Production of proton-rich nuclei in the vicinity of 100Sn via multinucleon transfer reactions

The production of new proton-rich nuclei in the vicinity of 100Sn is investigated via multinucleon transfer reactions within the framework of microscopic time-dependent Hartree-Fock (TDHF) and statistical model GEMINI++. The TDHF+GEMINI method has demonstrated the reliable description in the multinucleon transfer dynamics and the agreement between theoretical results and experimental data is quite satisfactory in the observed transfer reactions. We reveal the production cross sections of proton-rich nuclei in 100Sn region via multinucleon transfer reactions to be several orders of magnitude higher than those measured via fusion-evaporation and projectile fragmentation experiments. About 19 new proton-rich isotopes with cross sections of larger than 1 nb are predicted to be produced in multinucleon transfer reaction of 58Ni with 112Sn. The reaction mechanisms are discussed to lead the experimental production of these previously unreported nuclei. Multinucleon transfer reactions provide a fascinating possibility to reach the proton drip-line in 100Sn region and beyond.


INTRODUCTION
The proton-rich nuclei in the vicinity of 100 Sn, the heaviest self-conjugate doubly magic nucleus discovered in the chart of nuclei [1], exhibit unique nuclear structure features and play crucial role in the astrophysical rapid proton (rp) capture process. This region has been the subject of intense experimental and theoretical studies in the last two decades, see [2] and references therein. Recent β-decay spectroscopy of 100 Sn [3,4], establishing the superallowed nature of the 100 Sn Gamow-Teller decay and together with the isomeric decay spectroscopy below 100 Sn [5,6], provide strong evidence for the robustness of the N = Z = 50 shell closures. Due to the doubly magic nature of 100 Sn, an island of enhanced α emitters above 100 Sn [7,8] is expected and preliminary evidence for the super-allowed α decay to 100 Sn has been reported recently [9]. This fascinating region is crossing the proton drip-line and proton-unbound nuclei have been identified both below and above 100 Sn.
Despite continuous efforts, progress in the experimental investigation in this region has been slow over the years, because of the very low production yield of these very proton-rich nuclei. Nuclei in this region have been produced by projectile fragmentation of intermediate to high-energy heavy-ion beams and fusionevaporation reactions between low-energy ions. The most proton-rich nuclei below 100 Sn have mainly been produced in projectile fragmentation reactions [1]. For those above 100 Sn, fusion-evaporation reactions are still advantageous, e.g., the heaviest self-conjugate isotope 108 Xe has been identified via fusion-evaporation reaction and the production cross section has been measured to be less than 1 nb [9]. The prospects for extending experimental research towards the more exotic nuclei in this region are severely limited by the beam intensities available at current facilities. It is of great importance to explore an alternative reaction mechanism for the production of proton-rich nuclei in this region.
In recent years multinucleon transfer (MNT) reaction occurring in low-energy collisions has been considered as a promising method of enormous potential to produce new unstable nuclei, especially on the neutron-rich side, which are difficult to be produced by other reactions. For example, the hard-to-reach neutron-rich nuclei around N = 126 have been successfully produced via MNT reactions [10][11][12], and the measured cross sections are several orders of magnitude larger than those from projectile fragmentation of high-energy heavy-ion beams [12]. The MNT experiment with actinide projectile and target nuclei indicated the possible production of superheavy nuclei with atomic numbers as high as 116 [13], which provides an alternative pathway for the production of superheavy elements.
In present work, we propose to produce the protonrich nuclei in the vicinity of 100 Sn via MNT reactions in low-energy collision of 58 Ni with 112 Sn, and show its advantage over the conventional production through fusion-evaporation and projectile fragmentation reactions. The idea is to take advantage of transfer mechanisms and the stabilizing effect of the closed proton shells in projectile and target nuclei. The crucial transfer mechanisms determining the mass and charge of reaction products are charge equilibrium and neck evolution in transfer process. Our recent study of MNT dynamics [14] revealed that neck formation and abruption are expected to be dominant at the relatively small impact parameters leading to both neutrons and protons transferred in the same direction, while at large impact parameters the predominant mechanism of charge equilibrium leads to the opposite transfer of protons and neutrons. Concurrently, the shell effect plays a very important role in the transfer process. These reaction mechanisms have been automatically taken into account in our microscopic approaches [14]. To compare the production of nuclei around 100 Sn in different reaction systems, MNT reactions of 58 Ni with 106 Cd and 124 Xe are also investigated in this work.
Various theoretical models [15][16][17][18][19][20][21][22][23][24][25] have been developed to describe the multinucleon transfer process. In this work, we combine the microscopic time-dependent Hartree-Fock (TDHF) approach [26][27][28][29][30] with the stateof-the-art statistical model GEMINI++ [31,32] to describe such reaction process. The initial multinucleon transfer stage is depicted with TDHF approach, and particle-number projection technique is applied on TDHF wave functions to extract the transfer probability for each transfer channel. The subsequent deexcitation process including the emission of light particles and fission of heavy fragments is described with the statistical model GEMINI++. Very recently, the combined approach TDHF+GEMINI [14,[33][34][35][36][37] has been applied in multinucleon transfer reactions and reasonably reproduced the experimental cross sections for the final products. In addition, the combined method also well accounts for the experimental observation of isotopic dependence of fusion-evaporation cross sections for the production of superheavy elements [38]. These results are very remarkable since reaction calculations do not require additional parameters related to the reaction dynamics.
The article is organized as follows. The microscopic TDHF approach, particle-number projection, and statistical model GEMINI++ are briefly reviewed in Sec. 2. Section 3 presents the multinucleon transfer dynamics for the production of new proton-rich nuclei in 100 Sn region. A brief summary is given in Sec. 4.
In TDHF approach, the many-body wave function Φ is assumed to be a Slater determinant (1) in the dynamic process. The time-dependent singleparticle states φ λ (r, t) satisfy the TDHF equation where h denotes the single-particle Hamiltonian.
To extract the transfer probability P Z,N for each transfer channel, the particle-number projection method [55] is applied to project the TDHF wave functions into the eigenstates with good proton and neutron number Z, N. The cross section of primary product is obtained by integrating P Z,N over impact parameter b where b cut and b min are the upper and lower limits of the impact parameter at which the binary reaction occurs. The excited primary products will undergo the deexcitation process including the emission of light particles and fission of heavy fragments. This process is described with the Monte-Carlo-based statistical model GEMINI++. The decay probability P decay (Z, N; Z ′ , N ′ ) from primary product with Z protons and N neutrons to final product with Z ′ protons and N ′ neutrons is evaluated by where M Z ′ ,N ′ is the event number of final products with (Z ′ , N ′ ) in the total M event simulations. The production probability for final product with Z ′ protons and N ′ neutrons is the product of transfer and decay probabilities (5) The cross section for final fragments is calculated by integrating P (final) We solve TDHF equation in a symmetry-unrestricted three dimensional grid 56 × 24 × 46 with a grid spacing 1 fm. The reaction is in x-z plane and the collision axis is along x-axis. We utilize Skyrme effective interaction SLy5 [56], in which all the time-even and time-odd terms in the mean-field Hamiltonian are included in our code [14,36,38,46,48,49]. The static calculations are performed on three-dimensional grid 24 ×24 ×24 with a grid spacing 1 fm. We find that the HF ground states of 58 Ni and 106 Cd show prolate deformations, while 112 Sn and 124 Xe present triaxially deformed ground states. It should be noted that with the inclusion of pairing correlations, the ground-state deformations of the projectile and target nuclei exhibit spherical symmetry for 58 Ni and 112 Sn and axial symmetry for 106 Cd, while a coexistence of triaxially deformed ground state and prolately deformed local minimum with a small energy difference of 140 KeV is observed for 124 Xe. The ground-state deformation effect of pairing correlations may have an impact on the transfer dynamics and pairing could also affect the reaction mechanisms in the dynamical evolution. However, the effect of pairing correlations in multinucleon transfer reaction is still an open question, because the inclusion of pairing requires much more computational cost in the microscopic simulation of collision dynamics. In this work, we concentrate on the systematic TDHF studies and leave the inclusion of pairing correlation for future works.

Results and discussions
Since both projectile and target are deformed, the complete reaction dynamics should be done by the proper average over all the deformation orientations. However, since the microscopic TDHF computation is very time-consuming, we calculate the TDHF dynamics in two extreme collisions of tip and side, in which both projectile and target nuclei are in the tip and side orientations. The so-called tip (side) orientation for the deformed nucleus indicates that the long (short) axis of nucleus is initially set along the collision axis. For the collisions of the proposed reactions, we extract the fusion barrier from the nucleus-nucleus potential by using the frozen Hartree-Fock (FHF) method, in which both projectile and target nuclei are assumed to keep their ground-state densities [14,36,38,46,48,49]. The method and programming details to calculate the nucleus-nucleus potential by using the FHF approximation can be found in Ref. [57]. The fusion barrier is found to be V B =151.89, 158.40, 163.96 MeV for the reactions of 58 Ni with 106 Cd, 112 Sn, and 124 Xe, respectively, which becomes higher with the increase of mass number of target nuclei as expected.
We first compare our calculated cross sections with the available experimental data. The experiment performed in Argonne National Laboratory has measured the transfer cross sections in 58 Ni+ 112 Sn at center-ofmass (c.m.) energy of 217 MeV [58], which are shown by solid circles with errorbars in Fig. 1. The symbol MeV (corresponding to 1.4V B ) for the tip and side collisions are denoted by the red and blue histograms, respectively. We find that the production cross sections obtained from TDHF+GEMINI calculations are in good agreement with the experimental data, although the theoretical values are slightly higher than the experimental data. The minor overestimation may partly be attributed to the slightly higher incident energy in TDHF+GEMINI calculation. The agreement between theoretical results and experimental data is quite satisfactory, considering that TDHF has no parameter adjusted on the reaction dynamics. These results give us confidence in providing rather reliable predictions of transfer dynamics for the production of new proton-rich nuclei in the vicinity of 100 Sn. Since the low production cross sections of protonrich nuclei in 100 Sn region via projectile fragmentation and fusion-evaporation reactions restrict further experimental exploration toward and beyond the proton drip-line, multinucleon transfer reaction is proposed as an alternative reaction mechanism for the production of new proton-rich nuclei. Figure 2 shows the production cross sections of proton-rich Sn isotopes using multinucleon transfer (MNT), projectile fragmentation, and fusion-evaporation reactions. The MNT reac-tion 58 Ni+ 112 Sn is performed with TDHF+GEMINI approach at the energy of 1.4V B for the tip (open circles) and side (solid circles) collisions, respectively. The measured cross sections using a fragmentation of 124 Xe projectile on a Be target [1,59] (solid diamonds) and fusion-evaporation reaction 58 Ni+ 50 Cr [60] (open diamonds) are included for comparison. We see that the production cross sections present a rapid decrease with the decrease of neutron number for all three reaction mechanisms. For the nuclei closer to the stability line, the MNT, fragmentation, and fusion reactions provide the similar cross sections. However, the cross sections of the most proton-rich Sn isotopes via MNT reaction are found to be several orders of magnitude higher than the other two experimental measurements. For example, the cross section of 100 Sn nucleus, the most protonrich Sn isotopes produced so far, is around 10 −9 mb via the fragmentation of 124 Xe projectile [59], while MNT reaction predicts the cross section to be four orders of magnitude higher lying in between 10 −4 − 10 −6 mb. The production mechanism of proton-rich Sn isotopes in 58 Ni+ 112 Sn is mainly dominated by the neutron transfers due to the stabilizing effect of closed proton shells of 28 and 50 in projectile and target nuclei. Our results suggest that new proton-rich Sn isotopes beyond 100 Sn may be expected to be produced via MNT reactions at the current experimental facilities, which are difficult to be produced by other methods. These encouraging results clearly reveal the advantage of the proposed MNT reaction for the production of new proton-rich nuclei in the vicinity of 100 Sn.
To reveal the isotopic trend of new isotopes produced in multinucleon transfer reaction, the production cross sections for Z = 46 − 54 isotopes are shown in Fig. 3 at center-of-mass energies of 1.2, 1.4, and 2.0V B in the tip collision of 58 Ni+ 112 Sn. We observe that the cross sections, in particular for the proton-rich nuclei, are sensitive to the incident energy, while the cross sections of stable and neutron-rich isotopes weakly depend on the incident energy. The energy-dependent behavior may partly be attributed to the deexcitation of primary fragments by the evaporation of neutrons. On the other hand, we found that for proton stripping channels ((a)-(d) panels in Fig. 3) the production of proton-rich nuclei is dominated by the mechanism of neck formation leading to the transfer of both protons and neutrons in the same direction, while for proton pickup channels ((f)-(i) panels in Fig. 3) the charge equilibrium predominates the production of proton-rich nuclei to reduce the asymmetry of neutron-to-proton ratio N/Z between projectile and target nuclei. These two competing mechanisms leading to the production of Z = 46 − 54 proton-rich nuclei have been shown to be related to the incident energy [14]. For proton-rich nuclei, the cross sections at the energies of 1.4 and 2.0V B are similar, but several orders of magnitude higher than those at 1.2V B . This is consistent with the experimental evidence that the optimal incident energy in multinucleon transfer reactions lies in between 6 − 8.5 MeV/nucleon. The systematic studies of energy dependence of MNT dynamics suggest the optimal incident energy to be between 1.4 and 2.0V B for the production of proton-rich nuclei in 100 Sn region.
For a systematic comparison among the different reaction mechanisms, the measured cross sections in the projectile fragmentation [1,59] (solid diamonds) and fusion-evaporation reactions [60][61][62][63] (open diamonds) are also included in Fig. 3. We see that for the nuclei around the stability line the cross sections with MNT reaction are close to those measured via fragmentation and fusion-evaporation experiments, but those of proton-rich nuclei via MNT reactions gradually deviate from the measurements and are predicted to be several orders of magnitude higher than those from conventional experimental methods. For example, the recently measured cross section of 97 In nucleus via the fragmentation of a 345A MeV 124 Xe beam on a Be target is around 10 −10 mb [1] in contrast to 10 −6 mb in multinucleon transfer reaction 58 Ni+ 112 Sn. For Sb isotopes, 104 Sb is the most proton-rich nucleus measured so far and has the cross section of 10 −7 mb in fragmenta-tion reaction [59] as compared to the 10 −3 mb via MNT reaction. For the more proton-rich Sb isotopes, the recent experimental observation concerning the stability of 103 Sb against proton emission and half-lives [64] contradicts the previous experiment [65], while the MNT reaction predicts its production cross section to be the order of 100 nb. The 104 Te nucleus has been observed until very recently via fusion-evaporation reaction [9] due to a small cross section of less than 10 −6 mb , as compared to the 10 −4 mb via MNT reaction. Figure 4 further clarifies the dependence of production cross sections on reaction systems for Z = 46 − 54 isotopes. The MNT reactions of 58 Ni with 106 Cd (red), 112 Sn (green), and 124 Xe (blue) are performed with TDHF+GEMINI approach at center-of-mass energy of 1.4V B in the tip collision. The cross sections show complicated and notable dependence on the reaction systems. For stable and proton-rich nuclei, the cross sections are quite close for the reactions using 106 Cd and 112 Sn targets, but much smaller cross sections are observed with 124 Xe target for the proton-rich nuclei. For example, the cross sections of 100 Sn nucleus are predicted to be 10 −3 mb with both 106 Cd and 112 Sn targets as compared to 10 −7 mb with 124 Xe target. On the neutron-rich side, the cross sections exhibit the opposite behavior, i.e, the reaction with 124 Xe target presents the much larger cross sections than those with 106 Cd and 112 Sn targets. The remarkable target-nucleus dependence might be attributed to the competition of the two reaction mechanisms between charge equilibrium and neck formation. In 58 Ni+ 124 Xe reaction, the large asymmetry of neutron-to-proton ratio N/Z between projectile and target nuclei brings about the suppression of neck formation, leading to the lower cross sections for proton-rich nuclei and higher ones for neutron-rich nuclei. Our results suggest that about 19, 13, and 5 new proton-rich isotopes with cross sections of larger than 1 nb may be produced via multinucleon transfer reactions of 58 Ni with 112 Sn, 106 Cd, and 124 Xe, respectively. These previously unreported proton-rich nuclei with much higher production cross sections arise from the combined effect of several factors of stabilizing effect of shell structure, charge equilibrium, and neck formation. These results indicate that multinucleon transfer reaction is a more promising way for the production of new proton-rich nuclei near the doubly magic nucleus 100 Sn.

SUMMARY
To summarize, multinucleon transfer reactions in low-energy collisions are proposed for the production of new proton-rich nuclei in the vicinity of 100 Sn, which are of great importance for experimental studies related to astrophysical rp-process and shell evolution far from stability. The microscopic TDHF+GEMINI results are in good agreement with experimental data without introducing any parameter adjusted for reaction dynamics. The production cross sections of proton-rich nu-clei via multinucleon transfer reactions are found to be several orders of magnitude higher than those measured in projectile fragmentation and fusion-evaporation reactions. The large cross sections explicitly reveal the enormous advantage of multinucleon transfer reactions and provide a good opportunity to discover new protonrich isotopes in 100 Sn region. Our results suggest that about 19, 13, and 5 new proton-rich isotopes with cross sections of larger than 1 nb may be produced in multinucleon transfer reactions of 58 Ni with 112 Sn, 106 Cd, and 124 Xe, respectively. The reaction mechanisms for the production of these previously unreported nuclei arise from the mixed effects of shell structure, neck evolution, and charge equilibrium. With the development of new experimental techniques as well as the investigation of new reaction mechanisms, the unknown exotic nuclei with extreme neutron-to-proton ratio may become accessible, particularly in the region of the doubly magic 100 Sn nucleus.