stripping channels, this trend is more significant. In Fig. 5(a) and 5(b), there are 5 (for Z=75) and 3 (for Z=76) unknown neutron-rich nuclei produced by the ¹⁴⁸Xe+²⁰⁸Pb reaction. This is because the exchange of particles between the two colliding partners is primarily driven by the N/Z ratio equilibration. The very neutron-rich nuclei are easily produced by the reaction of ¹⁴⁸Xe+²⁰⁸Pb via neutrons transfer. It can be seen that the production cross sections for the new neutron-rich nuclei ^200-204Re are 1.710, 1.810, 0.173, 0.060, and 0.002 μb, ^204-206Os are 1.040, 0.041, and 0.007 μb, ^206-208Ir are 0.081, 0.006, and 0.001 μb, ^209,210Pt are 0.012 and 0.004 μb, and ²¹¹Au is 0.003 μb, respectively. Based on the above, the MNT reaction with the neutron-rich radioactive beams is a very effective method to produce new heavy neutron-rich nuclei.

Fig. 5Full-screen View

(Color online) Final isotopic production cross sections with charge numbers from Z=75 to 80. The dashed, solid and dotted lines denote the calculation results from ¹³⁶Xe+²⁰⁸Pb, ²³⁸U+²⁰⁸Pb and ¹⁴⁸Xe+²⁰⁸Pb systems at incident energy E_c.m.=1.10 VC. The open circles denote unknown neutron-rich nuclei.

4 Summary

In summary, the multinucleon transfer reactions of ¹³⁶Xe+²⁰⁸Pb, ²³⁸U+²⁰⁸Pb, and ¹⁴⁸Xe+²⁰⁸Pb are investigated using the DNS model. The isotopic production cross section of reaction ¹⁴⁸Xe+²⁰⁸Pb at near barrier incident energies is studied. We find that the final isotopic production cross section of reaction ¹⁴⁸Xe+²⁰⁸Pb in the neutron-deficient side is very sensitive to incident energies. For the neutron-rich nuclei, it is not sensitive to incident energies. The N/Z ratio equilibration plays an important role between the projectile and target during the collision. The cross sections of proton stripping channels are obviously larger than the pickup channels for the reaction of ¹⁴⁸Xe+²⁰⁸Pb on N=126 shell closure. The extremely neutron-rich projectile, ¹⁴⁸Xe, improves significantly the production cross sections of neutron-rich nuclei.