Introduction

The investigation of heavy-ion fusion reactions is important for the synthesis of new superheavy nuclei (SHN) [1-14] and extremely proton-rich nuclei [15-17] and the exploration of nuclear structures [18-22]. In the case of fusion reactions involving light and intermediate nuclei, approaches such as fusion-coupled channel calculations [23-24] and empirical barrier distribution methods [25-30] have been adopted to calculate capture (fusion) cross sections. These calculations are typically based on the static or dynamic nuclear potentials [31-37]. Static potentials are typically characterized using models, such as the liquid drop model, energy density functional, or double-folding concept, coupled with a sudden approximation. The precise characterization of the nucleus–nucleus potential, particularly at short distances, is essential for comprehending the fusion mechanism.

A previous study [27] proposed a static entrance channel nucleus-nucleus potential to describe heavy-ion fusion reactions. They used the Skyrme energy density functional [38] combining the extended Thomas-Fermi (ETF) approach [39-41] and the sudden approximation for densities. By introducing an empirical barrier distribution composed of a combination of two Gaussian (2G) functions to consider the dynamic effects in the fusion processes, the most probable barrier heights $V_{\text{B}}\approx 0.946B_0$ (with frozen barrier height B₀) and fusion excitation functions for various reactions can be described reasonably well [27, 28, 37]. Although certain measured fusion cross sections can be accurately reproduced, the realistic nucleus–nucleus potential in fusion processes, particularly at short distances, remains unclear. In addition, for certain fusion reactions related to the synthesis of super-heavy nuclei such as ⁶⁴Ni+²³⁸U [42, 43], the extracted capture cross sections from the measured mass-total kinetic energy (TKE) distributions at energies above the Bass barrier [32] are significantly smaller than the predicted results of the classic fusion cross-section formula ${\sigma }_{\text{cap}}=\pi R_{\text{B}}^2\left(1-V_{\text{B}}/E_{\text{c}\mathrm{.}\text{m}\text{.}}\right)$ and the results of ETF+2G approach mentioned above. Owing to the measurements being highly time-consuming for reactions yielding elements 119 and 120, an accurate prediction of the capture cross sections and the evaporation residue (EvR) cross sections [44-48] is required.

In addition to static nuclear potentials, certain microscopic dynamics models [49], such as the time-dependent Hartree–Fock (TDHF) [50-52] and improved quantum molecular dynamics (ImQMD) model [53, 54] have also been widely adopted in studies focused on heavy-ion fusion reactions, wherein the time evolution of the densities of the composite system can be self-consistently described. Recently, the capture thresholds for 144 fusion reactions induced by nearly spherical nuclei have been systematically studied using TDHF calculations [55]. In conjunction with the Siwek-Wilczyński (SW) cross-section formula, which incorporates the classic cross-section formula by folding it with a Gaussian barrier distribution, the experimentally measured fusion cross sections at energies around the barriers can be reproduced well for certain fusion reactions such as ¹³²Sn+^40,48Ca. Moreover, the reaction Q-value can influence the fusion cross sections at sub-barrier energies in reactions with nearly spherical nuclei [55]. However, for fusion reactions involving strongly deformed nuclei, such as ¹⁵⁴Sm, ²³⁸U, and ²⁴³Am, the effective consideration of both the Q-value and nuclear static deformations in the calculation of capture cross sections remains an unresolved issue. Furthermore, as microscopic TDHF calculations are extremely time-consuming, a time-saving nucleus-nucleus potential with high accuracy needs to be developed for the systematic study of fusion reactions.

In this work, we attempt to proposed an effective nucleus-nucleus potential based on the Skyrme energy density functional (EDF) and the reaction Q-value for a systematic description of heavy-ion fusion reactions, particularly reaction systems with well-deformed nuclei.

Effective nucleus-nucleus potential

In this study, we first calculated the frozen nucleus–nucleus potential based on Skyrme EDF. The entrance-channel nucleus–nucleus potential V(R) between two nuclei is expressed as [27, 56] $V(R)=E_{\text{tot}}\left(R\right)-E_1-E_2\mathrm{,}$ (1) where R is the center-to-center distance between the two fragments, $E_{\text{tot}}\left(R\right)$ denotes the total energy of the nuclear system, and E₁ and E₂ denote the energies of the reaction partners at an infinite distance. The total energy of a nuclear system can be expressed as an integral of the Skyrme EDF $\mathcal{H}(\text{r})$ using the frozen density approximation: $\begin{array}{l}{E}_{\text{tot}}\left(\text{R}\right)\\ ={\displaystyle \int }\mathcal{H}\left[{\rho }_{1p}(\text{r})+{\rho }_{2p}\left(\text{r}-\text{R}\right)\mathrm{,}{\rho }_{1n}\left(\text{r}\right)+{\rho }_{2n}\left(\text{r}-\text{R}\right)\right]\text{d}\text{r}\mathrm{.}\end{array}$ (2) The energies E₁ and E₂ are expressed as $E_1={\displaystyle \int }\mathcal{H}[{\rho }_{1p}\left(\text{r}\right)\mathrm{,}{\rho }_{1n}\left(\text{r}\right)]\text{d}\text{r}\mathrm{,}$ (3) $E_2={\displaystyle \int }\mathcal{H}[{\rho }_{2p}\left(\text{r}\right)\mathrm{,}{\rho }_{2n}\left(\text{r}\right)]\text{d}\text{r}\mathrm{.}$ (4) where ρ_1p, ρ_2p, ρ_1n, and ρ_2n are the frozen proton and neutron densities of the projectile and target described by spherically symmetric Fermi functions. When calculating the energies and corresponding densities of the reaction partners, Skyrme EDF with the parameter set SkM*[57] was adopted. Further, the extended Thomas-Fermi (ETF2) approach was used to describe both the kinetic energy density and spin-orbit density within the EDF.

Based on the entrance nucleus–nucleus potential V(R), the frozen barrier height B₀ and depth of the capture pocket B_cap can be obtained [56]. The solid curve in Fig. 1(a) denotes the calculated V(R) value for the reaction ⁴⁰Ca+¹⁴⁴Sm. Certain microscopic dynamics simulations have indicated that the frozen potential barrier is reduced in the fusion process owing to the dynamic deformations of the reaction partners and nucleon transfers. In a systematic study of ¹⁶O-induced fusion using the ImQMD model in [54], the dynamic barrier height was lower than the frozen barrier height by approximately 5%.

Fig. 1Full-screen View

(a) Nucleus-nucleus potential for ⁴⁰Ca + ¹⁴⁴Sm. The solid and dot-dashed curves denote the frozen and effective potentials, respectively. The short dashed lines denote the position of the frozen barrier R₀ and that of the pocket R_s. (b) Empirical barrier distribution for ⁴⁰Ca + ¹⁴⁴Sm proposed in [27]

To consider the influence of the dynamic effect, we proposed an effective potential (EP) V_D (R) for fusion systems in the regions before the contact of the two nuclei: $V_{\text{D}}\left(R\right)=V(R)\left[1-k\text{erfc}\left(\frac{R-R_0}{s}-1\right)\right]\mathrm{,}$ (5) with two parameters of k=0.027 and s=1.0 fm. With an increase in R between the two nuclei, the Coulomb excitation becomes negligible, and V_D(R) is consequently close to V(R). The dot-dashed curve in Fig. 1(a) denotes the effective nucleus-nucleus potential for ⁴⁰Ca+¹⁴⁴Sm. The barrier height was reduced from B₀=150.57 MeV to V_B=143.32 MeV, and in the region R >13.5 fm, one had $V_D\left(R\right)\approx V(R)$. Figure 1(b) shows the empirical barrier distribution with the superposition of the two Gaussian functions proposed in [27]. Notably, the value of V_B approached the average barrier height owing to the distribution function. Further, the effective barrier radius R_B was slightly larger than the frozen radius R₀.

Following projectile-target contact at energies around the barrier height V_B, the frozen density approximation is no longer applicable because of the dynamic evolution of the neck. For a light fusion system, the compound nucleus is formed directly after the fusion barrier is overcome because the fission barrier is sufficiently high to render fission an improbable decay mode at incident energies close to the fusion barrier, and the potential V_D should approach -Q when the distance between the two fragments becomes very small. For heavy systems, such as the reactions leading to SHN, the influence of quasi-fission becomes evident. This implies that the effective entrance-channel nucleus-nucleus potential $V_{\text{D}}\left(R\right)$ for heavy systems could be significantly larger than the value of -Q at short distances. Moreover, nucleon transfer through the neck becomes an important method to form the compound nucleus, as described in the dinuclear system (DNS) model [45, 47, 58-61]. In the regions following the projectile-target contact, the effective potential (EP) is expressed as $V_{\text{D}}=V_{\text{s}}+\frac{\text{Δ}U-Q-V_{\text{s}}}{1+\exp \left[\left(R-R_2\right)/s\right]}+\left(V_1-V_{\text{s}}\right)\exp \left(\frac{R-R_1}{s}\right)\mathrm{,}$ (6) where V_s and V₁ are the corresponding effective potentials obtained using Eq. (5) at distance $R=R_{\text{B}}-\text{Δ}R$ and at the touching point R=R₁, respectively. Further, $\text{Δ}R=R_0-R_{\text{s}}$ and $R_2=R_{\text{s}}/2$. R₀ and R_s are the barrier radius and position of the capture pocket in the frozen nucleus-nucleus potential V(R), respectively (indicated by the dashed lines in Fig. 1). In this study, the touching point R₁ was considered as $R_1=\{\begin{array}{ll}{R}_0\hfill & :\text{Δ}R<1.5\text{fm}\hfill \\ R_0-\text{Δ}R/2\hfill & :\text{Δ}R\ge 1.5\text{fm}\hfill \end{array}$ (7) where $\text{Δ}U=\left(V_{\text{B}}+Q\right)-\left(V_{\text{B}}^{\text{sym}}+Q_{\text{sym}}\right)$ denotes the difference between the driving potential at the entrance channel and that of the corresponding symmetric system (i.e., the mass asymmetry of the projectile-target combination is approximately zero). According to the DNS model, the nucleon transfer is primarily governed by the driving potential. Here, we introduce the truncation for fusion reactions, that is, $\text{Δ}U\ge 0$ and $V_{\text{s}}\ge -Q$, considering that the energy of the composite system after the projectile-target contact should be larger than that of the compound nucleus at its ground state for fusion reactions with heavy nuclei.

Figure 2 shows the calculated effective potentials for the reactions ¹⁶O + ²⁰⁸Pb, ⁴⁸Ca + ²⁰⁸Pb, ³⁰Si + ²³⁸U, and ⁷⁰Zn + ²⁰⁹Bi. The frozen nucleus–nucleus potentials (solid curves) are also presented for comparison. In a previous study [62], the fusion barrier parameters for 367 reaction systems were systematically extracted based on 443 datasets of measured fusion/fission cross sections. The green squares denote the extracted barrier heights [62]. As evident, the capture barrier heights V_B obtained from the effective potential (EP) were consistent with the experimental values. In this study, the calculated VB with the proposed EP was systematically compared with the extracted barrier heights. The root-mean-square (RMS) deviations with respect to the 443 extracted barrier heights was 1.53 MeV, which is smaller than the Bass [32, 33] and BW91 [34] potentials. For ¹⁶O + ²⁰⁸Pb and ⁴⁸Ca + ²⁰⁸Pb, the EP results approached the corresponding TDHF capture thresholds [55]. The RMS deviations with respect to the 144 capture barrier heights [55] predicted by TDHF calculations was only 1.05 MeV. Considering that microscopic TDHF calculations are extremely time-consuming, the proposed effective nucleus-nucleus potentials with similar accuracy would be useful for the systematic study of fusion reactions.

Fig. 2Full-screen View

Similar to the situation presented in Fig. 1(a), albeit for ¹⁶O + ²⁰⁸Pb, ⁴⁸Ca + ²⁰⁸Pb, ³⁰Si+ ²³⁸U and ⁷⁰Zn + ²⁰⁹Bi. The green squares denote the extracted barrier heights [62]. The gray lines in (b) and (d) indicate the energies of V_E=0.99 V_B. $V_{\text{B}}^{\text{TDHF}}$ indicates the predicted capture barrier height in the TDHF calculations [55]

Figure 2 shows that for the intermediate fusion system ¹⁶O + ²⁰⁸Pb, the EP evidently decreased with a decrease in R after the projectile-target contact and gradually approached the value of -Q=46.5 MeV. However, for the heavy fusion system ⁷⁰Zn + ²⁰⁹Bi, the EP slightly decreased by $B_{\text{cap}}^{\text{D}}=1.71$ MeV after projectile-target contact, and at very short distances, the potentials were even higher than V_B by approximately 2 MeV. This implied that the formation of the compound nuclei was a relatively slow process, and competition among fusion, quasi-fission, and deep-inelastic scattering was evident for this system. The gray lines in (b) and (d) denote the energies V_E=0.99 V_B below which elastic scattering was the dominant process. Figure 2(d) also indicates the difficulty in forming compound nuclei in the reaction ⁷⁰Zn + ²⁰⁹Bi at energies lower than V_E based on the barrier penetration concept. This is because the potential barrier becomes extremely thick. In addition, compared with the EP for ³⁰Si + ²³⁸U, wherein the depth of the capture pocket $B_{\text{cap}}^{\text{D}}=9.38$ MeV was considerably larger than that of ⁷⁰Zn + ²⁰⁹Bi, the quasi-fission and deep-inelastic scattering events were considerably greater in ⁷⁰Zn + ²⁰⁹Bi at energies around the capture barriers.

Capture cross sections

The couplings between the relative motion of colliding nuclei and the intrinsic degrees of freedom play an important role in heavy-ion fusion reactions. To consider these couplings, Stelson introduced the distribution of barrier heights D(B) in the calculation of the fusion excitation function [63]. A well-known example is the Gaussian distribution of barrier heights predicted from different orientations of colliding nuclei that undergo slow deviations from sphericity [63]. In a previous study [26], an analytical cross section formula was proposed for describing the capture excitation function at energies around the Coulomb barrier by Siwek-Wilczyńska and Wilczyński (SW) under the Gaussian distribution assumption, ${\sigma }_{\text{cap}}\left(E_{\text{c}\text{.m}\text{.}}\right)=\pi R_{\text{m}}^2\frac{W}{\sqrt{2}{E}_{\text{c}\mathrm{.}\text{m}\text{.}}}\left[X\text{erfc}(-X)+\frac{1}{\sqrt{\pi }}\exp \left(-X^2\right)\right]\mathrm{,}$ (8) where $X=\left(E_{\text{c}\text{.m}\text{.}}-V_{\text{B}}\right)/\sqrt{2}W$, V_B and W denote the centroid and the standard deviation of the Gaussian function, respectively, and R_m denotes the average barrier radius, which is typically set as $R_{\text{m}}=R_{\text{B}}$. In this study, the average barrier radius was $R_{\text{m}}=\frac{{\displaystyle \int }\left(V_{\text{D}}-V_{\text{E}}\right)R\text{d}R}{{\displaystyle \int }\left(V_{\text{D}}-V_{\text{E}}\right)\text{d}R}.$ (9) This was calculated using the effective potential V_D at energies near the capture barrier height $E_{\text{c}\text{.m}\text{.}}=(1\pm 0.01)V_{\text{B}}$. Notably, $R_{\text{m}}\approx R_{\text{B}}$ for most fusion systems (indicated by the positions of the green squares in Fig. 2). In contrast, for reaction systems with very shallow capture pockets, R_m is significantly smaller than R_B, which is discussed later.

For fusion reactions induced by heavy target nuclei with a quadrupole deformation of β₂, the range of barrier heights $\text{Δ}B\propto {\beta }_2{V}_{\text{B}}{R}_{\text{T}}/R_{\text{B}}$ owing to the different orientations of the deformed nuclei can be derived using an average target radius R_T [22]. Therefore, the VB and β₂ dependences of W are expected for reactions with deformed nuclei. In contrast, the excitation energy of the compound nuclei (related to the reaction Q-value) at energies around the barrier in the reaction with deformed nuclei is typically larger than that with neighboring spherical nuclei. For example, the Q-value of the reaction ¹⁶O+¹⁵⁴Sm is higher than that of ¹⁶O+¹⁴⁴Sm by approximately 12 MeV (which is further discussed later). In addition, for heavy nuclei with large deformations, the excitation threshold ${\epsilon }_{\text{th}}$, defined as the energy of the lowest excited state of the reaction partner, is typically very small. For example, ε_th =0.082 MeV for ¹⁵⁴Sm and ε_th =0.045 MeV for ²³⁸U. The deformation parameter β₂ is model-dependent [64, 29]. Thus, it would be interesting to modify the barrier height effects with W having a Q-value and excitation threshold dependence.

Recently, the SW formula was applied to study the fusion excitation functions for reactions involving nearly spherical nuclei with three parameters determined by TDHF calculations [55]. The standard deviation of the Gaussian function W was observed to be related to the reaction Q-value and deformation effects. A relatively higher excitation energy at the capture position can facilitate stronger impacts on the dynamic deformations and nucleon transfer during the capture process, thereby broadening the width of the barrier distribution. Based on the concepts described in [55], we extended the expression of W to intermediate and heavy fusion reactions induced by nuclei with large deformations. The value of W for the fusion reactions induced by nuclei with large deformations is parameterized as $W=\frac{2}{3}{W}_0+\frac{1}{3}{W}_1,$ (10) where $W_0=0.052\left(V_{\text{B}}+Q\right)$ for reactions induced by nearly spherical nuclei [55]. In a systematic study of the fusion barrier parameters, Chen et al. determined the value of $W\approx \left(0.014+0.135{\lambda }_{\text{B}}\right)V_{\text{B}}$ with the reduced de Broglie wavelength ${\lambda }_{\text{B}}=\hslash /\sqrt{2\mu V_{\text{B}}}$ [62]. Considering the systematic behavior of W, we write $W_1=\left(0.048+\eta {\epsilon }_{\text{1}}/{\epsilon }_{\text{2}}\right)V_{\text{B}}$. Here, $\eta =|A_1-A_2|/\left(A_1+A_2\right)$ denotes the mass asymmetry of the reaction system, where A₁ and A₂ are the mass numbers of the projectile and target, respectively. Further, ε₁ and ε₂ denote smaller and larger energies of the lowest excited states for the reaction partners, respectively. Simultaneously, we introduce a truncation of the value of W, that is, $W\le 2.5W_0$. Equation (10) shows that the V_B dependence of the barrier distribution width is considered in W₁ and the deformation effects are indirectly considered using the Q-value in W₀ and the energies of the lowest excited states in W₁.

Table 1 lists the calculated barrier parameters of the reactions under consideration. The barrier height V_B and average barrier radius R_m were obtained using Eqs. (5) and (9), respectively. The standard deviation of the Gaussian function W was obtained using Eq. (10). The reaction Q-value for unmeasured super-heavy system was obtained from the prediction of the Weizsäcker-Skyrme (WS4) mass model [65] with which the known masses can be reproduced with an RMS error of ~0.3 MeV [66] and the known α-decay energies of SHN can be reproduced with deviations smaller than 0.5 MeV [6, 12].

Figure 3 presents the effective potentials and fusion excitation functions for the reactions ¹⁶O + ^144,154Sm. As shown in Fig. 3(a), the calculated barrier heights were consistent with the experimental values. The fusion barrier height for the neutron-rich system ¹⁶O + ¹⁵⁴Sm was lower than that of ¹⁶O + ¹⁴⁴Sm by 1.09 MeV. At very short distances, the potentials approached the corresponding values of -Q. From Fig. 3(b), it can be observed that the measured fusion cross-sections for both ¹⁶O + ¹⁴⁴Sm and ¹⁶O + ¹⁵⁴Sm were well reproduced.

Fig. 3Full-screen View

(a) Similar to situation presented in Fig. 2, albeit for ¹⁶O + ^144,154Sm. The data for the barrier heights are obtained from [62]. (b) Fusion excitation functions for ¹⁶O + ^144,154Sm. The scattered symbols denote the experimental data taken from [67] and the curves denote the predicted results with Eq. (8)

Figure 4 shows the calculated capture excitation functions for reactions with the deformed nuclei ¹⁵⁴Sm and ²³⁸U. The scattered symbols represent experimental data. The solid curves and short dashes denote the results of the effective potential coupled with the SW formula (Eq. (8)) and the results of the ETF+2G approach [27], respectively. Notably, at energies near the capture barriers, the experimental data were reasonably well reproduced by both approaches. For ⁴⁸Ca + ²³⁸U, the measured cross-sections at the sub-barrier energies were reproduced considerably better using the EP+SW approach. Here, we emphasize that the SW formula was obtained from the folding of the Gaussian barrier distribution and the classic over-barrier fusion cross-section expression, which is not applicable to deep sub-barrier energies because it does not consider the barrier penetration effect. Figure 3 and 4 indicate that the measured capture cross sections at energies near the barrier were well reproduced by the SW formula. Thus, the folding of a suitable barrier distribution plays a role in describing the capture cross sections around barrier energies. In addition, the Wong formula [31], which considers the quantum mechanical tunneling effect, was used to fold the two-Gaussian barrier distribution in the ETF+2G calculations. As evident, the results from the two approaches were similar for the reactions ³²S + ¹⁵⁴Sm, ³⁰Si + ²³⁸U, and ⁴⁸Ca + ¹⁵⁴Sm, which also indicates the importance of the barrier distribution.

Fig. 4Full-screen View

Capture excitation functions for the reactions ³²S + ¹⁵⁴Sm [68], ³⁰Si + ²³⁸U [69], ⁴⁸Ca + ¹⁵⁴Sm [70], and ⁴⁸Ca + ²³⁸U [71, 43, 72]. The solid curves and short dashes denote the results of the effective potential coupled with the SW formula (Eq. (8)) and the results of the empirical two Gaussian (2G) barrier distribution approach [27], respectively

Simultaneously, the proposed effective potential (EP) was tested to describe fusion reactions leading to the synthesis of superheavy nuclei. Figure 5 shows the calculated EP for the reactions ³⁰Si,⁴⁸Ca,⁶⁴Ni + ²³⁸U, and ⁵⁴Cr+²⁴³Am. To observe the depth of the capture pockets more clearly, the potential and distance were scaled by V_B and R_B, respectively. Notably, the depths of the capture pockets significantly decreased with an increase in the product of the projectile-target charge numbers Z₁ Z₂. Very recently, Yao et al. found that the yields of fission-like events in the measured mass-TKE distributions and the ratio of capture to deep-inelastic scattering events evidently decreased with a decrease in the depths of the capture pockets B_cap [56]. To understand the underlying physics, we studied the average barrier radius, R_m. Table 1 shows that for the reactions induced by ^52,54Cr and ⁶⁴Ni, the average barrier radii were significantly smaller than those of the other reactions, although the contact distances were larger. With smaller average barrier radii for reactions with ^52,54Cr and ⁶⁴Ni, the corresponding capture cross-sections at the above barrier energies are expected to be reduced. Figure 6 shows the predicted capture cross-sections for the reactions ⁵²Cr+ ²³²Th, ⁵²Cr+ ²⁴⁸Cm, ⁵⁴Cr+²⁴³Am, and ⁶⁴Ni+ ²³⁸U. The green dashed and blue solid curves denote the results predicted using the empirical 2G barrier distribution and EP+SW approaches, respectively. The solid circles denote the experimental data obtained from [43]. As evident, for ⁵²Cr+ ²⁴⁸Cm and ⁶⁴Ni+ ²³⁸U, the capture cross-sections were significantly overpredicted by the ETF+2G approach. The dot-dashed curves denote the results with the ETF+2G approach but adopt the average barrier radius R_m in the calculations. When using R_m rather than R_B in the calculations, the capture cross sections at the above barrier energies were considerably better reproduced. The predicted capture cross sections at sub-barrier energies with the effective potential were evidently smaller than those with the 2G approach for ^52,54Cr induced reactions. This implies that the predicted 2n EvR cross sections would be considerably smaller than those obtained using the ETF+2G approach for the reaction ⁵⁴Cr+²⁴³Am.

Fig. 5Full-screen View

Effective nucleus-nucleus potentials for ³⁰Si,⁴⁸Ca,⁶⁴Ni + ²³⁸U and ⁵⁴Cr+²⁴³Am

Fig. 6Full-screen View

(Color online) Similar to the situation presented Fig. 4, albeit for ⁵²Cr+²³²Th,²⁴⁸Cm, ⁵⁴Cr+²⁴³Am and ⁶⁴Ni+²³⁸U. The dot-dashed curves denote the results with the 2G approach but with the average barrier radius R_m rather than R_B in the calculations. The data are obtained from [43]

Summary

Based on the frozen nucleus–nucleus potential from the Skyrme energy density functional, coupled with the extended Thomas-Fermi approach and sudden approximation of densities, this study proposed an effective approach to obtain the capture barrier heights and average barrier radii for heavy-ion fusion reactions. The 443 extracted barrier heights were well reproduced with an RMS error of 1.53 MeV. The effective potential results were very close to the corresponding TDHF capture thresholds for ¹⁶O + ²⁰⁸Pb and ⁴⁸Ca + ²⁰⁸Pb. The RMS deviation with respect to the 144 capture barrier heights predicted using the TDHF calculations was only 1.05 MeV, which provides a useful balance between accuracy and computation cost, facilitating numerous fusion systems in a simple uniform manner. Combined with Siwek-Wilczyński formula, wherein the three parameters are determined by the proposed effective potentials, the measured capture cross-sections at energies around the barriers were well reproduced for several fusion reactions induced by both spherical and well-deformed nuclei. The shallow capture pockets and small values of the average barrier radii led to the reduction of the capture cross-sections for ^52,54Cr-and ⁶⁴ Ni-induced reactions. Moreover, the decrease in the capture pocket depth for the heavy reaction system resulted in an increase in deep inelastic collisions at energies around the capture barrier, which consequently suppressed the production of the compound system.