Introduction

The synthesis and investigation of unknown superheavy elements (SHEs) can reveal the shell structure and decay properties near the predicted "island of stability" [1-5]. Over recent decades, remarkable advancements have been achieved in the synthesis of new SHEs with Z = 114–118 through fusion reactions with the ⁴⁸Ca projectile and the actinide targets [6-11]. To date, modern accelerators coupled with sensitive detection techniques have made notable progress in the synthesis of new superheavy nuclei [12-20]. Despite these advances, a significant gap remains in the superheavy nuclei region, necessitating continued experimental and theoretical investigations.

For synthesizing new superheavy nuclei, the ⁴⁸Ca-induced fusion reactions encounter constraints due to the limited amount of experimental feasible Bk and Cf targets. Consequently, employing combinations of Cm targets and heavier stable projectiles can be an alternative for future experiments. The mixed-Cm target material was produced by irradiating the plutonium target at the Savannah River Site, with subsequent recovery at the Oak Ridge National Laboratory [21]. With long half-lives ranging from decades to millions of years, many Pu and Cm isotopes have been extensively applied as target materials in fusion reactions aimed at synthesizing new isotopes [14, 22-37]. In 2022, the reaction ⁵¹V+²⁴⁸Cm was attempted to synthesize a new element with Z = 119, and the optimal reaction energy for this reaction was estimated [38]. These experiments revealed the critical influence of the target neutron excess on the maximal ER cross sections, indicating the necessity for further research into the isotopic dependence of target materials.

Based on the experimental results, a variety of theoretical models, including both the macroscopic [39-44] and the microscopic approaches [45-51], have been developed. Among these, the dinuclear system (DNS) model has been proven to be reliable in investigating the fusion-evaporation reactions [52-62]. Within the framework of the DNS model, the fusion-evaporation process is divided into three stages: initial formation of the DNS upon overcoming the Coulomb barrier by the colliding nuclei; subsequent production of a compound nucleus through nucleon transfer from the lighter projectile to the heavier target; and finally, the synthesis of a superheavy nucleus via the evaporation of neutrons by the excited compound nucleus to reach the ground state.

Experiments have revealed that the evaporation residual (ER) cross sections of fusion reactions exhibit remarkable sensitivity to the selection of the target material [14, 63]. The isotopic dependence of the target not only affects the ER cross sections for synthesizing new superheavy nuclei but also influences their decay properties and the feasibility of observation for a sufficient duration to study their chemical and physical properties. Therefore, to search for the optimal projectile-target combinations, it is necessary to investigate the isotopic dependence of the target materials. Based on the DNS model, this study discusses the isotopic dependence of the targets and investigates the potential for extending the superheavy nuclei region with Cm targets.

The remainder of this article is organized as follows: In Sect. 2, we describe the theoretical framework of the DNS model. In Sect. 3, the reliability of the DNS model is examined based on the ample experimental results of the fusion reactions ⁴⁸Ca+^{239,240,242,244}Pu, with an investigation into the influence of the target isotope on the capture, fusion, and survival stages. Additionally, the experimental results of the reactions ⁴⁸Ca+^245,248Cm are compared with the theoretical calculations. The combinations of the ^242-248Cm target and the ⁴⁵Sc, ⁵⁰Ti, ⁵¹V, ⁵⁴Cr, and ⁵⁵Mn projectiles were investigated for extending the superheavy nuclei region with Z = 117–121, presenting the maximal ER cross sections and corresponding incident energies. The isotopic dependence of the Cm targets was discussed in detail for the capture, fusion, and survival stages. In Sect. 4, a summary of this study is provided.

Theoretical descriptions

Within the framework of the DNS model, the ER cross section as a function of the center-of-mass energy E_c.m. is calculated by summing the contributing partial waves J [64]: (1)In this formula, is the transmission probability for the formation of the dinuclear system. denotes the probability of complete fusion into a compound nucleus [65]. represents the probability that the excited compound nucleus survives fission [66]. The interaction potential of the colliding nuclei is given as follows: (2)Here, indicates the corresponding static deformations, which are usually considered the quadrupole deformation parameters of the nucleus [1]. denotes the stiffness of the nuclear surface predicted by the liquid drop model [67, 68]. Assuming that the deformation energy is proportional to the mass number , the dynamical deformations of the two nuclei can be expressed in terms of the total dynamical deformation parameter . represents the additional deformation beyond the static values, which is distributed between the projectile and target according to the above relation derived from the energy minimization condition: (3) (4)with is a weighting coefficient.

The Coulomb potential V_C is calculated using the Wong formula [69]: (5)The nuclear potential V_N is described by a sudden-approximation double-folding potential [70]: (6)with (7)Here C₀ = 300 MeV·fm³, f_in=0.09, f_ex=-2.59, and [70]. The nuclear densities are expressed as two-parameter Woods–Saxon forms with a deformed radius: (8) (9)with (10)Here ρ₀=0.17 fm^-3 and a_1,2 = 0.55 fm. Ri denotes the spherical radii of the nuclei, and is the spherical harmonic.

The transmission probability, which represents the capacity of colliding nuclei to surpass the Coulomb barrier, is determined as: (11) is given by the Ahmed formula [71-73]. The barrier distribution function f(B) is described by an asymmetric Gaussian form [73]: (12)Here, B represents the position of the Coulomb barrier. is given by a weighted combination of the two barrier values. refers to the Coulomb barrier of the spherical nuclei, whereas denotes that at the saddle-point. The width parameters Δ₁ and Δ₂ are defined as and , respectively. The deformation parameters correspond to the projectile and target nuclei in the saddle-point configuration. For deformed nuclear systems, the empirical parameters are set as a = 0.37, b = 0.12, and c = 1.12 [74].

The capture cross section is expressed as follows [75]: (13)In the DNS model, it is assumed that the two touching nuclei maintain their individual identities, along with their ground-state characteristics and deformations [76]. The nucleon transfer process occurs along the mass asymmetry degree . The nucleon transfer process, involving the dissipation of kinetic energy and angular momentum, occurs at the valley of the potential energy surface, defined as the driving potential [77]. To form a compound nucleus, the dinuclear system must possess sufficient energy to overcome the inner fusion barrier B_fus, which is the energy difference between the incident point and the point of the maximal driving potential (B.G. point). Therefore, with the interaction time τ_int(J) given by the deflection function method [78], the fusion probability is obtained by summing the distribution probabilities of the fragments that successfully overcome the inner fusion barrier: (14)Here, the distribution probability is determined by resolving a set of two-dimensional master equations [73]: (15) denotes the mean transition probability from (Z₁, N₁) to [79], is the microscopic dimension associated with (Z₁, N₁). Λ_qf and Λ_fis represents the quasi-fission and fission probabilities obtained using the one-dimensional Kramers formula [80].

For the evaporation of x neutrons, the survival probability is given by the statistical model: (16) is the realization probability of emitting x neutrons at excitation energy [81]. The realization probability for emitting one neutron is expressed as: (17)Here, σ = 2.5 MeV is the half-height width of the excitation function. For multiple neutron emissions, the realization probability can be expressed using the Jackson formula as follows: (18)where I and Δ are given by (19) (20)Here, is the separation energy of the i-th neutron. denotes the excitation energy before evaporating the i-th neutron. The partial decay widths for neutron evaporation Γ_n and fission decay Γ_f were determined using the Weisskopf-Ewing theory [82] and the Bohr-Wheeler transition-state method [83], respectively. The level density parameter a can be taken as a = A/12 MeV, and a_f/a = 1.1. In the calculation of Γ_f, the fission barrier of the rotating nucleus is expressed as: (21)Here, denotes the macroscopic part determined by the liquid-drop model. is the microscopic shell correction [1]. and represent the temperature-dependent parameter and shell damping energy, respectively, as defined in Ref. [84]. The moments of inertia of the compound nucleus in the ground state and at the saddle point are given by , respectively [85, 86], where k = 0.4 is the correction factor for the rigid-body approximation. The quadrupole deformation at the saddle point is taken as , with denoting the quadrupole deformation parameter at the ground state [1]. In this work, we analyze uncertainties arise from the parameterized E_D [73, 87, 88].

Results and discussion

3.1

The isotopic dependence of the reactions with ^{239,240,242,244}Pu targets

Given the large amount of available experimental data, Pu-based hot fusion reactions are crucial for assessing theoretical models. Typically, a higher neutron excess in the target leads to an enhanced maximal ER cross section. Figure 1 presents both experimental and theoretical maximal ER cross sections of the 3n- and 4n-emission channels for the reactions ⁴⁸Ca+^{239,240,242,244}Pu. An ascending trend was observed in both the experimental and theoretical maximal ER cross sections with an increase in the neutron number in the target. Notably, the maximal ER cross sections of the 4n-emission channel increase more rapidly than those of the 3n-emission channel as the neutron number of the target increases. To understand the isotopic dependence of the target, thorough investigations of the capture, fusion, and survival stages are required.

Fig. 1Full-screen View

(Color online) The experimental [22-27, 34, 35] and calculated maximal ER cross sections in the 3n and 4n-emission channel of the reactions ⁴⁸Ca+^{239,240,242,244}Pu→ ^{287,288,290,292-xn}Fl+xn

Figure 2(a) presents the capture cross sections of the reactions ⁴⁸Ca+^{239,240,242,244}Pu at , 40 and 45 MeV. This reveals that the capture cross sections increase with increasing , owing to the enhanced probability at higher for the colliding nuclei to overcome the Coulomb barrier. A slight declining trend in the capture cross sections was observed as the neutron number of the target increased. This can be attributed to the effect of the Coulomb barriers. Figure 2(b) illustrates the excitation energies associated with the Coulomb barriers for these reactions. It is evident that the values exhibit an upward trend as the neutron number of the target increases, resulting in a decrease in the capture cross section.

Fig. 2Full-screen View

(Color online) (a) The calculated capture cross sections for the reactions ⁴⁸Ca+^{239,240,242,244}Pu→ ^{287,288,290,292-xn}Fl+xn with MeV, 40 MeV and 45 MeV. (b) The excitation energies of the corresponding Coulomb barriers of the reactions ⁴⁸Ca+^{239,240,242,244}Pu→ ^{287,288,290,292-xn}Fl+xn

For the fusion stage, Fig. 3(a) illustrates the fusion probabilities for the reactions ⁴⁸Ca+^{239,240,242,244}Pu at , 40 and 45 MeV. The fusion probabilities exhibited a slight decreasing trend with increasing neutron number of the Pu target. Moreover, as the increases, the fusion probabilities are enhanced, and the isotopic effect on the fusion probabilities gradually diminishes. This can be attributed to the influence of the inner fusion barriers. The formation of the compound nucleus requires overcoming the inner fusion barrier; otherwise, quasi-fission occurs. At higher , the impediment from the inner fusion barrier fades, leading to a higher fusion probability and diminished isotopic dependence. In Fig. 3(b), the inner fusion barrier height B_fus values of the Pu-based reactions are presented. The B_fus exhibits a slight variation of approximately 0.1 MeV, resulting in minimal differences in the fusion probabilities, less than an order of magnitude.

Fig. 3Full-screen View

(Color online) (a) The calculated fusion probabilities for the reactions ⁴⁸Ca+^{239,240,242,244}Pu→ ^{287,288,290,292-xn}Fl+xn with MeV, 40 MeV and 45 MeV. (b) The B_fus values of the reactions ⁴⁸Ca+^{239,240,242,244}Pu→ ^{287,288,290,292-xn}Fl+xn

Within the framework of the dinuclear system model, the survival process is independent of the formation of the compound nucleus. The isotopic dependence of the target primarily affects the neutron richness of the formed compound nucleus. Figures 4(a) and 4(b) display the survival probabilities of the compound nuclei formed via the reactions ⁴⁸Ca+^{239,240,242,244}Pu in the 3n- and 4n-emission channels at different . It is observed that as the increases, the survival probabilities diminish, suggesting that the compound nucleus becomes less stable and is more likely to undergo fission at high . Additionally, both Fig. 4(a) and Fig. 4(b) reveal an enhancement in the survival probability with the neutron-rich target. This trend is due to the enhanced stability of the formed compound nucleus as it approaches a closed neutron shell.

Fig. 4Full-screen View

(Color online) (a) The calculated survival probabilities for the reactions ⁴⁸Ca+^{239,240,242,244}Pu→ ^{287,288,290,292-xn}Fl+xn with MeV, 40 MeV and 45 MeV. (b) The calculated survival probabilities for the same reactions with MeV, 45 MeV and 50 MeV. (c) The values of the corresponding emission channel for the reactions ⁴⁸Ca+^{239,240,242,244}Pu→ ^{287,288,290,292-xn}Fl+xn

Notably, the variation in the survival probabilities in the 3n-emission channel is approximately an order of magnitude, whereas in the 4n-emission channel, the variation is approximately two orders of magnitude. This can be attributed to the influence of the fission barrier height . Within the DNS model, the survival probability is determined by the competition between fission and neutron emission. In Fig. 4(c), the values are presented. This reveals that as the formed compound nucleus approaches the closed neutron shell N = 184, the value increases, resulting in a suppressed possibility of fission, thereby enhancing the survival probability. The impact of the fission barrier is more pronounced in the calculation of survival probabilities in the 4n-emission channel owing to the additional competition between neutron emission and fission. Hence, with an increase in the neutron number of the target nucleus, a more significant enhancement in the survival probability was observed in the 4n-emission channel. The investigation of the capture, fusion and survival stages reveals a substantial enhancement in the survival probabilities of the compound nuclei formed by the neutron-rich Pu targets. This enhancement significantly contributes to the observed high ER cross section.

3.2

The synthesis of new superheavy nuclei with ^242-248Cm targets

In Fig. 5, the comparative analysis between calculated and experimental results is extended to the reactions ⁴⁸Ca+²⁴⁵Cm→^293-xnLv+xn and ⁴⁸Ca+²⁴⁸Cm→^296-xnLv+xn. The theoretical results for both reactions are in agreement with the experimental results in the 3n-emission channel. For the 4n-emission channel, the calculated ER cross sections were consistent with the experimental results for the reaction ⁴⁸Ca+²⁴⁵Cm, while the calculated maximal ER cross section for the reaction ⁴⁸Ca+²⁴⁸Cm exceeded the currently available experimental results. Investigations into the Pu- and Cm-based reactions prove not only the reliability of the DNS model but also its potential in identifying the optimal projectile-target combinations for the synthesis of new superheavy nuclei with Cm targets.

Fig. 5Full-screen View

(Color online) Comparison of the calculated results with the available experimental data of the reactions ⁴⁸Ca+^245,248Cm→ ^293,296-xnLv+xn [6, 34, 35]

In Table 1, the optimal reaction systems, alongside the corresponding E_c.m., and the maximal ER cross sections of the reaction systems with ⁴⁵Sc, ⁵⁰Ti, ⁵¹V, ⁵⁴Cr, ⁵⁵Mn projectiles and ^242-248Cm targets are presented. This indicates an exponential decrease in the maximal ER cross sections with an increase in the charge number of the formed compound nuclei. The comprehensive analysis in Table 1 reveals a dominance of the 3n-emission channel in the synthesis of new superheavy nuclei via the Cm-based reactions. An exception is observed in the synthesis of the isotope ²⁹³Og, where the reaction ⁵⁰Ti+²⁴⁷Cm → ²⁹³Og+4n exhibits a calculated maximal ER cross section of 114 fb. This value slightly exceeds the maximal ER cross section of 102 fb for the reaction ⁵⁰Ti+²⁴⁶Cm → ²⁹³Og+3n. For the synthesis of SHE with Z = 119, the maximal ER cross section of 144 fb appears in the reaction ⁵¹V+²⁴⁵Cm → ²⁹³119+3n. However, the synthesis of SHE with Z = 120 and 121 exhibits significantly reduced maximal ER cross sections of 0.877 fb for the reaction ⁵⁴Cr+²⁴⁷Cm → ²⁹⁸120+3n and 0.052 fb for the reaction ⁵⁵Mn+²⁴⁷Cm → ²⁹⁹121+3n, respectively, which can be attributed to the increase in the mass number of the projectile.

Table 1Full-screen View

The optimal Cm-based reaction systems for producing new superheavy nuclei with Z=117–121

Isotope	Reaction	Ec.m. (MeV)	(MeV)	(fb)
284Ts	242Cm(45Sc,3n)	209.0	38.0
285Ts	243Cm(45Sc,3n)	205.7	36.0
286Ts	244Cm(45Sc,3n)	204.6	36.0
287Ts	245Cm(45Sc,3n)	202.2	35.0
288Ts	246Cm(45Sc,3n)	202.0	36.0
289Ts	247Cm(45Sc,3n)	199.7	35.0
290Ts	248Cm(45Sc,3n)	200.3	37.0
289Og	242Cm(50Ti,3n)	226.1	36.0
290Og	243Cm(50Ti,3n)	224.3	35.0
291Og	244Cm(50Ti,3n)	223.5	35.0
292Og	245Cm(50Ti,3n)	221.4	34.0
293Og	247Cm(50Ti,4n)	224.8	39.0
295Og	248Cm(50Ti,3n)	220.0	35.0
290119	242Cm(51V,3n)	237.0	36.0
291119	243Cm(51V,3n)	234.9	35.0
292119	244Cm(51V,3n)	233.9	35.0
293119	245Cm(51V,3n)	230.8	33.0
294119	246Cm(51V,3n)	231.7	35.0
295119	247Cm(51V,3n)	228.8	33.0
296119	248Cm(51V,3n)	230.0	35.0
293120	242Cm(54Cr,3n)	248.6	37.0
294120	243Cm(54Cr,3n)	246.9	36.0
295120	244Cm(54Cr,3n)	246.7	37.0
296120	245Cm(54Cr,3n)	245.0	36.0
297120	246Cm(54Cr,3n)	244.1	36.0
298120	247Cm(54Cr,3n)	242.3	35.0
299120	248Cm(54Cr,3n)	242.1	34.0
294121	242Cm(55Mn,3n)	259.1	38.0
295121	243Cm(55Mn,3n)	256.9	37.0
296121	244Cm(55Mn,3n)	255.8	37.0
297121	245Cm(55Mn,3n)	253.6	36.0
298121	246Cm(55Mn,3n)	253.7	37.0
299121	247Cm(55Mn,3n)	251.5	36.0
300121	248Cm(55Mn,3n)	250.7	36.0

Show more

The isotopes, reaction systems, optimal incident energy E_c.m., and maximal calculated ER cross sections are listed in columns 1–5, respectively. Theoretical computational uncertainties arise from the E_D range

3.3

The isotopic dependence of the reactions with ^242-248Cm targets

To further investigate the isotopic dependence of the ^242-248Cm target, the calculated maximal ER cross sections of the Cm-based reactions in the 3n-emission channel are plotted in Fig. 6. An increasing trend in the maximal ER cross sections was observed as the neutron excess in the Cm target increased. This trend is accompanied by an apparent odd-even stagger, indicating the advantage of neutron-rich Cm targets with odd neutron numbers. To further discuss this phenomenon, the capture, fusion, and survival stages are discussed in Fig. 7.

Fig. 6Full-screen View

(Color online) The calculated maximal ER cross sections in the 3n-emission channel for synthesizing new superheavy nuclei with Z = 117–121 via the combinations of the ⁴⁵Sc, ⁵⁰Ti, ⁵¹V, ⁵⁴Cr and ⁵⁵Mn projectiles and the ^242-248Cm targets

Fig. 7Full-screen View

(Color online) (a) The capture cross sections, (b) the fusion probabilities and (c) the survival probabilities at MeV for synthesizing new superheavy nuclei with Z = 117–121 via the combinations of the ⁴⁵Sc, ⁵⁰Ti, ⁵¹V, ⁵⁴Cr and ⁵⁵Mn projectiles and the ^242-248Cm targets

In Fig. 7(a), the capture cross sections of the reaction systems with ⁴⁵Sc, ⁵⁰Ti, ⁵¹V, ⁵⁴Cr, ⁵⁵Mn projectiles and ^242-248Cm targets are presented. Consistent with the aforementioned discussion, the capture cross sections displayed a decreasing trend as the neutron number of the target increased. For ⁴⁵Sc-induced reactions, the isotopic effect on the capture was relatively significant. For the other reactions, the capture cross section exhibited considerably less variation. Figure 7(b) presents the fusion probabilities of the corresponding reactions. A significant decline in the fusion probabilities was observed between the ⁴⁵Sc- and ⁵⁰Ti-induced reactions, as well as the ⁵¹V- and ⁵⁴Cr-induced reactions. This can be ascribed to the varying mass asymmetry values influenced by the projectile mass numbers. In Fig. 7(b), a decreasing trend in the fusion probabilities with an increasing neutron number of the target can be observed. This isotopic-dependent decreasing trend in the fusion probability, along with the same trend discussed in Fig. 3, can be attributed to the deviation in the neutron number of the target from the closed neutron shell [89]. Furthermore, an odd-even effect can also be noted, which can be attributed to the pairing effect in the fusion stage.

In Fig. 7(c), the survival probabilities of the corresponding reactions in the 3n-emission channel are presented. As the neutron number of the formed compound nucleus approaches the predicted closed neutron shell, N = 184, the stability of the compound nucleus is enhanced, leading to ascending survival probabilities with increasing neutron excess in the target. The odd-even effect is also significant, which primarily arises from the influence of the variance of the fission barrier. For reactions synthesizing superheavy nuclei with Z = 117, the survival probability increases rapidly as the neutron number of the target increases. This trend can be attributed to the neutron number of the formed compound nuclei approaching the semi-closed neutron shell at N = 178. It was also observed that for reactions involving even-even projectiles such as ⁵⁰Ti and ⁵⁴Cr, the odd-even effect was suppressed. This reduction in the odd-even effect can be ascribed to the pairing effects. It is evident that, in the synthesis of new superheavy nucleus via the ^242-248Cm targets, the isotopic dependence on the maximal ER cross section mainly arises from the survival stages. Here, the odd-even effect, coupled with the high neutron excess of the target, strongly enhances the stability of the formed compound nucleus. Consequently, the ²⁴⁷Cm target is favorable for the future synthesis of superheavy nuclei.

Summary

In this study, the predictive reliability of the DNS model on isotopic dependence was examined using experimental results for the reactions ⁴⁸Ca+^{239,240,242,244}Pu. The impact of the target isotope is discussed in the capture, fusion, and survival stages, revealing a strong enhancement in the survival probabilities owing to the influence of the fission barrier height. The feasibility of applying the ^242-248Cm targets and the stable projectiles ⁴⁵Sc, ⁵⁰Ti, ⁵¹V, ⁵⁴Cr, ⁵⁵Mn for synthesizing new superheavy nuclei ^284-290Ts, ^289-293,295Og, ^290-296119, ^293-299120, ^294-300121 is investigated. To synthesize new superheavy elements with Z = 119–121, the optimal reaction systems are predicted to be the reactions ⁵¹V+²⁴⁵Cm → ²⁹³119+3n, ⁵⁴Cr+²⁴⁷Cm → ²⁹⁸120+3n, and ⁵⁵Mn+²⁴⁷Cm → ²⁹⁹121+3n, with the maximal ER cross sections of 144 fb, 0.877 fb, and 0.052 fb, respectively. The isotopic dependence of the maximal ER cross sections of the ^242-248Cm-based reactions was investigated in detail, indicating that the isotopic dependence of the Cm targets mainly arises from the survival stages. This is attributed to the enhanced stability of the formed compound nuclei, influenced by a higher neutron number when approaching the predicted neutron shell closure N=184. The odd-even effect coupled with the high neutron excess renders the ²⁴⁷Cm target promising for the future synthesis of superheavy nuclei.