Introduction

The synthesis of superheavy elements has attracted considerable attention in the field of Nuclear Physics. Earlier it has been shown that superheavy elements can be produced in explosive stellar events, for example, the element with the proton number 110 was expected to be found in cosmic rays [1]. Since 1970, several attempts have been made to synthesize superheavy elements with the atomic number Z ≥ 110 [2]. Using cold fusion reactions, elements with atomic numbers Z=107-113 were synthesized using ²⁰⁸Pb and ²⁰⁹Bi as targets [3]. Element Z=112 was synthesized by the bombardment of ²⁰⁸Pb with ⁷⁰Zn [4]. The superheavy element tennessine ${}_{117}^{294}\text{Ts}$ was synthesized via a ⁴⁸ Ca-induced reaction with ²⁴⁹Bk at the Dubna Gas-Filled Recoil Separator (DGFRS) in Dubna, Russia [5]. Further experiments were conducted to synthesize new superheavy elements from Z=113 to 118 using the hot-fusion technique [6, 7].

Many microscopic and macroscopic theoretical attempts were observed on the prediction of production cross-sections. For instance, previous researchers studied the production cross-section using the dynamic cluster decay model (DCM) for the superheavy element Z=116 [8]. Using the dinuclear system (DNS) model [9], superheavy elements with Z=119 and Z=120 were investigated using Ca as a projectile with different target nuclei. Even though the systems ⁴⁴Ca+²⁵²Es and ⁴⁰Ca+²⁵⁷Fm yield larger production cross-sections [10], there are experimental difficulties in preparing the targets. Therefore, it is challenging to synthesize superheavy elements greater than Z=118 using ^40,48Ca as a projectile. Recent detailed studies on alpha decay and fusion between different projectile and target combinations have shown the production of superheavy elements Z=121 in the mass number range of 265 to 316 [11]. Studying a combined dynamic and statistical model, previous researchers [12] predicted the production cross-section for superheavy elements in the atomic number range 104 ≤ Z ≤ 112. Earlier researchers predicted the production cross-sections of superheavy elements Z ≥ 104 using a statistical model [13-17].

By considering the angular orientations of the projectile and target nuclei in the reactions, the production cross-sections can be maximized [18]. ⁴⁸ Ca-induced hot fusion reactions, such as ⁴⁸Ca+²³⁶Np, ²⁴²Am,²⁴⁸Bk, produce odd superheavy elements with atomic numbers 113, 115, and 117 [19]. ⁴⁸Ca has been extensively used in the synthesis of isotopes with Z=115, and its decay properties have been measured [7]. Furthermore, studies have been conducted on actinide targets, such as ²³⁸U, ²⁴²Pu, ²⁴⁴Pu, ²⁴³Am, ²⁴⁵Cm, ²⁴⁹Cf, ²⁴⁹Bk, and ⁴⁸Ca beams, as projectiles to synthesize superheavy elements from Z=113 to 118 using the hot fusion technique [20]. The microscopic approach based on the time-dependent Hartree-Fock theory (TDHF) [21] and the Langevin approach have been developed to synthesize the superheavy element Z=120 with hot fusion reaction systems such as ⁴⁸Ca+²⁵⁷Fm, ⁵¹V+²⁴⁹Bk, ⁵⁴Cr+²⁴⁸Cm[22]. Earlier researchers anticipated the production cross-sections, optimal energy, quasifission, and fusion-fission lifetimes of superheavy elements in the region 104 ≤ Z ≤ 120 using a dinuclear system model [23-26].

Recent investigations have shown that instead of using ⁴⁸Ca as a projectile, Ti, Cr, and Fe beams can be used as projectiles with actinide targets to produce superheavy elements through processes such as accelerated fission fragments process and multi-nucleon transfer process [27]. Hence, based on detailed investigations, we were motivated to explore ^50,52-54Cr-induced fusion reactions on targets from Hg to Cf for the formation of superheavy elements in the atomic number range 104 ≤ Z ≤ 122 using a statistical model.

The theory used to predict evaporation residue cross-sections using the statistical model is given in Sect. 2. The results obtained using ^50,52-54Cr-induced fusion reactions are presented in Sect. 3. The conclusions drawn from this study are presented in Sect. 4.

Theoretical Framework

The total potential for ^50,52-54Cr-induced fusion reactions is evaluated as follows: $V\left(R\right)=V_{\text{C}}\left(R\right)+V_{\text{N}}\left(R\right)+\frac{\mathcal{l}\left(\mathcal{l}+1\right)}{2\mu \times R^2}\mathrm{.}$ (1) The Coulomb interaction potential (V_C(R)) and nuclear interaction potential (V_N(R))[23] are expressed as $V_{\text{C}}\left(R\right)=\frac{e^2{Z}_1{Z}_2}{R}$ (2) and $V_{\text{N}}\left(R\right)=\frac{V_0}{1+\exp \left[\left(R-R_0\right)/a\right]}$ (3) In equations 2 and 3, Z₁ and Z₂ are the atomic numbers of the projectile and the target, respectively. $e^2\approx 1.44$, a is the diffuseness parameter, and R₀ is the minimum nuclear potential distance. R₀ and V₀ are evaluated as described in literature [28]. The above potential focuses on fusion-fission reactions, minimizing the role of quasi-fission. It employs a modified Woods-Saxon potential based on the Skyrme energy density functional and an extended Thomas-Fermi approach. The modified Woods (MWS) potential model, built on previous successful descriptions of fusion reactions, transitioned from a numerically computed entrance channel potential to a practical analytical expression. This analytical MWS potential streamlines the investigation of the fusion and fission barriers, thereby improving their practical utility. $\frac{\mathcal{l}(\mathcal{l}+1)}{2\mu \times R^2}$ denotes the centrifugal potential. The average angular momentum $\langle J\rangle$ is deduced from [29]: $\langle \mathcal{l}\rangle =\{\begin{array}{lll}\frac{2}{3}\sqrt{2\mu R_{\text{B}}^2\left(E_{\text{cm}}-V_{\text{B}}\right)/{\hslash }^2}\hfill & \text{for}\hfill & E_{\text{cm}}\ge V_{\text{B}}\hfill \\ \frac{4}{3}\sqrt{2\mu R_{\text{B}}^2ϵ/{\hslash }^2}\hfill & \text{for}\hfill & E_{\text{cm}}<V_{\text{B}}\hfill \end{array}$ (4) where μ is the reduced mass of the projectile and target nuclei and R_B is the barrier radius. E_cm are the center of mass energy and the fusion barrier height, respectively.

The boundary conditions used to determine the determination of fusion barrier position (R_B) and height (V_B) are explained in [30]. The evaporation cross section of the superheavy nuclei with consequent light particle emission is represented as follows: ${\sigma }_{\text{ER}}^{xn}=\frac{\pi }{k^2}{\displaystyle \sum _{\mathcal{l}=0}^{\infty }\left(2\mathcal{l}+1\right)}T\left(E\mathrm{,}\mathcal{l}\right)P_{\text{CN}}\left(E^{*}\mathrm{,}\mathcal{l}\right)P_{\text{sur}}^{xn}\left(E^{*}\mathrm{,}\mathcal{l}\right),$ (5) where k, $\mathcal{l}$ and $T_{\mathcal{l}}\left(E_{\text{cm}}\right)$ have the same notation. In the above equation, P_CN is evaluated as follows: $P_{\text{CN}}\left(E^{*}\mathrm{,}\mathcal{l}\right)=\frac{\exp \left[-c\left({\chi }_{\text{eff}}-{\chi }_{\text{thr}}\right)\right]}{1+\exp \left(\frac{E_{\text{B}}^{*}-E^{*}}{\text{Δ}}\right)}.$ (6) where the compound nucleus excitation energy is denoted by E^* and $E_{\text{B}}^{*}$, when E_cm (the center of mass energy) is equal to the Coulomb and proximity barriers. Δ, χ_thr and c are adjustable parameters, χ_eff is the effective fissility [17] is as follows: ${\chi }_{\text{eff}}=\left[\frac{\left(Z^2/A\right)}{{\left(Z^2/A\right)}_{\text{crit}}}\right]\left[1-\alpha +\alpha f(\varphi )\right],$ (7) where ${\left(Z^2/A\right)}_{\text{crit}}$, f(ϕ), and ϕ are expressed as: ${\left(Z^2/A\right)}_{\text{crit}}=50.883\left[1-1.7286\left(\frac{{\left(N-Z\right)}^2}{A}\right)\right],$ (8) $f(\varphi )=\frac{4}{{\varphi }^2+\varphi +\frac{1}{\varphi }+\frac{1}{{\varphi }^2}},$ (9) and $\varphi ={\left(A_1+A_2\right)}^{1/3}$ (10) A, N, and Z are the mass, neutron number, and atomic number of the compound nuclei, respectively. A₁ and A₂ are the masses of the projectile and target nuclei, respectively. $T_{\mathcal{l}}\left(E_{\text{cm}}\right)$ is evaluated as $T_{\mathcal{l}}\left(E_{\text{cm}}\right)={\left[1+\exp \left(\frac{2\pi }{\hslash {\omega }_{\mathcal{l}}}\left(V_{\text{B}}-E_{\text{cm}}\right)\right)\right]}^{-1},$ (11) where $\hslash {\omega }_{\mathcal{l}}$ is the inverted parabola and V_B is the fusion barrier height. Both $\hslash {\omega }_{\mathcal{l}}$ and V_B are evaluated using a set of equations explained in the literature [31]. E_cm is the center of mass energy and the compound nucleus probability P_CN is evaluated as explained in the literature [32]. The survival probability $P_{\text{sur}}^{xn}(E^{*}\mathrm{,}\mathcal{l})$ is expressed as $P_{\text{sur}}^{xn}\left(E_{\text{CN}}^{*}\mathrm{,}\mathcal{l}\right)=P_{xn}\left(E^{*}\right){\displaystyle \prod _{i=1}^{i_{\text{max}}=x}}{\left(\frac{{\text{Γ}}_n\left(E_{\text{CN}}^{*}\mathrm{,}\mathcal{l}\right)}{{\text{Γ}}_n\left(E_{\text{CN}}^{*}\mathrm{,}\mathcal{l}\right)+{\text{Γ}}_f\left(E_{\text{CN}}^{*}\mathrm{,}\mathcal{l}\right)}\right)}_{i\mathrm{,}{E}^{*}},$ (12) where ${\text{Γ}}_n\left(E_{\text{CN}}^{*}\mathrm{,}\mathcal{l}\right)$ is the decay width of neutrons and ${\text{Γ}}_f\left(E_{\text{CN}}^{*}\mathrm{,}\mathcal{l}\right)$ is the fission decay width. The decay width was calculated as follows: ${\text{Γ}}_i=\frac{R_{C{N}_i}}{2\pi \rho \left(E_{\text{CN}}^{*}\right)},$ (13) the level density at saddle point is denoted by ${\rho }_{\text{f}}\left(E_{\text{CN}}^{*}-B_{\text{f}}-ϵ\mathrm{,}\mathcal{l}\right)$, $\hslash \omega =2.2$ MeV [33], and B_f is the fission barrier [34]. The level density [35] is expressed as follows: $\begin{array}{c}\rho \left(E^{*}\mathrm{,}\mathcal{l}\right)=K_{\text{vib}}\left(E^{*}\right)K_{\text{rot}}\left(E^{*}\right)\\ \times \frac{2\mathcal{l}+1}{24\sqrt{2}{\sigma }_{\text{eff}}^3{\left[a\left(A\mathrm{,}{E}^{*}-E_{\text{c}}\right){\left(E^{*}-E_{\text{c}}\right)}^5\right]}^{1/4}}\\ \times \exp \left[2\sqrt{a\left(A\mathrm{,}{E}^{*}-E_{\text{c}}\right)\left(E^{*}-E_{\text{c}}\right)}-\frac{{\left(\mathcal{l}+1/2\right)}^2}{2{\sigma }_{\text{eff}}^2}\right],\end{array}$ (14) where E_c, σ_eff, K_rot and K_vib are the usual notations explained in detail in [35]. The perpendicular (${\Im }_{\perp }$) and parallel (${\Im }_{\parallel }$) moments of inertia are evaluated as previously described [23].

The level density ($a\left(A\mathrm{,}{E}^{*}-E_{\text{c}}\right)$) is expressed as $a\left(A\mathrm{,}{E}^{*}-E_{\text{c}}\right)=\tilde{a}\left(A\right)\left[1+\frac{1-\exp \left[-\left(E^{*}-E_{\text{c}}\right)/{E^{\prime }}_{\text{D}}\right]}{E^{*}-E_{\text{c}}}\delta W\right].$ (15) Here, we take the values ${E^{\prime }}_{\text{D}}=18.5$ MeV and $\tilde{a}\left(A\right)=0.114A+0.162A^{2/3}$.

Results and Discussions

Using a ^50,52-54Cr projectile, a search was conducted to find acceptable targets with a longer half-life for the synthesis of the superheavy elements in the region 104 ≤ Z ≤ 122. In this regard, we observed more stable Hg-Cf isotopes with longer half-lives. Consequently, in subsequent studies, we explored the fusion reactions using Hg-to-Cf isotopes as the target and ^50,52-54Cr as the projectile. The fusion cross-section [28] is evaluated as follows. ${\sigma }_{\text{fus}}^{\text{Wang}}\left(E_{\text{cm}}\mathrm{,}B\right)=\frac{\hslash \omega R_{\text{B}}^2}{2E_{\text{cm}}}\ln \left(1+\exp \left[\frac{2\pi }{\hslash \omega }\left(E_{\text{cm}}-V_{\text{B}}\right)\right]\right)$ (16) where E_cm, V_B, R_B and ℏω are the center of mass energy, barrier height, barrier radius, and barrier curvature, respectively. Furthermore, the evaporation residue cross-sections were evaluated, as explained in Sect. 2.

The evaporation residue cross-sections are validated by comparing them with those from available experiments. Table 1 shows a comparison of ⁵⁴ Cr projectiles on ²⁰⁸Pb and ²⁰⁹Bi targets with the available experimental values [36, 37]. The prediction of the theoretical model was successful when the findings agreed with the experimental values. Table 1 shows the σ_EVR for ²⁰⁸Pb and ²⁰⁹Bi targets using ⁵⁴Cr projectile. Notably, the agreement between the predicted and experimental values is good for ⁵⁴Cr+²⁰⁸Pb and ⁵⁴Cr+²⁰⁹Bi. Hence, the current model is more reliable for the prediction of cross-sections in the superheavy element region 104 ≤ Z ≤ 122 using Cr projectiles. Therefore, with the confidence of reproducing the experimental evaporation residue cross-sections, we extended our studies to ^50,52-54 Cr projectiles on different targets. Therefore, we considered the stable isotopes of targets ranging from mercury to californium. In each fusion reaction case, the evaporation residue cross-section was evaluated, and its optimal energy was identified, as explained in the literature [26]. Optimal energy is the energy corresponding to the maximum evaporation residue cross-section. Hence, in each Cr-induced fusion reaction, the optimal energy at which the maximum evaporation residue cross-section was considered.

Furthermore, we plotted fusion cross-section as a function of the atomic number of compound nuclei and it is illustrated in Fig. 1. The fusion cross-sections for ⁵⁰Cr projectile on different targets at optimal energies are illustrated in Fig. 1(a). The figure shows that a larger fusion cross-section is observed for Z_c=106 and the minimum fusion cross-section is observed for Z_c=112. From Fig. 1(a), it is clear that the fusion cross-sections increase and are maximum for Z_c=106, and then gradually decrease and reach a minimum when Z_c=112. In addition, σ_fus gradually increases. Similar results were observed for ^52,53,54Cr-induced fusion reactions, as shown in Fig. 1(b-d). In all these cases, σ_fus is maximized when Z_c= 104 and Z_c=106 for ^52,53Cr and ⁵⁴Cr, respectively. The maximum values of σ_fus are owing to the presence of numerous atomic/neutron compound nuclei.

Fig. 1Full-screen View

A plot of fusion cross-sections for ^50,52-54Cr-induced fusion reactions at optimal energies as a function of the atomic number of compound nuclei leading to the formation of superheavy elements in the region 104 ≤ Z ≤ 122

First, the value of P_CN is determined by the competition between complete fusion and quasifission. The formed compound nuclei were excited because the beam energy of the projectile (E_cm) was often higher than the Q value for the production of the compound nuclei. Hence, we investigated the effect of magic numbers on P_CN for superheavy elements in the region $104\le Z_{\text{c}}\le 122$, as shown in Figure 2(a-d). Despite the effect of the atomic number on the fusion cross sections, we also investigated P_CN as a function of the target nuclei. From Fig. 2(a), it can be observed that P_CN is the maximum when A_T=248 for which Z_T=96 and N_T=152. Similarly, for ⁵² and ⁵⁴Cr-induced fusion reactions, we observed a larger P_CN when A_T=248. However, we observe a larger P_CN for A_T=249, with N_T=151 and Z_T=98.

Fig. 2Full-screen View

A plot of compound nucleus formation probability for ^50,52-54Cr-induced fusion reactions at optimal energies as a function of the mass number of compound nuclei leading to the formation of superheavy elements in the region 104 ≤ Z ≤ 122

In the literature [38, 30], remarkable contribution has been observed related to the Coulomb interaction parameter. Hence, we investigated the effect of the Coulomb interaction parameter $\left(z=\frac{Z_1{Z}_2}{A_1^{1/3}+A_2^{1/3}}\right)$ on the Cr-induced fusion reactions, leading to the formation of compound nuclei in the region 104 ≤ Z ≤ 122 as shown in Fig. 3(a-d). For each fusion reaction, we considered P_CN-value at the optimal beam energy. The P_CN value increases with the Coulomb interaction parameters. The value of P_CN was found to be smaller when z was approximately 344 and larger when Z=353 in the case of ^50,52-54Cr-induced fusion reactions. Hence, P_CN gradually increases with an increase in the Coulomb interaction parameter.

Fig. 3Full-screen View

A plot of compound nucleus formation probability for ^50,52-54Cr-induced fusion reactions at optimal energies as a function of Coulomb interaction parameter leading to the formation of compound nuclei in the region 104 ≤ Z ≤ 122

In the second step of the process, the compound nucleus loses excitation energy predominantly by light particles and γ-emission. One of the most important considerations in the production of heavy and superheavy elements is the probability of the compound nucleus surviving fission during the de-excitation process. Hence, we further investigated the survival probability for each fusion reaction and considered P_sur at the optimal energies. P_sur is evaluated as explained in Eq. 12, where the neutron decay width and fission decay width are estimated as explained in the Theory section. Figure 4(a-d) shows a plot of survival probability as a function of the atomic number of superheavy elements in the region 104 ≤ Z ≤ 122 using ^50,52-54Cr-induced fusion reactions. Here, the value of P_sur increases with Z_C. Furthermore, additional stability was observed when the formed compound nuclei acquired an even atomic number, as shown in the figure. A compound system with an even number of protons or neutrons exhibits comparatively high stability [39]. An even nucleus will have a more symmetric distribution of protons and neutrons, leading to enhanced binding energy and contributing to greater stability and a higher survival probability than the neighboring odd-numbered nuclei. In agreement with these results, it was also observed that the survival probability for even atomic numbers of compound nuclei is comparatively larger than that of their neighboring odd-number nucleons in the compound nucleus.

Fig. 4Full-screen View

A plot of survival probability for ^50,52-54Cr-induced fusion reactions at optimal energies as a function of the atomic number of superheavy elements in the region 104 ≤ Z ≤ 122

Finally, superheavy nuclei will be formed with the liberation of light particles such as neutrons/gamma/alpha particles. Figure 5(a-d) shows a plot of evaporation residue cross-sections for 2n, 3n, and 4n evaporation channels as a function of the center of mass energy for ⁵³Cr+²⁴³Am, ⁵⁴Cr+²⁴⁷Cm, ⁵³Cr+²⁴⁸Bk, and ⁵³Cr+²⁵⁰Cf respectively. In all cases, we recognized a larger cross-section for the 3n evaporation channel, and the energy at which the maximum cross section was observed was the optimal energy. The predicted cross-sections were 42fb at 236 MeV for ⁵³Cr+²⁴³Am, 23.2 fb at 236 MeV for ⁵⁴Cr+²⁴⁷Cm, 95.6 fb at 240 MeV for ⁵³Cr+²⁴⁸Bk, and 1.33 fb at 242 MeV for ⁵³Cr+²⁵⁰Cf. The larger cross-sections for ⁵³Cr and ⁵⁴Cr induced fusion reactions owing to their stability, which is advantageous for experimental purposes. They did not undergo radioactive decay during the experiment, thus providing a more stable environment for the experimentalist. Additionally, these isotopes are readily available, making them practical choices for experimental setups. Furthermore, we tabulated the predicted cross sections for the unexplored isotopes of superheavy elements Z=119 and 120, as provided in Table 2. Additionally, the optimal energy obtained in the current study was compared with the prediction from Eq.(8) in Ref. [40].

Fig. 5Full-screen View

A plot of evaporation residue cross-sections as a function of center of mass energy for the fusion reactions of (a) ⁵³Cr+²⁴³Am, (b) ⁵⁴Cr+²⁴⁷Cm, (c) ⁵³Cr+²⁴⁸Bk, and (d) ⁵³Cr+²⁵⁰Cf leading to the formation of superheavy elements Z=119, 120, 121 and 122 respectively

Furthermore, we plotted evaporation residue cross-sections as a function of the atomic number of compound nuclei during the 3n evaporation channel and it is shown in Fig. 6. The evaporation residue cross-sections decreased with an increase in the atomic number of the compound nuclei. However, a larger cross-section was observed for Z=121. Hence, these predicted cross-sections with excitation and beam energies are important for future experiments on Cr-induced fusion reactions.

Fig. 6Full-screen View

A plot of evaporation residue cross-sections for the 3n channel as a function of atomic number of compound nuclei

Conclusion

We studied ^50,52-54Cr-induced fusion reactions for the synthesis of superheavy element in the 104 ≤ Z ≤ 122 range. The barrier height and position were determined using boundary conditions, in which the total potential was equal to the sum of the Coulomb, nuclear, and centrifugal potentials. The cross-sections obtained in this study using ⁵⁴Cr projectiles were compared to those obtained in previous studies. The projected cross-sections from this study are in good agreement with the results of previous investigations. Detailed investigations revealed that the survival probability is more stable when the atomic number of the compound nuclei is even. Consequently, these projected cross-sections with excitation and beam energies will be useful in future Cr-induced fusion reaction investigations.