Impact of quadrupole and hexadecapole deformations and associated orientations on a variety of asymmetric nuclear reactions

A great deal of experimental as well as theoretical efforts have been made in the last few decades to understand the synthesis and decay of the heavy and superheavy nuclei via the heavy-ion fusion reactions [1−8]. These studies focus on the influence of various parameters and properties of interacting nuclei, on the dynamics of heavy-ion-induced reactions. Numerous methods are available in the literature to study the synthesis as well as decay dynamics of such reactions. For instance, various theoretical models like the dinuclear system (DNS) model [9, 10], diffusion model [11], macro-microscopic model [12], dynamical Langevin model [13, 14], fluctuation-dissipation model [15] are used as the fusion-ER decay model to study the decay by evaporation residue (ER) through fusion reactions. The fusion-ER model takes into account various processes that lead to the formation of compound nuclei (CN) and their decay via the emission of a cascade of light particles.

The heavy-ion-induced reactions can be studied via three stages [5, 16−18]. In the first stage, the projectile and target (P-T) come in contact after overcoming the barrier and the process is quantized via the capture cross-section ($ \sigma_{cap} $). In the second stage, these interacting nuclei fuse to form the compound nucleus (CN). The extent of the formation of CN against the non-compound nucleus (nCN) processes can be determined by the compound nucleus formation probability ($ P_{CN} $). The formed CN is then studied via the fusion cross-section ($ \sigma_{fus} $). When the incident energy of the projectile typically exceeds the energy required for the production of an intermediate nucleus, the resulting nucleus is called a compound nucleus (excited nuclear state). This excited compound nucleus (CN) attains stability by decaying to the ground state via the fission channel or emission of light particles like protons, neutrons, $ \alpha $-particles, and gamma radiation. The extent of survival of an excited CN against the fission process and decay via the evaporation of light particles is measured in terms of survival probability ($ W_{sur} $). This process provides a stable residual nucleus in the ground state. Collectively, these quantities $ \sigma_{cap} $, $ P_{CN} $ and $ W_{sur} $ provide the fusion-ER cross-section ($ \sigma_{ER} $).

In literature [19], it has been observed that the fusion reaction dynamics depend on the properties of the colliding nuclei. Several theoreticians have examined that along with the other properties like mass and charge number, the deformation and orientation degrees of freedom, angular momentum, and excitation energy, etc., of the colliding nuclei impart significant influence on fusion reaction dynamics [20−25]. Along with that, it has also been stated that the incorporation of higher order deformations like octupole $ \beta_3 $ and hexadecapole $ \beta_4 $ deformation induces significant changes in the fusion barrier characteristics and provides a better explanation of the experimental results at their respective compact and elongated configurations [26, 27]. For example, in the reactions ¹⁶O+^174,176Yb [28, 29] and ^14,15C+²³²Th [30], the authors have found evidence of modification in the fusion barrier after the incorporation of hexadecapole deformations $ \beta_4 $ that leads to significant enhancements in the capture cross-sections. Moreover, extensive studies in the literature have analyzed fusion cross-sections by incorporating higher-order deformations via microscopic models [31−34]. The optimum orientations for the deformed interacting nuclei are determined with respect to the barrier height ($ V_B $) and barrier separation or interaction radius ($ R_B $). For instance, the configuration with the smallest interaction radius and maximum barrier height is termed as a compact configuration also known as belly-to-belly configuration, while the configuration with the largest interaction radius and the minimum barrier height is depicted as elongated configuration or the pole-to-pole configuration[20].

Numerous efforts with different theoretical approaches are used to investigate the ER cross-sections through fusion reactions. However, it would be interesting to study the relevance of $ \beta_2 $, $ \beta_4 $ deformations along with their respective 'compact or elongated' orientation degrees of freedom for the evaluation of fusion-ER cross-sections. So, in the present work, the fusion-ER cross-sections are investigated for reactions of hexadecapole ($ \beta_4 $) deformed targets $ ^{148,150} $Nd, ¹⁶⁵Ho, ¹⁹⁴Pt, $ ^{188,192} $Os and ¹⁷⁰Er with spherical $ ^{16, 18} $O and $ \beta_4 $-deformed ¹⁹F, ³⁰Si projectile beams for center of mass energies ($ E_{c.m.} $) spread across Coulomb barrier. The fusion-ER cross-sections are calculated for the considered reactions by incorporating the $ \beta_2, \beta_4 $ deformations at their respective compact and elongated configurations. Additionally, cross-sections integrated over all orientations are also calculated to provide more extensive insight.The obtained fusion-ER cross-sections are then compared with the available experimental data [18, 35−39].

In the present work, the calculations are made to obtain the fusion-ER cross-sections for a set of reactions by using the semi-classical approach. In this approach, the fusion-ER cross-sections are determined by using the capture cross-sections ($ \sigma_{cap} $), compound nucleus formation probability ($ P_{CN} $), and survival probability ($ W_{sur} $). The capture cross-sections ($ \sigma_{cap} $) are calculated by using the extended $ \ell $-summed Wong model incorporating deformations $ \beta_2 $, $ \beta_4 $ at compact, elongated configurations with the proximity potential [40]. The compound nucleus formation probability ($ P_{CN} $) dependent on the Coulomb parameter and center of mass energies ($ E_{c.m.} $) is determined using the Siwek-Wilczyńska formula [41]. To calculate the survival probability ($ W_{sur} $) of the CN against fission, statistical model is used [42]. In addition to the deformations and orientations, the influence of the level density parameter ratio between the fission channel to neutron evaporation channel ($ a_f/a_n $) is studied to determine the fusion-ER cross-sections.

The manuscript is organized as follows. Section II provides the formalism for the calculation of the fusion-ER cross-section via the capture cross-section, compound nucleus formation probability, and survival probability. Section III discusses the results and the results are summarized in section IV.

The fusion-ER cross-sections calculated by using the semi-classical approach, after the emission of x-number of neutrons from the excited compound nucleus is defined by the product of capture cross-section ($ \sigma_{cap} $), compound nucleus (CN) formation probability ($ P_{CN} $) and survival probability of the compound nucleus ($ W_{sur} $) is given as

and the fusion cross-section of the formed CN is determined as

The capture cross-section of deformed and oriented nuclei, colliding with the center of mass energies ($ E_{c.m.} $) is determined by using the extended $ \ell $-summed Wong Model [43]. This includes the sum of the cross-section corresponding to each $ \ell $-partial wave and is expressed as:

with $ k=\sqrt{\dfrac{2\mu E_{c.m.}}{\hbar^2}} $ and $ \mu $ as the reduced mass. Here, the sharp cut-off approximation [44, 45] is used to calculate the maximum angular momentum, denoted by $ \ell_{max} $.

Wong approximated the penetration of barrier for different $ \ell $-waves by inverted harmonic oscillator potential with barrier height $ V_{B}^{\ell}(E_{c.m.}) $, interaction radius $ R_{B}^{\ell}(E_{c.m.}) $ and curvature $ \hbar\omega_{B}^{\ell}(E_{c.m.}) $. So, penetration probability $ P_{\ell} $ across the barrier for each '$ \ell $' is given by Hill-Wheeler approximation [46] as

The $ V_{B}^{\ell}(E_{c.m.}) $, $ R_{B}^{\ell}(E_{c.m.}) $ and $ \hbar\omega_{B}^{\ell}(E_{c.m.}) $ are obtained from the total interaction potential between the two interacting nuclei which is the sum of repulsive Coulomb ($ V_C $) and centrifugal ($ V_{\ell} $) potentials, and attractive nuclear potential ($ V_N $). The total potential $ V_T(R_i,A_i,\beta_{\lambda i},\theta_i) $ is therefore calculated as follows:

The radius vector $ R_i(\alpha_i) $ can be used to express the shape of the deformed nuclear surface, which involves the spherical harmonic wave functions $ Y_{\lambda}^{(0)}(\alpha_i) $ of higher-order deformations $ \beta_{\lambda i} $, i.e. $ \lambda=2,3,4 $ are orders for quadrupole, octupole and hexadecapole deformed shapes, respectively [47−50], as given below

In this case, the projectile and target nuclei are denoted by i = 1, 2, respectively.

$ R_{0i} = 1.28A^{1/3}_i-0.76+0.8A^{-1/3}_i $ in fm [40] represents the radius of the spherical nuclei. The value for the static deformation parameter $ \beta_{\lambda i} $ is taken from the Data Table of M$ \ddot{o} $ller et al [49]. $ P_{\lambda}(\cos{\alpha_i}) $ is the Legendre polynomial.

The Coulomb potential ($ V_C $) in Eq.(5) is defined as follows for deformed-deformed colliding nuclei [43]:

While for the spherical-deformed cases, $ \beta_{21},\beta_{41}=0.0 $ are considered. The centrifugal potential ($ V_{\ell} $) is represented in the form of rotational kinetic energy and is given as

In Eq.(5), the attractive nuclear potential ($ V_{N} $) among the two interacting nuclei is determined by the generalized theorem for proximity forces, given by Blocki and collaborators [40]. According to this theorem the nuclear potential is the product of $ 4\pi\bar{R}\gamma b $, and $ \Phi(s_0) $, is a function of the shape and geometry of the colliding nuclei, and universal function depending on the shortest distance between the interacting nuclei respectively. Thus, $ V_{N}(R,A_i,\beta_{\lambda i},\theta_i) $ called the 'Proximity potential' is given as [40, 44, 51]

where, 'b' is the surface thickness, and the value is considered as 0.99 fm.

The mean curvature radius ($ \bar{R} $) is calculated in terms of radius vector of curvatures $ R_{i1} $ and $ R_{i2} $ for projectile and target nuclei resp., as follows:

The principal radii of curvature ($ R_{i1} $ and $ R_{i2} $) at the angle of projection w.r.t. to collision axis ($ \alpha_i $) for deformed-deformed or spherical-deformed cases is given [51] as

It is important to note here that for spherical projectiles, $ R_{11(\alpha_1)}=R_{12}(\alpha_1)=R_1(\alpha_1) $. In Eq.(9), the surface energy constant, $ \gamma $ is expressed in terms of asymmetry terms as,

In Eq.(9), the universal function dependent on shortest distance parameter $ s_0 $ is given as,

The equations (5) - (14) that outline the total interaction potential yield valuable information regarding the properties of the interacting nuclei. This includes barrier characteristics such as the barrier height $ V_B $, the barrier position $ R_B $, and the barrier curvature $ \hbar\omega_B $. These parameters are influenced by the degrees of freedom associated with deformation and orientation, introduced in the radius vector represented in Eq.(6). Consequently, these variables are utilized in the calculations of capture cross-sections.

In addition, the integrated or average capture cross-sections are obtained by integrating over the range of orientations $ \theta_i $ of the deformed nuclei having $ \beta_2 \beta_4 $ deformations, which is represented as

The compound nucleus formation probability $ P_{CN} $ gives the information about the chances of formation of the compound nucleus, where the fusion cross-section of CN is given by $ \sigma_{fus} = \sigma_{cap}\times P_{CN} $. The phenomenological approach for $ P_{CN} $ was given by Siwek-Wilczyńska [41] is

where $ k $ is taken as 3.0, $ z $ is the Coulomb parameter,

and $ Z_1 $, $ A_1 $, and $ Z_2 $, $ A_2 $ are the atomic number and mass numbers of projectile and target, respectively. Parameter $ b $ depends on the $ E_{c.m.} $ and mean interaction barrier $ B $ and reads as [52, 53]

Where $ B $ is given by

The excited CN gets cooled down by the emission of neutrons instead of undergoing the fission process. The survival probability ($ W_{sur} $) is used to measure CN's resistance to the fission process and is given by the statistical approach as [42, 54]

where $ \Gamma_n $ and $ \Gamma_f $ are decay widths for neutron emission and fission respectively and $ P_{xn}(E_{CN}^*) $ the 'Jackson factor'[55] gives the probability of emission of $ x $ neutrons from the excited CN at excitation energy $ E_{CN}^* $. $ B_{i} $ is the separation energy of the $ i^{th} $ neutron. $ T $ is the temperature (in MeV), obtained by using the statistical method [56] as $ E_{CN}^* $ = $ {a}{T^{2}} $ - $ T $ = $ E_{c.m.} $+$ Q $, where $ Q $ represents the $ Q $-value and a= $ A $/ 10 is the level density parameter for CN having mass number $ A $. The $ P_{xn} $ in terms of $ B_{i} $, $ T $ and $ E_{CN}^* $ is given as

where $ I(\Delta_x, 2x-3) $ is Pearson's incomplete gamma function [57], and $ \Delta_x $ = $ \dfrac{(E_{CN}^*-\sum_{i=1}^{x}B_{i})}{T} $. The classical approach, to determine the ratio of decay widths for $ i^{th} $ neutron to fission $ \frac{\Gamma_n}{\Gamma_f} $ is suggested by Vandenbosch and Huizenga [58] is given as

where $ B_f $ is the fission barrier height which decides the chances of fission or evaporation of neutrons. The value of constant $ K_0 $ = $ \hbar^2 $/ $ 2mr_{0}^2 $ is considered as 10 MeV. The level density parameter of the neutron evaporation channel is $ a_n $ = $ A/10 $ and that for the fission channel is $ a_f $ = $ 1.1a_n $, respectively. The values of $ a_f $ are also considered from 1.0 to 1.05 w.r.t $ a_n $. The fission barrier height $ B_f $ before the emission of $ i^{th} $ neutron is calculated as the sum of liquid drop fission barrier $ B_f^{LD} $ and the shell correction energy $ B_f^{shell} $ as given below [54]

here the $ B_f^{LD} $ is determined from the approximate analytic method given in reference [59] and value of $ B_f^{shell} $ is taken from the M$ \ddot{o} $ller and Nix table [49]. The $ E_d $ = $ 5.48A^{1/3}/(1+1.3A^{-1/3}) $ is the damping factor, which characterizes the decrease in shell effects with an increase of excitation energy of the compound nucleus.

In the present work, the study is focused on analyzing the relevance of the higher-order deformations ($ \beta_2, \beta_4 $) on the fusion-ER cross-sections at their respective compact, elongated configurations. In order to investigate the impact of deformations and orientation degrees of freedom on the fusion-ER cross-section, we have considered heavy-ion-induced reactions involving different projectile and target nuclei. For the chosen reactions the experimental data of ER cross-sections is available in [18, 35−39]. These reactions include interactions of hexadecapole ($ \beta_4 $) deformed targets $ ^{148,150} $Nd, ¹⁶⁵Ho, ¹⁹⁴Pt and ¹⁹²Os with spherical projectiles $ ^{16, 18} $O, and $ \beta_4 $-deformed projectile-target combination ¹⁹F and ¹⁸⁸Os.

The compound nuclei (CN) formed using these reactions have the mass range $ A $ = $ 150 $ - $ 212 $. The reactions are studied at the center of mass energies ($ E_{c.m.} $) lying across the Coulomb barrier, shown in Table 1. The static deformations are taken from the Table of P. M$ \ddot{o} $ller[49].

Reaction	CN	$ E_{c.m.} $	$ E_{CN}^* $	$ V_{B} $	$ z $	$ \chi $
18O+148Nd	166Er	55-69	41-56	57.47	60.68	0.581
16O+150Nd	166Er	55-73	42-59	58.061	61.27	0.581
16O+148Nd	164Er	58-72	42-56	58.208	61.46	0.584
16O+165Ho	181Re	66-96	43-73	63.948	66.96	0.644
18O+192Os	210Po	73-113	52-92	69.896	72.46	0.711
18O+194Pt	212Rn	75-97	48-70	71.649	74.20	0.732
16O+194Pt	210Rn	69-96	39-66	72.541	75.101	0.735
19F+188Os	207At	72-95	42-65	79.097	81.45	0.727

Table 1.  Tabulation of the entrance channel properties such as the center of mass energies ($ E_{c.m.} $), interaction barrier ($ V_{B} $), Coulomb factor ($ z $), along with the fissility parameter ($ \chi $) of the formed compound nucleus ($ CN $) at excited energies ($ E_{CN}^* $) for all the considered reactions

In section A we have discussed the behaviour of capture cross-section of these reactions as a function of center of mass energy ($ E_{c.m.} $). Section B is dedicated to the fusion cross-sections and section C covers the analysis of evaporation residue component.

In literature, it has been shown that the deformations and orientations (compact and elongated) associated with the interacting nuclei alter the barrier characteristics $ V_B $, $ R_B $, $ \hbar\omega $, etc., which further influence the capture cross-sections ($ \sigma_{cap} $). We have evaluated $ \sigma_{cap} $ by incorporating the deformations (up to $ \beta_4 $) of the interacting nuclei chronologically (i.e. $ \beta_2 $ and $ \beta_2\beta_4 $) at respective compact, elongated orientations ($ \theta_i $) as well as by integrating over all the orientations for the considered P-T combinations. The $ \sigma_{cap} $ are calculated within the same plane ($ \phi $=0) for the center of mass energies ($ E_{c.m.} $) lying across the Coulomb barrier, as depicted in Fig. 1. The $ \ell $-values used to incorporate angular momentum effects are calculated using the sharp cut-off approximation. From the Fig. 1, one can observe different cross-section trends for different combinations of deformations at their respective compact/elongated configurations along with the cross-sections integrated over all the orientations in comparison to the spherical system. For example, the presence of $ \beta_{2}^+ $ deformation leads to reduced cross-sections in the compact configuration due to small interaction radius with highest barrier height. However, in elongated configurations of $ \beta_{2}^+ $, the cross-sections are enhanced relative to the spherical case due to the larger interaction radius with smallest barrier height, as illustrated in Fig. 1(a-h). Further incorporation of $ \beta_4^+ $ deformation in $ \beta_2^+ $ provides elongation to the nuclear shape increasing the interaction radius. Therefore, the cross-sections are further enhanced for both configurations (i.e. compact and elongated) of $ \beta_2^+ $$ \beta_4^+ $ as compared to $ \beta_2^+ $ only, which is illustrated in Fig. 1(a-d). On the other hand, for the reactions shown in Fig 1(e-h), the inclusion of $ \beta_4^- $ deformations to $ \beta_2^+ $ of the target nuclei results in a suppression of cross-sections in both compact and elongated configurations as compared to cross-sections obtained by $ \beta_2^+ $ deformation alone. This suppression is attributed to compressive effects of $ \beta_4^- $, reducing the interaction radius relative to $ \beta_2^+ $. The integrated cross-sections ($ \sigma_{int} $) obtained for the considered $ \beta_2\beta_4 $ deformed nuclei, represent the average effect of all orientations. These integrated capture cross-sections ($ \sigma_{int} $) lies in between the cross-sections obtained by compact and elongated configurations, reflecting the combined effects of nuclear shapes and interaction dynamics across all orientations.

Figure 1.  (color online) Schematic diagram of Capture cross-sections ($ \sigma_{cap} $) obtained by extended $ \ell $-summed Wong model as function of $ E_{c.m.} $ with spherical, compact and elongated configurations along with the cross-sections integrated over all orientations for considered P-T combinations. The black arrow denotes the Coulomb barrier energy corresponding to each P-T combinations.

The obtained capture cross-sections $ \sigma_{cap} $ for the various configurations of the deformed nuclei are then compared with experimental ER cross-sections ($ \sum\sigma_{xn}^{ER} $), as shown in Fig. 1. This comparison highlights the significance of nuclear deformation and orientation, which subsequently affect the ER cross-sections. The additional parameters for predicting the ER cross-sections such as CN formation probability and survival probability, are discussed in the subsequent sections.

The previous section emphasizes the importance of incorporating the deformation and orientation degrees of freedom in the capture cross-section as seen in Fig. 1. At the same time, Fig. 1 provides the idea that the capture cross-sections obtained with the incorporation of $ \beta_4 $ deformation are enhanced. To estimate the fusion-ER cross-section mentioned in Eq.(1), we incorporate the role of the compound nucleus formation probability ($ P_{CN} $). This term determines the extent to which the interaction between projectile and target (P-T) can lead to a fusion state, and the creation of compound nuclei (CN) after the capture process. The fusion cross-section, described in Eq.(2), examines the extent of the compound nucleus's formation. In the present study, we have analyzed $ P_{CN} $ using the Siwek-Wilczyńska formula [41] given in Eq.(16). The $ P_{CN} $ is influenced by the charge and mass number of the entrance channel. These parameters collectively calculate the Coulomb parameter '$ z $'. This '$ z $' varies the mean interaction barrier for fusion, denoted as '$ B $', which is further used for the calculation of parameter '$ b $' at given $ E_{c.m.} $.

Referring to Fig. 2, we observe the variation of the probability of the formation of compound nuclei ($ P_{CN} $) with respect to the center of mass energy ($ E_{c.m.} $). The calculated $ P_{CN} $ for the considered reactions is plotted for the same energy range. One can also observe that $ P_{CN} $ increases with an increase in the $ E_{c.m.} $. Implying the increase in the fusion process for the formation of the compound nucleus. This value approaches to 1 with the increase in the energy. This variation in the $ P_{CN} $ for all the considered P-T combinations depends on the Coulomb parameter '$ z $', which hinders the complete formation of CN. For example, the reactions or the P-T combinations with a smaller value of the Coulomb parameter have higher $ P_{CN} $ than those with a higher value of the Coulomb parameter. The $ ^{16, 18} $O induced reactions, exhibit a higher $ P_{CN} $ within the range of 0.7-0.9 for the energy range given in Table 1. However, in the case of the ¹⁹F induced reaction, the addition of one proton in ¹⁹F compared to $ ^{16, 18} $O leads to increased Coulomb parameter, resulting in a decrease in $ P_{CN} $ and obtained values lie within the range of 0.5-0.7. The above discussion makes it evident that the entrance channel plays a crucial role in the formation of CN. This repulsion by the Coulomb parameter is compensated by increasing $ E_{c.m.} $, allowing the P-T combination to facilitate the formation of a CN.

Figure 2.  (color online) Schematic diagram of a variation of CN formation probability ($ P_{CN} $) with the center of mass energies ($ E_{c.m.} $) for considered P-T combinations.

The product of the compound nucleus formation probability ($ P_{CN} $) with the capture cross-sections ($ \sigma_{cap} $) gives us the fusion cross-sections ($ \sigma_{fus} $). Investigation of the formation of the compound nucleus is carried out by analyzing the ($ \sigma_{fus} $), as mentioned in Eq.(2). The $ \sigma_{fus} $ values for all the reactions under consideration are compared to the experimental ER cross-section data depicted in Fig. 3. From the figure, it can be observed that there is a significant contribution of the fusion component in the capture process, as the value of $ P_{CN} $$ \ge 0.7 $. To achieve an explicit understanding and better explanation of the ER cross-sections, we need to introduce the missing term to examine the relevance of the fission process, i.e., the survival probability of the formed compound nucleus ($ W_{sur} $). This part is addressed in the next section.

Figure 3.  (color online) Schematic diagram of Fusion cross-sections ($ \sigma_{fus} $ = $ \sigma_{cap} $ $ \times $ $ P_{CN} $) as function of $ E_{c.m.} $ with spherical, compact and elongated configurations along with the cross-sections integrated over all orientations for considered P-T combinations. The black arrow denotes the Coulomb barrier energy corresponding to each P-T combinations.

The fusion process results in the formation of a compound nucleus (CN), in an excited nuclear state. The CN stabilizes itself either by emitting light particles or forming fragments through fission. In this study, we examine the possibility of fission by calculating the survival probability $ W_{sur} $ of the formed compound nucleus. The survival probability $ W_{sur} $ in Eq.(1) represents the chances of a compound nucleus transitioning to the ground state by emitting neutrons instead of undergoing fission. The survival probability is calculated using the statistical model described in Eq.(20), which takes into account the probability of neutron emission from the excited state $ P_{xn} $ and the ratio of neutron decay width $ \Gamma_n $ to fission decay width $ \Gamma_f $ at a specific excitation energy $ E_{CN}^* $ of the CN. The factor $ P_{xn} $ depends on the temperature ($ T $), excitation energy ($ E_{CN}^* $) of the CN, and separation energies of the neutrons $ B_{i} $, but is independent of the atomic mass of nuclei. However, the ratio of decay widths depends on the mass number $ A $, fission barrier height $ B_{f} $, and neutron separation energy of the CN at given $ E_{CN}^* $. The survival of the CN against fission is determined by the fission barrier $ B_{f} $ of the compound nucleus at that specific $ E_{CN}^* $. The fission barrier $ B_{f} $ is calculated as the sum of the macroscopic part or liquid drop term ($ B_f^{LD} $) and the microscopic shell correction energy ($ B_f^{Shell} $) as described in Eq.(23). The $ B_f^{Shell} $ decreases exponentially with increasing $ E_{CN}^* $, resulting in a decrease in $ B_{f} $ with increasing $ E_{CN}^* $ enhancing the fission probability. On the contrary, the values of $ B_f^{LD} $ and $ B_f^{Shell} $ depend on the mass and atomic number of the formed CN and exhibit different magnitudes of $ B_f^{LD} $ and $ B_f^{Shell} $. The $ B_f^{LD} $ component arises from the competition between the Coulomb and surface forces, which is determined by the fissility parameter ($ \chi $). This parameter $ \chi $ is dependent on the mass number '$ A $' and the charge '$ Z $' of the CN. The magnitudes of $ \chi $ obtained for CN in the considered P-T combinations are provided in Table 1. The fissility parameter decreases as the atomic number decreases, causing an increase in $ B_f^{LD} $. As a result, the fission barrier $ B_{f} $ becomes considerably higher compared to $ B_{i} $ for compound nuclei in reactions having high $ B_f^{LD} $ values and very low shell correction energies, such as ¹⁶O+^148,150Nd, ¹⁶⁵Ho, ¹⁸O+ ¹⁴⁸Nd, leading to a higher likelihood of neutron emissions demonstrated by higher values of $ W_{sur} $. Meanwhile, the fission barrier $ B_{f} $ obtained for compound nuclei having low $ B_f^{LD} $ values and significant shell correction in reactions like ¹⁶O+ ¹⁹⁴Pt, ¹⁸O+¹⁹⁴Pt, ¹⁹²Os, ¹⁹F+¹⁸⁸Os, is comparable to $ B_i $. Although the higher $ B_f^{Shell} $ values of these nuclei having neutron number around $ N=126 $ do not have a significant impact on $ B_f $ [18, 60], as $ B_f^{Shell} $ values decrease with increase in $ E_{CN}^* $. Therefore, the decrease in the fission barrier and the $ W_{sur} $ implies a higher likelihood of the fission process dominating over neutron evaporation channels.

As mentioned in Eq.(1), the fusion-ER cross-section is obtained by multiplying the survival probability $ W_{sur} $ with the fusion cross-section $ \sigma_{fus} $. In the previous discussion, we discussed the various factors affecting both the $ W_{sur} $ and $ \sigma_{fus} $ and determined their values for the considered P-T combinations. The total fusion-ER cross-sections calculated by summing all the fusion-ER cross-sections for all the neutron channels are portrayed in comparison with experimental data for the considered reactions in Fig. 4. We can see that, the calculated fusion-ER cross-sections obtained for the '$ \beta_{2}\beta_{4} $' case for reactions ¹⁶O+^148,150Nd,¹⁶⁵Ho, ¹⁸O+ ¹⁴⁸Nd show good agreement with the experimental data presented in Fig. 4(a-d). However, these calculations do not yield satisfactory results for other reactions ¹⁶O+ ¹⁹⁴Pt, ¹⁸O+¹⁹⁴Pt,¹⁹²Os, ¹⁹F+¹⁸⁸Os shown in Fig. 4(e-h) due to the presence of a small fission barrier, indicating the dominance of fission.

Figure 4.  (color online) Schematic diagram of total fusion-ER cross-sections ($ \sum\sigma_{xn}^{ER} $) as function of $ E_{c.m.} $ with spherical, compact and elongated configurations along with the cross-sections integrated over all orientations for considered P-T combinations. The black arrow denotes the Coulomb barrier energy corresponding to each P-T combinations.

In order to attain the desired cross-sections, it has been observed that, apart from the fission barrier, the $ W_{sur} $ of the nuclei is also affected by the ratio of level density parameters between the fission channel and the neutron evaporation channel ($ a_f/a_n $) [61, 62]. This ratio determines the likelihood of fission in the nuclei. An investigation of $ W_{sur} $ for all the contributing neutron channels is conducted in terms of $ a_f/a_n $ for the compound nuclei (CN) ²¹⁰Po, ²⁰⁷At, and ²¹⁰Rn, at common excitation energy $ E_{CN}^* $. The findings of this analysis are presented in Fig. 5. From the figure, one can observe that as the $ a_f/a_n $ ratio increases, the $ W_{sur} $ of neutrons decreases and the nucleus is likely to undergo the fission process. The variation of $ W_{sur} $ at $ E_{CN}^* $ = 52 MeV is associated with the fission barrier height of the CN, as depicted in Fig. 5(a). While 3n, 4n, and 5n are common neutron decay channels in all of the formed CN, the contribution of neutron channels 6n and 7n is found in ²¹⁰Po, and the contribution of 6n is found in ²¹⁰Rn as the excitation energy increases. This result is depicted in Fig. 5(b-d). It can be concluded that to achieve better agreement with experimental results and gain a deeper understanding of the cases with lower fission barriers, the calculation of the survival probability should incorporate lower values for this ratio, as the lower value of the ratio shows the dominance of decay via evaporation of neutrons.

Figure 5.  (color online) Schematic diagram of survival probability ($ \Sigma W_{sur} $) of all the contributing neutron channels corresponding to CN as the function of level density parameter ratio of fission to neutron channel ($ a_f/a_n $) at different magnitudes of $ E_{CN}^* $.

For instance, the results represented in Fig. 4 are obtained by considering the ratio $ a_f $/$ a_n $ = 1.1. The sensitivity of the $ a_f $/$ a_n $ ratio in the fusion-ER cross-sections is investigated for the reactions ¹⁶O+¹⁹⁴Pt, ¹⁸O+¹⁹⁴Pt, ¹⁸O+¹⁹²Os and ¹⁹F+¹⁸⁸Os, by varying the value of $ a_f $/$ a_n $ from 1.0 to 1.05 and compared with the cross-sections obtained with $ a_f $/$ a_n $=1.1, as shown in Fig. 6(a-d) respectively. The figure demonstrates that the fusion-ER cross-sections vary significantly with the change in the value of $ a_f $/$ a_n $ ratio. Based on the graph shown in Fig. 6, it can be observed that the $ a_f/a_{n} $ = 1.03 - 1.05 provides better agreement with the experimental data for all the considered reactions. On the other hand, the cross-sections for the reactions ¹⁶O+ ^148,150Nd, ¹⁶⁵Ho, ¹⁸O+ ¹⁴⁸Nd persist even when the value of $ a_f $/$ a_n $ varies from 1.0 to 1.1. This can be attributed to the higher $ B_{f} $ of CN. Therefore, the total fusion-ER cross-sections for all the considered reactions are collectively obtained with $ a_f $/$ a_n $ = 1.05 with $ \beta_2\beta_4 $ deformations.

Figure 6.  (color online) Schematic diagram of total fusion-ER cross-sections ($ \sum\sigma_{xn}^{ER} $) as functions of $ E_{c.m.} $ by varying $ a_f $/$ a_n $ ratio from 1.0 to 1.05 and compared with 1.1 for (a) ¹⁶O+¹⁹⁴Pt, (b) ¹⁸O+¹⁹⁴Pt (c) ¹⁸O+¹⁹²Os, (d) ¹⁹F+¹⁸⁸Os reactions under the influence of $ \beta_2\beta_4 $ with elongated configuration. The black arrow denotes the Coulomb barrier energy corresponding to each P-T combinations.

The present analysis utilized the parameters $ a_f $/$ a_n $ = 1.05 and $ E_d $ = $ 5.48A^{1/3}/(1+1.3A^{-1/3}) $, in conjunction with the $ \beta_2\beta_4 $ deformation, to calculate the Fusion-ER cross-sections. Along with $ a_f $/$ a_n $, the damping factor ($ E_d $) plays a crucial role in modifying the fission barrier, as shown in Eq. 23. It measures the decrease in shell correction energy that occurs with the increase in excitation energy of the compound nucleus. In the literature [63], the value of $ E_d $ is also considered to be a constant value i.e. $ E_d $=25.65 MeV. To investigate the influence of the $ E_d $ on the cross-sections, we have conducted a comparison between the outcomes obtained from a constant $ E_d $ value of 25.65 MeV, and those obtained using the formula-based $ E_d $ expressed as $ 5.48A^{1/3}/(1+1.3A^{-1/3}) $. The calculated ER cross-sections using both approaches facilitate a similar analysis for all the reactions under consideration, as illustrated in Fig. 7.This demonstrates that the formula-based evaluation of $ E_d $ used here aligns well with the fixed value employed in the previous studies, ensuring consistency of the results. Therefore, predicted total ER cross-sections associated with $ \beta_2\beta_4 $ along with the observed neutron channels, are presented for all reactions at $ a_f $/$ a_n $ = 1.05 and formula based $ E_d $, as illustrated in Fig. 8.

Figure 7.  (color online) Schematic diagram of total fusion-ER cross-sections ($ \sum\sigma_{xn}^{ER} $) as function of $ E_{c.m.} $ by considering ratio $ a_f $/$ a_n $=1.05 for P-T combinations under the influence of average orientations and the elongated configuration of $ \beta_2\beta_4 $ deformed nuclei with different values of $ E_d $.

Figure 8.  (color online) Schematic diagram of total fusion-ER cross-sections ($ \sum\sigma_{xn}^{ER} $) with x-neutrons emission channels as function of $ E_{c.m.} $ by considering ratio $ a_f $/$ a_n $=1.05 for all chosen P-T combinations under the influence of $ \beta_2 \beta_4 $ with elongated configuration. The black arrow denotes the Coulomb barrier energy corresponding to each P-T combinations.

To validate the results further, we have performed a comparative analysis among the cross-sections obtained from the elongated configuration and cross-sections integrated over all orientations of the $ \beta_2\beta_4 $ deformed nuclei, as shown in Fig. 9. This analysis revealed that, in most cases, the cross-sections from both the elongated configuration and the integrated approach yield comparable outcomes.

Figure 9.  (color online) Schematic diagram of total fusion-ER cross-sections ($ \sum\sigma_{xn}^{ER} $) as function of $ E_{c.m.} $ by considering ratio $ a_f $/$ a_n $=1.05 for P-T combinations under the influence of average orientations and the elongated configuration of $ \beta_2\beta_4 $ deformed nuclei. The black arrow denotes the Coulomb barrier energy corresponding to each P-T combinations.

In addition to the $ O $-induced reactions, we have examined an additional reaction- $ ^{30}Si $ + $ ^{170}Er $, to assess the aforementioned results. For this reaction, the total evaporation residue (ER) cross-sections are calculated by taking into account the $ \beta_2 $ and $ \beta_2\beta_4 $ deformations of both the projectile and target nuclei in their respective compact and elongated configurations. The cross-sections are also calculated by integrating across all orientations ($ \theta_i $), assuming that $ a_f $/$ a_n $ = 1.05. The calculated results are subsequently compared with experimental data [64] for ER cross-sections, as illustrated in Fig. 10. It is observed that, similar to previous findings the $ \beta_2\beta_4 $ deformed nuclei provide better agreement with the experimental data. The results highlight the significant influence of $ \beta_2 $ and $ \beta_4 $ deformations, their corresponding orientations, and the $ a_f $/$ a_n $ parameter in determining the fusion-ER cross-sections in this study.

Figure 10.  (color online) Schematic diagram of total fusion-ER cross-sections ($ \sum\sigma_{xn}^{ER} $) for reaction $ ^{30}Si $+ $ ^{170}Er $ calculated by incorporating the $ \beta_2 $, $ \beta_2\beta_4 $ and the integrated effect of all the orientations of $ \beta_2\beta_4 $ deformations at $ a_f/a_n $ = 1.05 as function of $ E_{c.m.} $. The black arrow denotes the Coulomb barrier energy corresponding to P-T combination.

The analysis focuses on studying the influence of deformations and associated orientations on Fusion-ER cross-sections in ¹⁶O + ^148,150Nd, ¹⁶⁵Ho, ¹⁹⁴Pt, ¹⁸O+ ¹⁴⁸Nd, ¹⁹⁴Pt, ¹⁹²Os, ¹⁹F + ¹⁸⁸Os, and $ ^{30}Si $ + $ ^{170}Er $ reactions consisting of quadrupole ($ \beta_2 $) and hexadecapole ($ \beta_4 $) deformed target nuclei. The influence of these parameters has been studied in terms of capture cross-section ($ \sigma_{cap} $), CN formation probability ($ P_{CN} $), survival probability ($ W_{sur} $), and corresponding fusion-ER cross-sections for the center of mass energies ($ E_{c.m.} $) lying across the barrier. The capture cross-sections ($ \sigma_{cap} $) are studied within the $ \ell $-summed Wong model. The $ \beta_2 $ and $ \beta_4 $ deformations within the compact configuration provided the hindered cross-sections as compared to the spherical, whereas enhanced cross-sections are obtained at the elongated configuration. The extent of the contribution of the fusion component in the capture cross-section is studied using the compound nucleus formation probability ($ P_{CN} $). It is observed that an increase in the $ E_{c.m.} $ leads to higher values of $ P_{CN} $, enhancing the fusion cross-sections. Besides this, the survival probability ($ W_{sur} $) of the compound nucleus is calculated to segregate the contribution of ER and fission components. The observed discrepancy in fusion-ER reactions is attributed to the influence of the fission barrier and neutron separation energy. The survival probability of CN and subsequent ER process are addressed by adopting the appropriate level density ratio ($ a_f/a_n $). The $ a_f/a_n $ ratio plays rather dominant role for the reactions with lower fission barrier. The fusion-ER cross-sections are calculated by including $ \beta_2 $$ \beta_4 $ deformations at $ a_f/a_n $ = 1.05, finds decent agreement with experimental data.