Introduction

The synthesis of heavy and super-heavy nuclei (SHN) has drawn considerable interest in nuclear physics [1-6]. Notably, nuclei with N=184 and Z=114 are predicted double magic nuclei at the center of the island of stability [7], which also is predicted by early calculations. Up to now, this landscape of nuclear charts now includes elements up to Z = 118 element. Fusion-evaporation reaction, particularly effective for neutron-deficient nuclei, have become a crucial method for producing heavy and SHN [8-11]. For instance, such as ²⁶²Bh, ²⁶⁵Hs, ²⁶⁷Mt, ²⁶⁹Ds, and ²⁷²Rg have been synthesized using ²⁰⁸Pb and ²⁰⁹Bi targets [12-16]. Over recent decades, ongoing research to produce neutron-deficient actinide nuclei has been carried out at facilities including Lawrence Berkeley National Laboratory (LBNL, Berkeley) [17], Flerov Laboratory of Nuclear Reactions (FLNR, Dubna) [18], and the Institute of Modern Physics (IMP, Lanzhou) [9, 19, 20]. Recently, ²⁴⁹No was detected via the α-decay of the new isotope ²⁵³Rf [21]. Although fusion-evaporation reactions continue to be a promising approach for synthesizing neutron-deficient nuclei in the heavy-mass region [22, 23], the resulting mass spectrum remains relatively narrow. The multinucleon transfer (MNT) reaction, characterized by its wide mass range due to broad excitation functions within the transfer process, offers an alternative approach for extending the nuclear chart landscape [24-27].

Since the 1970s, transfer reactions and deep inelastic heavy-ion collisions have been extensively studied, resulting in the identification of new neutron-deficient actinide nuclei and neutron-rich isotopes of light nuclei [28-34]. The products of MNT reactions are closely related to the structure of the reaction system and the projectile-target mass asymmetry [35-37]. Typically, neutron-deficient beams paired with actinide targets are used to produce neutron-deficient actinide isotopes through MNT reactions. For instance, five new neutron-deficient isotopes, ²¹⁶U, ²¹⁹Np, ²²³Am, ²²⁹Am, and ²³³Bk, were identified in the reaction ⁴⁸Ca +²⁴⁸Cm [38]. Recently, a new isotope, ²⁴¹U, was synthesized, and systematic atomic mass measurements of 19 neutron-rich Pa-Pu isotopes were conducted using the MNT reaction of the ²³⁸U+¹⁹⁸Pt system at the KISS facility [39]. Consequently, the MNT reaction mechanism provides an accessible pathway to synthesize previously unknown actinide nuclei.

Various theoretical techniques have been developed to describe transfer reactions in low-energy heavy-ion collisions. These include semi-classical models such as the Grazing model [40, 41], Grazing-F model [42], the dinuclear system (DNS) model [5, 37, 43-50], the dynamical model based on multidimensional Langevin equations [51-56], the improved quantum molecular dynamics (ImQMD) model [57-61], and the time-dependent Hartree-Fock (TDHF) approach [62, 63, 63, 64]. These models have enabled extensive investigation of topics such as the production cross-sections of new isotopes [65], shell effect on fragment formation [66], total kinetic energy spectra of transfer fragments [60], and angle distributions for MNT products [67]. Notably, MNT reactions reveal unique features, including the ability to manually control nucleon transfer, the interaction mechanisms of projectile-target combinations, and the impact of incident energy on the production cross-sections of neutron-deficient actinide nuclei in theoretical studies.

Due to isospin relaxation, MNT reactions using neutron-deficient beams offer advantages in accessing neutron-deficient isotopes within the actinide region [68]. The properties of these neutron-deficient heavy isotopes are essential for investigating shell evolution and proton driplines. In addition to fusion-evaporation reactions, MNT reactions may present a viable pathway for producing neutron-deficient heavy isotopes [68]. The primary goal of studying MNT reactions with actinide targets is to probe an island of stability. This study focuses on the production of unknown neutron-deficient isotopes with Z = 99-106 using the DNS model via ¹²⁴Xe+²⁴⁸Cm, ¹²⁴Xe+²⁴⁹Cf, and ¹²⁹Xe+²⁴⁹Cf reaction systems. The de-excitation process is addressed using the GEMINI + + statistical model.

The remainder of this paper is organized as follows. In Sec. 2, we provide a brief overview of the theoretical framework of the DNS model. The results and discussion are presented in Sec. 3. Finally, a summary and outlook are presented in Sec. 4.

Theoretical framework

The DNS model describes a diffusion process that takes place along the relative distance between the centers of interacting nuclei and in terms of mass asymmetry. The probability $P\left(Z_{\text{1}}\mathrm{,}{N}_{\text{1}}\mathrm{,}{E}_{\text{1}}\mathrm{,}t\right)$ for the mass distribution of a fragment with proton number Z₁ and neutron number $N_{\text{1}}$ can be determined by solving the master equation [69]. $\begin{array}{c}\frac{\text{d}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}t\right)}{\text{d}t}={\displaystyle \sum _{Z_1^{\text{'}}}{W}_{Z_1\mathrm{,}{N}_1;Z_1^{\text{'}}\mathrm{,}{N}_1}}\left(t\right)\\ \left[{\text{d}}_{Z_1\mathrm{,}{N}_1}P\left(Z_1^{\text{'}}\mathrm{,}{N}_1\mathrm{,}{E}_1^{\text{'}}\mathrm{,}t\right)-{\text{d}}_{Z_1^{\text{'}}\mathrm{,}{N}_1}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}t\right)\right]\\ +{\displaystyle \sum _{N_1^{\text{'}}}{W}_{Z_1\mathrm{,}{N}_1;Z_1\mathrm{,}{N}_1^{\text{'}}}}\left(t\right)\\ \left[{\text{d}}_{Z_1\mathrm{,}{N}_1}P\left(Z_1\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{E}_1^{\text{'}}\mathrm{,}t\right)-{\text{d}}_{Z_1\mathrm{,}{N}_1^{\text{'}}}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}t\right)\right],\end{array}$ (1) where $W_{Z_1\mathrm{,}{N}_1;Z_1^{\text{'}}\mathrm{,}{N}_1}\left(t\right)$ and $W_{Z_1\mathrm{,}{N}_1;Z_1\mathrm{,}{N}_1^{\text{'}}}\left(t\right)$ are the mean transition probabilities from channel (Z₁,N₁) to (${Z^{\prime }}_1$, N₁) and (Z₁,N₁) to (Z₁, ${N^{\prime }}_1$) at the time t, respectively. The quantity ${\text{d}}_{Z_1\mathrm{,}{N}_1}$ denotes the microscopic dimension, that is, the microscopic state number of the fragment for the macroscopic state (Z₁, N₁, E₁) [70]. The locally excited energy E₁ of the composite system is provided by the dissipation of the relative kinetic energy. The relationship between the transition probability and local excitation energy can be expressed as follows: $\begin{array}{l}{W}_{Z_1\mathrm{,}{N}_1;Z_1^{\text{'}}\mathrm{,}{N}_1}\left(t\right)\\ =\frac{{\tau }_{\text{mem}}\left[Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\left(Z_1\mathrm{,}{N}_1\right);Z_1^{\text{'}}\mathrm{,}{N}_1\mathrm{,}{E}_1\left(Z_1^{\text{'}}\mathrm{,}{N}_1\right)\right]}{{\text{d}}_{Z_1\mathrm{,}{N}_1}{\text{d}}_{Z_1^{\text{'}}\mathrm{,}{N}_1}{\hslash }^2}\\ \times {\displaystyle \sum _{i{i}^{\text{'}}}{\left|\langle Z_1^{\text{'}}\mathrm{,}{N}_1\mathrm{,}{E}_1\left(Z_1^{\text{'}}\mathrm{,}{N}_1\right)\mathrm{,}{i}^{\text{'}}|V(t)|Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\left(Z_1\mathrm{,}{N}_1\right)\mathrm{,}i\rangle \right|}^2}.\end{array}$ (2) where i denotes the remaining quantum numbers. During the diffusion process, a single-nucleon transfer at each time step was assumed in DNS model as follows: ${Z^{\prime }}_1=Z_1\pm 1$ or ${N^{\prime }}_1=N_1\pm 1$. Memory time τ_mem is expressed as ${\tau }_{\text{mem}}\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1;{Z^{\prime }}_1\mathrm{,}{N}_1\mathrm{,}{E^{\prime }}_1;t\right)=\hslash {\left[\frac{2\pi }{{\displaystyle {\sum }_{K{K}^{\prime }}\langle V_{K{K}^{\prime }}{V}_{K{K}^{\prime }}^{*}\rangle }}\right]}^{1/2}.$ (3) More details regarding the memory time τ_mem are discussed in Ref. [71]. The excitation energy in the DNS model can be expressed as $\begin{array}{c}{E}_1\left(Z_1\mathrm{,}{N}_1\right)=E_{\text{diss}}-[U\left({\text{Z}}_1\mathrm{,}{N}_1\mathrm{,}{Z}_2\mathrm{,}{N}_2\right)\\ -U\left(Z_P\mathrm{,}{N}_P\mathrm{,}{Z}_T\mathrm{,}{N}_T\right)]-\frac{M^2}{2{\zeta }_{\text{int}}}\mathrm{,}\end{array}$ (4) where E_diss is the energy dissipated into the composite system and is associated with the incident energy and entrance angular momentum J [72]. M and ζ_int denote the intrinsic angular momentum of the DNS and intrinsic moment of inertia, respectively, [72]. The fragment distributions are strongly influenced by E_diss $E_{\text{diss}}\left(t\right)=E_{\text{c}\text{.m}\text{.}}-V_{\text{cont}}\left(Z_{\text{p}}\mathrm{,}{N}_{\text{p}}\mathrm{,}{R}_{\text{cont}}\right)-\frac{{\langle J(t)\rangle }^2}{2{\zeta }_{\text{rel}}}-\langle E_{\text{rad}}\left(J\mathrm{,}t\right)\rangle \mathrm{.}$ (5) The dissipation of the relative angular momentum $\langle J(t)\rangle$ can be calculated as $\langle J(t)\rangle =J_{\text{st}}+\left(J-J_{\text{st}}\right)\exp [-t/{\tau }_J)]$, where the angular momentum the relaxation time ${\tau }_J=12\times {10}^{-22}\text{ s}$. J_st denotes angular momentum at the stick limit. The radial energy at time t is given by $\langle E_{\text{rad}}\left(J\mathrm{,}t\right)\rangle =E_{\text{rad}}\left(J\mathrm{,0}\right)\exp \left(-t/{\tau }_R\right)$, where τR(${\tau }_R=2\times {10}^{-22}\text{ s}$) is the relaxation time of the radial kinetic energy [73].

The potential energy surface (PES) of a projectile-target system is critical in governing the nucleon transfer process [74, 75], which can be expressed as $\begin{array}{c}U\left(Z_1\mathrm{,}{N}_1\mathrm{,}{R}_{\text{cont}}\right)=\text{Δ}\left(Z_1\mathrm{,}{N}_1\right)+\text{Δ}\left(Z_2\mathrm{,}{N}_2\right)\\ +V_{\text{cont}}\left(Z_1\mathrm{,}{N}_1\mathrm{,}{R}_{\text{cont}}\right)\mathrm{.}\end{array}$ (6) where Δ(Z₁, N₁) and Δ(Z₂, N₂) are the mass excesses of fragments (Z₁, N₁) and (Z₂, N₂), respectively. $V_{\text{cont}}\left(Z_1\mathrm{,}{N}_1\mathrm{,}{R}_{\text{cont}}\right)$ is the effective nucleus-nucleus interaction potential for the two fragments, and $R_{\text{cont}}$ is the location of the potential pocket if the nucleus-nucleus potential contains a potential pocket [73]. The position at which the nucleon transfer process occurs for the reaction without potential pockets can be obtained using the equation $R_{\text{cont}}=R_1\left[1+{\beta }_2^{\left(1\right)}{Y}_{20}\left({\theta }_1\right)\right]+R_2\left[1+{\beta }_2^{\left(2\right)}{Y}_{20}\left({\theta }_2\right)\right]+0.7\text{fm}$ [69]. Here, $R_{\mathrm{1,2}}=1.16A_{\mathrm{1,2}}^{1/3}$ fm and ${\beta }_2^{\left(i\right)}$ (i=1, 2) is the quadrupole deformation parameter of fragment i which can be obtained from Ref. [76].

The effective nucleus-nucleus interaction potential between the two fragments can be expressed as $\begin{array}{l}{V}_{\text{cont}}\left(Z_1\mathrm{,}{N}_1\mathrm{,}{R}_{\text{cont}}\right)\\ =V_{\text{C}}\left(Z_1\mathrm{,}{N}_1\mathrm{,}{R}_{\text{cont}}\right)+V_{\text{N}}\left(Z_1\mathrm{,}{N}_1\mathrm{,}{R}_{\text{cont}}\right)\mathrm{.}\end{array}$ (7) The Coulomb potential is expressed as follows [77]: $\begin{array}{c}{V}_{\text{C}}\left(\text{r}\mathrm{,}{\theta }_i\right)=\frac{Z_1{Z}_2{e}^2}{r}+{\left(\frac{9}{20\pi }\right)}^{1/2}\left(\frac{Z_1{Z}_2{e}^2}{r^3}\right)\\ \times {\displaystyle \sum _{i=1}^2{R}_i^2}{\beta }_2^{\left(i\right)}{P}_2\left(\cos {\theta }_i\right)+\left(\frac{3}{7\pi }\right)\left(\frac{Z_1{Z}_2{e}^2}{r^3}\right)\\ \times {\displaystyle \sum _{i=1}^2{R}_i^2}{\left[{\beta }_2^{\left(i\right)}{P}_2\left(\cos {\theta }_i\right)\right]}^2\mathrm{.}\end{array}$ (8) Here, r denotes the centroid distance between the projectile and target nuclei and Ri is the nuclear radius of fragment i. The nuclear potential can be written as [78] $\begin{array}{l}{V}_{\text{N}}\left(r\mathrm{,}\theta \right)=C_0\{\frac{F_{\text{in}}-F_{\text{ex}}}{{\rho }_0}[{\displaystyle \int }{\rho }_1^2\left(r\right){\rho }_2\left(r-R\right)\text{d}r\\ +{\displaystyle \int }{\rho }_1\left(r\right){\rho }_2^2\left(r-R\right)\text{d}r]+F_{\text{ex}}{\displaystyle \int }{\rho }_1\left(r\right){\rho }_2\left(r-R\right)\text{d}r\}\mathrm{,}\end{array}$ (9) with $\begin{array}{l}{F}_{\text{in}}=f_{\text{in}}+{f^{\prime }}_{\text{in}}\frac{N_1-Z_1}{A_1}\frac{N_2-Z_2}{A_2}\mathrm{,}\\ F_{\text{ex}}=f_{\text{ex}}+{f^{\prime }}_{\text{ex}}\frac{N_1-Z_1}{A_1}\frac{N_2-Z_2}{A_2}\mathrm{.}\end{array}$ where C₀=300 MeV fm³ and the values of the amplitudes used in this study were taken from Ref. [79]; that is, f_in=0.09, f_ex=-2.59, ${f^{\prime }}_{\text{in}}=0.42$, and ${f^{\prime }}_{\text{ex}}=0.5$. The mean nuclear density at the center of the composite system is ${\rho }_0=0.16{\text{fm}}^{-3}$. The expressions ${\rho }_1\left(r\right)$ and ${\rho }_2\left(r-R\right)$ represent the density distributions of the two nuclei, which are expressed in Woods-Saxon form.

The following equation expresses the production cross-section of the primary fragment with proton number Z and mass number A: $\begin{array}{c}{\sigma }_{\text{pr}}\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_{\text{c}\mathrm{.}\text{m}\mathrm{.}}\right)=\frac{\pi {\hslash }^2}{2\mu E_{\text{c}\mathrm{.}\text{m}\mathrm{.}}}{\displaystyle \sum _J\left(2J+1\right)}\\ \times \left[P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}t={\tau }_{\text{int}}\right)\right],\end{array}$ (10) where P is regarded as the fragment distribution probability of nuclei with charge number Z and neutron number N, and t=τ_int is the interaction time. The sequential statistical evaporation of excited fragments was calculated using the GEMINI++ code [80, 81]. The GEMINI++ code is an improved version of the GEMINI sequential decay code, capable of handling light particle evaporation, symmetric fission, and all potential binary-decay modes. To simulate nucleon and light particle evaporation, the Hauser-Feshbach formalism is applied. The Moretto formalism is used to model the emission of heavier fragments and asymmetric fission of heavy systems, while symmetric fission is predicted using the Bohr-Wheeler formalism, accounting for structural evolution. GEMINI++ is commonly employed to treat both light particle evaporation and various binary-decay modes effectively.

Results and Discussions

3.1

Comparison with experimental data

To inspect the reliability of DNS model in reproducing the transfer cross-sections of actinide nuclei, the production cross-sections with Z = 95–100 isotopes were investigated through the MNT reaction ¹²⁹Xe+²⁴⁸Cm at an incident energy E_{c. m.} = 513.10 MeV.

As illustrated in Fig. 1, the cross-sections of target-like fragments (TLFs) as a function of the mass number are explicitly presented for the reaction ¹²⁹Xe+²⁴⁸Cm. The black line denotes the distribution of final fragments obtained from the DNS model, while the solid circle represents the experimental data from Ref. [29]. The systematic trends observed in the theoretical results calculated using the DNS model closely align with the experimental data. The peak position of the theoretically calculated cross-section is in proximity to that of the experimental section. Additionally, the calculated isotopic distribution cross-sections exhibit a decrease with increasing proton numbers in the TLFs. For instance, the peak production cross-section for Am [see Fig. 1 (a)] is two orders of magnitude larger than that of Fm [Fig. 1 (f)]. Furthermore, the nuclides investigated in these reactions are distributed in the transuranic region. This indicates that the combination of the DNS model and the GEMINI++ model can reliably extrapolate the multinucleon transfer reactions of actinide nuclei.

Fig. 1Full-screen View

(Color online) Target-like fragment (TLF) cross sections depicted as a function of mass number in reaction ¹²⁹Xe+²⁴⁸Cm at the incident energy 513.10 MeV. The solid lines denote the results obtained by the DNS and GEMINI++ models, while the solid circle represents the experimental data taken from Ref. [29]

3.2

Charge equilibrium

To identify the two optimal colliding partners, collisions of the projectiles ^124,129Xe with the targets ²⁴⁸Cm and ²⁴⁹Cf were investigated for the production of trans-target nuclei at an incident energy of E_{c. m.} = 1.1 V_B (shown in Fig. 2). The V_B values for the ¹²⁴Xe+²⁴⁸Cm, ¹²⁴Xe+²⁴⁹Cf, and ¹²⁹Xe+²⁴⁹Cf reactions are 497.4, 508.2, and 504.9 MeV.

Fig. 2Full-screen View

(Color online) At an incident energy E_{c. m.}=1.2 V_B, the production of neutron-deficient trancalifornium nuclei with Z = 99 -106 is investigated through the multinucleon transfer Reactions, including ¹²⁴Xe+²⁴⁸Cm (solid line), ¹²⁴Xe+²⁴⁹Cf (dashed line) and ¹²⁹Xe+²⁴⁹Cf (dash-dotted line). The open symbols indicate the new nuclide

From the ¹²⁴Xe+²⁴⁸Cm (solid line), ¹²⁴Xe+²⁴⁹Cf (dashed line) and ¹²⁹Xe+²⁴⁹Cf (dash-dotted line) reactions, it can be seen that target nuclei with a smaller N/Z value are more favorable for producing neutron-deficient final fragments when the projectile is identical. In deep inelastic collisions, the trend of charge equilibration significantly influences the nucleon transfer process [82]. The N/Z ratios of ²⁴⁸Cm and ²⁴⁹Cf are 1.58 and 1.54, respectively. Compared to the ²⁴⁸Cm target, ¹²⁴Xe (1.30) is more likely to transfer protons to the ²⁴⁹Cf target.

When comparing the cross sections generated by different Xe nuclei bombarding the same target nucleus ²⁴⁹Cf in the MNT reactions, specifically the systems ^124,129Xe+²⁴⁹Cf, neutron-deficient isotopes of einsteinium (Es), fermium (Fm), mendelevium (Md), nobelium (No), lawrencium (Lr), rutherfordium (Rf), dubnium (Db), and seaborgium (Sg) can be synthesized by transferring one to eight protons from the projectile to the target. The calculated results indicate that the cross sections for the neutron-deficient isotopes with Z = 99-106 increase as the N/Z value for Xe isotopes decreases (with N/Z ratios are 1.30 and 1.38 for ¹²⁴Xe and ¹²⁹Xe, respectively). Thus, more neutron-deficient nuclei are favorable for producing unknown neutron-deficient isotopes. In summary, a reaction system with a low N/Z value, such as ¹²⁴Xe+²⁴⁹Cf, represents the optimal combination of projectile and target for producing neutron-deficient isotopes with proton numbers Z = 99–106.

In principle, the MNT process process can be described as a reaction occurring on the so-called potential energy surface (PES), where the dynamic evolution of a dinuclear system is viewed as an exchange process of independent particles between the projectile and target [50]. The PES of the fragments produced in the ^124,129Xe+²⁴⁹Cf reactions as functions of Z₁ and N₁ are illustrated in Fig. 3, where open stars indicate the locations of injection points in the nucleus. The PES structure reveals that the fragments tend to form symmetric paths, characteristic of quasi-fission processes.

Fig. 3Full-screen View

(Color online) The PES as a function of Z₁ and N₁ of the fragment 1 in the reactions ¹²⁴Xe+²⁴⁹Cf (a) and ¹²⁹Xe+²⁴⁹Cf (b)

Figure 4 shows the driving potential of the ^124,129Xe+²⁴⁹Cf reaction system, reaction system, corresponding to the minimum trajectory of the PES depicted in Figs. 3, where the valley shape of the driving potential is clearly evident. Open circles in the figure denote the positions of the injection points for the projectile, while the dotted and dash-dotted lines represent the ^124,129Xe+²⁴⁹Cf reaction systems. The position of ¹²⁹Xe is located within the valley of the driving potential, whereas the position of ¹²⁴Xe is situated at the opposite end. The transfer of a nucleon (proton or neutron) from the injection point to either side is influenced by the direction of the lower potential energy surface.

Fig. 4Full-screen View

(Color online) The driving potential of the ^124,129Xe+²⁴⁹Cf reaction system corresponds to the minimum trajectory of the potential energy surface

To explain the charge equilibration effect depicted in Fig. 2, the driving potential that must be overcome during nucleon transfer, defined as (ΔU= $U\left(Z_1\mathrm{,}{N}_1\mathrm{,}{Z}_2\mathrm{,}{N}_2\right)-U\left(Z_{\text{P}}\mathrm{,}{N}_{\text{P}}\mathrm{,}{Z}_{\text{T}}\mathrm{,}{N}_{\text{T}}\right)$), is extracted and presented in Fig. 5(a). In a pure proton pickup channel, the value of ΔU is positive and increases with the number of protons picked up, indicating that a significant amount of potential energy must be surpassed during proton pickup. Conversely, in the pure proton-stripping channel, ΔU remains positive but decreases slightly with an increasing number of stripped protons, making nucleon transfer in the proton-stripping channel easier due to the reduction in potential energy. Additionally, it is evident that the value of ΔU in the neutron pickup channel is smaller than that in the neutron stripping channel. Consequently, the nucleus ¹²⁴Xe tends to gain neutrons over protons in the reaction ¹²⁴Xe+²⁴⁹Cf. ΔU comprises $Q_{gg}\left(=M_{\text{P}}+M_{\text{T}}-M_{\text{PLF}}-M_{\text{TLF}}\right)$ and a variation in the interaction potential energy.

Fig. 5Full-screen View

(Color online) In ¹²⁴Xe+²⁴⁹Cf reaction, the values of the driving potential to be overcome (a) and Qgg (= M_P+M_T-M_PLF-M_TLF) values (b) during nucleon transfer as a function of the number of transferred nucleons (N_tr), M_P, M_T, M_PLF and M_TLF are the masses of the nucleus, target nucleus, projectile-like fragments and target-like fragments, respectively

Figure 5(b) shows the relationship between Qgg during the nucleon transfer process and the number of nucleons transferred in the ¹²⁴Xe+²⁴⁹Cf reaction (N_tr). It can be observed that the value of Qgg in the pure proton-stripping channel is negative, and its absolute value increases with the number of stripped protons. This indicates that a considerable amount of energy must be absorbed in the pure proton-stripping channel. However, the formation probability in the neutron pickup channel is significantly higher than that in the proton pickup channel. From this, the following conclusions can be drawn: The MNT reaction within neutron-deficient beams is advantageous for accessing neutron-deficient isotopes along the trans-target region.

3.3

Incident energy dependence

The incident energy plays a critical role in MNT reactions, as shown in Fig. 6, which presents the production cross sections for Es, Md, Lr, and Db isotopes in the ¹²⁴Xe+²⁴⁹Cf reaction at different incident energies. The solid, dashed and dash-dotted lines denote 1.05 V_B, 1.10 V_B and 1.20 V_B, respectively.

Fig. 6Full-screen View

(Color online) Production cross sections for Es, Md, Lr and Db isotopes in the ¹²⁴Xe+²⁴⁹Cf reaction at various incident energies. The solid, dashed and dash-dotted lines indicate 1.05, 1.10, and 1.20 V_B, respectively

It is clear that the primary fragment cross sections are significantly influenced by the incident energy, with higher energies leading to increased cross sections.

For actinide nuclei with an abundance of neutrons, the final fragment cross sections also depend heavily on the incident energy, due to the higher likelihood of fission. Compared to the primary cross section, the mass region of the final cross section becomes narrower after the de-excitation process. Additionally, a double-peak phenomenon is observed in both the primary and final isotopic cross sections for the Es isotopes. This effect is mainly attributed to the neutron subshell influence identified at N = 152 [83]. Comparing these three different incident energies reveals that the production cross sections on the neutron-deficient side are not sensitive to the incident energy variations. To minimize the chance of fission, the incident energy was set to 1.05 V_B (533.64 MeV), which is optimal for producing new neutron-deficient nuclei.

In the DNS model, the interaction time directly affects the probability distribution and excitation energy of fragments during the transfer process. Fig 7 illustrates the evolution of interaction time with “effective” angular momentum (where “effective” refers to the distance at which the target interacts with the projectile) at different energies in the ¹²⁴Xe+²⁴⁹Cf reaction, where the dashed-dotted, solid, dashed-dotted, and dashed lines indicate 1.05, 1.10, 1.15, and 1.20 V_B, respectively. Here, the reaction time is calculated by deflection function [84].

Fig. 7Full-screen View

(Color online) Evolution of the interaction time with angular momentum under different energies in the ¹²⁴Xe+²⁴⁹Cf reaction, where the dash-dotted, solid, dash-dot-dotted, dashed lines indicate 1.05, 1.10, 1.15, and 1.20 V_B, respectively

As the incident energy increases, the range of “effective” angular momentum also expands, while the rate of change in “effective” angular momentum diminishes with higher energy. For the energy differences E_{c. m.} = 1.05, 1.10, 1.15 and 1.20 V_B, it is observed that interaction time decreases as “effective” angular momentum increases. Simultaneously, the range of interaction time increases with the rising incident energy. This occurs because, with higher incident energy, the internal excitation energy dissipated into the dinuclear system increases, leading to a greater interaction distance and extended evolution time between the two nuclei.

Landscape of the neutron-deficient transcalifornium nuclei

Charge balance and energy effects were also examined, indicating that the ¹²⁴Xe+²⁴⁹Cf reaction is most favorable for producing new neutron-deficient nuclei at an incident energy of 1.05 V_B. Beam ¹²⁴Xe has an intensity of six ×10⁹ ions/s [85]. The thickness of ²⁴⁹Cf target was 0.34 mg/cm² in Dubna [86]. In Fig. 8, the production cross sections for several unknown neutron-deficient isotopes with Z = 99-106 in the ¹²⁴Xe+²⁴⁹Cf reaction are presented, along with their distributions in the nuclide diagram. This highlights the potential for detecting these neutron-deficient isotopes through time-and position-correlated α- decay chains, which is a common experimental approach [87, 88]. From the data presented, a total of 25 new nuclei are predicted to be produced. The DNS model provides the following cross section predictions for specific isotopes: ^237-240Es of 0.009, 0.004, 3.504 and 3.980 nb; ^238-240Fm of 0.001, 0.014 and 0.678 nb; ^242-244Md of 0.002, 0.036 and 0.066 nb cross sections for ^244-248No are 0.001, 0.003, 0.040, 0.049 and 0.468 nb; cross sections for ^249-251Lr are 0.107, 0.108 and 1.080 pb; cross sections for ^250-252Rf are 0.013, 0.060 and 1.290 nb; ^253,254Db with cross sections of 0.015 and 0.005 nb; and ^256,257Sg with cross sections of 0.017 and 0.097 nb, respectively.

Fig. 8Full-screen View

(Color online) Landscape of nuclide diagrams with Z = 99–106 isotopes produced in the ¹²⁴Xe+²⁴⁹Cf reaction. Yellow, red, and green squares indicate α-decay, β⁺-decay, and spontaneous fission, respectively, and blank squares represent the predicted new nuclei

Summary and outlook

The properties of neutron-deficient heavy isotopes are essential for exploring shell evolution and the proton drip line. The synthesis of neutron-deficient isotopes within the range of Z = 99–106 was investigated through MNT reactions, utilizing the combination of the DNS model and GEMINI++. A comparison of the calculated results with experimental data from the reaction ¹²⁹Xe+²⁴⁸Cm at an incident energy of 513.10 MeV demonstrates that the DNS model, in conjunction with the GEMINI++ code, effectively describes the MNT reactions in heavy-mass systems. To produce exotic neutron-deficient transcalifornium nuclei, MNT reactions involving ¹²⁴Xe+²⁴⁸Cm, ¹²⁴Xe+²⁴⁹Cf, and ¹²⁹Xe+²⁴⁹Cf were studied. Owing to the competition between the sub-shell effect at N = 152 and charge equilibrium, the cross sections of neutron-deficient trancalifornium nuclei are enhanced. Along the pure neutron and proton-stripping channels, it is evident that the pure neutron-stripping channel exhibits a larger absolute value of Qgg, indicating that transferring neutrons from the projectile to the target is more challenging than transferring protons. Additionally, the effect of incident energy on the yield of TLFs in the ¹²⁴Xe+²⁴⁹Cf reaction was also explored. The optimal incident energy for producing neutron-deficient isotopes, with Z = 99–106, was identified as E_{c. m.} = 533.64 MeV.