Role of Polar vs Non-polar Configurations in the Decay of 268Sg* Compound Nucleus Within the Skyrme Energy Density Formalism

Received: January 17, 2021 Accepted: April 19, 2021 Published Online: August 31, 2021 The effect of polar and non-polar configurations is investigated in the decay of 268Sg* compound nucleus formed via spherical projectile (30Si) and prolate deformed target (238U) using the dynamical cluster decay model. The SSK and GSkI skyrme forces are used to investigate the impact of polar and nonpolar (equatorial) configurations on the preformation probability P0 and consequently on the fission cross-sections of 268Sg* nucleus. For non-polar configuration some secondary peaks corresponding to magic shells Z=28 and N=50 are observed, whose magnitude is significantly suppressed for the polar counterpart. The effect of polar and non-polar configurations is further analyzed in reference to barrier lowering parameter ΔVB. The calculated fission cross-section find adequate agreement with experimental data for chosen set of skyrme forces.


Introduction
For the production of super heavy elements the choice of appropriate target projectile combination is extremely important which in-turn leads to cold and hot fusion reactions. In cold fusion reactions, Pb and Bi targets are used at below barrier energies, whereas the use of actinide target leads to hot fusion reactions which take place at relatively higher excitation energies [1][2]. The relatively stable superheavy nuclear systems are synthesized via highly asymmetric hot fusion reactions where 238 U, 242,244 Pu, 243 Am, 245,248 Cm, 249 Bk and 249 Cf actinide targets are collided with the doubly magic 48 Ca projectile. Consequently, the hot fusion reactions have been introduced as a tool to understand the dynamics of super-heavy elements and provide a better option for the formation of super heavy systems with Z ≥ 112.
One of the major factor that influence such reactions is the contribution of the target deformation. The actinide nuclei are generally prolate deformed, so the reaction may proceed either via the equatorial or polar side of the target nucleus. The Coulomb barrier is significantly affected by the orientation and deformation of the interacting nuclei. The effect of orientation is such that, equatorial collision preferred at higher projectile energy results in compound nucleus formation, whereas the polar one leads the non compound nucleus process called quasi-fission. It has been observed that, 48 Ca+ 238 U [3] and 48 Ca+ 242,244 Pu [4] reactions have higher fusion probability and measured evaporation residue cross-section are maximum when equatorial configuration is used. The orientation effects on fusion were analyzed for the reaction 30 Si+ 238 U in ref. [5].
Earlier we had worked on the decay of 30 Si+ 238 U reaction forming 268 Sg * compound nucleus [6] using the dynamical cluster-decay model [7][8][9]. In ref. [6], the role of polar (cold) and non-polar (equatorial) collisions was studied across the Coulomb barrier region using the spin-unsaturated Blocki based potential [10]. At above-barrier energies the calculations were carried out by considering hot-equatorial configuration of nuclei, whereas at sub-barrier energies the cold-polar orientations were included. We had observed significant variations in the mass distribution across the barrier as the fission fragment distribution was symmetric at above barrier energies (for non-polar configurations) and became asymmetric when the nuclei approached pole-topole configuration at sub-barrier energies.
As an extension of above work, the present study is carried out where the nuclear potential is obtained from the skyrme energy density formalism (SEDF) [11][12]. p.62 SEDF has contribution from both the spin saturated and spin unsaturated parts of the potential. In this analysis, SSK [13] and GSkI [13] skyrme forces are used which differ significantly in their contribution to the spin. For the study, only below barrier region is chosen and the effect of polar and non-polar configuration and different skyrme forces is inspected. Within the DCM, the decay of 30 Si+ 238 U is studied by incorporating the quadrupole deformations of the nuclei along with their optimum orientations of [14].
In order to understand the significance of skyrme forces on the mass distribution the decay of 268 Sg * nucleus is studied in terms of preformation probability. It is found that for polar configurations, symmetric fragments show dominance over asymmetric one at lowest incident energies, whereas the asymmetric fragments overtake at higher energies. However in case of non-polar collision, independent of the E c.m. , the symmetric fragment show higher magnitude with broader asymmetric peaks. Within both the configurations, the SSK and GSkI skyrme forces are addressed the below barrier fission data.
The paper is organized as follows: the methodology used for the calculation is mentioned in section 2. The calculation and results are discussed in section 3 and finally the summary of the work is presented in section 4.

Skyrme Energy Density Formalism (SEDF)
Within SEDF [11][12], the nucleus-nucleus interaction potential is defined as the difference of energy expectation value E of colliding nuclei that are overlapping (at a finite separation distance R) and are completely separated (at H( , ,J) 2m Here ρ q , τ q and J q (q=n,p) are the nucleonic, kinetic energy and spin-orbit densities, respectively. m is the nucleon mass. x i , t i , α i, W 0 are the skyrme force parameters, fitted by different authors to obtain better descriptions of various ground state properties of nuclei. In this work GSkI and SSk skyrme forces are used [13].

Dynamical Cluster Decay Model (DCM)
The dynamical cluster decay model (DCM) [7][8][9] is a nonstatistical approach based on the quantum mechanical fragmentation theory (QMFT). In which all decay modes such as light particle emission (LPs), intermediate mass fragments (IMFs) and fusion-fission (ff) etc are estimated simultaneously in one set of calculation and bring out the nuclear structure information in terms of preformation probability P 0 of decaying fragments. According to DCM, the decay cross-sections in terms of the partial wave analysis is define as: With μ as the reduced mass, m the nucleon mass and E c.m. the canter of mass energy of the compound nucleus. l max is the maximum angular momentum fixed for the vanishing of the fusion barrier of the incoming channel or the light particle cross section σ LP → 0. P 0 is given by the solution of stationary Schrodinger equation in η co-ordinate which reads as: The penetrability P is calculated as the WKB tunneling probability, solved analytically in Ref. [15], as  with V (R a ,T ) = V (R b ,T ) = TKE(T) = Q eff for the entry and exit points of the potential barrier. Q eff is the effective Q value of the decay process and the entry point In above equation, α i (i = 1, 2) is an angle that the radius vector R i of the colliding nuclei makes with the symmetry axis, measured clockwise. ΔR refers to the neck-length parameter which assimilates the neck formation effects. The choice of ΔR for a best fit to the data allows us to define the effective "barrier lowering" parameter ΔV B (l) as [8] The deformation and orientation dependent fragmentation potential V(R) at any temperature is given as where the T-dependent terms V C , V N and V l are defined as follows: V c is the Coulomb potential, V N is the nuclear potential for hot deformed nuclei, and V l is the potential due to rotational motion of hot deformed nuclei. In this work V N is calculated using the skyrme energy density formalism explained briefly in section 2.1.

Calculation and Result
In this section, we investigate the decay of 268 Sg * compound nucleus within the center-of-mass energies E c.m. = 125-134 MeV (below barrier region) using polar and non-polar configurations of the colliding nuclei. For the study, the SEDF approach with two skyrme forces such as SSK and GSkI are applied to understand the fusion-fission dynamics of the 30 Si+ 238 U→ 268 Sg * reaction. The result related to preformation probability (P 0 ) and the barrier modification parameters (∆V B ) are outlined in the present work. Fig. 1 shows the variation of preformation probability (P 0 ) as a function of fragment mass (A i ) for the decay of 286 Sg * compound nucleus. Left and right panels respectively illustrate the result for non-polar and polar configurations using GSkI and SSk skyrme forces. The graph is plotted for below barrier center of mass energies i.e. E c.m. =125 MeV to 134 MeV. On comparing the non-polar and polar configurations at same center of mass energy i.e. at E c.m. =125 MeV, we observe the following: (i) for non-polar approach, both symmetric and asymmetric peaks are visible; although the probability of symmetric fission is higher than the asymmetric one. In asymmetric fission region significant contribution from the secondary peaks at 66 Ni, 72 Zn and 84 Se (and their complementary fragments) are also observed, which corresponds to the shell closure at Z 2 =28, Z 2 =40 and N 2 =50. Hence the role of the magic shell closures play significant role in the decay dynamics of 268 Sg * nucleus. (ii) On the other hand, when polar configuration is applied, the magnitude of the secondary peak around 66 Ni decreases at the cost of enhanced magnitude of preformation probability at deformed magic shell around Z=40 (isotope of 98-100 Zr) as shown in Fig.1(d). It is to be noted here that, above result is true for either choice of the skyrme forces. Further Fig. 1(a) to (c) shows that, when center of mass energy increases from 125 to 134 MeV, similar fragments are formed for non-polar approach. However in polar configuration, the fragment mass distribution of decaying fragments changes significantly with increase in incident energy. While looking at Fig. 1(d) to 1(f ), it is observed that with increase in energy, the probability of symmetric fission decreases and an asymmetric peak around 84 Se (N 2 =50) starts appearing. Thus we find that, the decay pattern of compound nucleus 286 Sg * is greatly influenced by the orientational configuration and the incident energy.
We have addressed the fission cross-sections of 286 Sg * compound nucleus within both the configuration and using SSK and GSkI skyrme forces. " Table 1" and "Table 2" illustrate the DCM calculated fission cross-sections, angular momentum values ( l ) and center of mass energy (E c.m. ) along with the experimentally given data. Fig. 2(a) and Fig. 2(b) depict the fitting parameter 'ΔR' as a function of E c.m. plotted with GSkI and SSK forces. We observe in Fig. 2(a) that, independent of the configuration, ΔR increases with increase in E c.m. .
It is to be observed that, in non-polar approach higher neck length is required to fit the fission data (as compared to polar configuration). Similar results are obtained when SSK force is used (as shown in Fig. 2(b)).
For sub-barrier energies, barrier modification parameter ΔV B is a crucial factor, has a direct dependence on the corresponding values of neck-length ΔR (see equation (6) of section 2) used to address the fission data. Fig. 3 shows the variation of ΔV B with center of mass energy E c.m. plotted for non-polar and polar configuration. From Fig. 3(a) one may observe that, using GSkI force, larger modification in barrier is required for non-polar configuration in comparison to polar configuration. Similar observations are drawn with SSk skyrme force.

Summary
The purpose of this work is to investigate the role of polar and non-polar configurations on the fission dynamics of 268 Sg * compound nucleus at below the Coulomb barrier energy.
The study is carried out by estimating the preformation probability P 0 and fission cross-sections within the framework of dynamical cluster decay model. It is to be observed that, within the scope of skyrme based nuclear potential, the probability of asymmetric fragments increases with increase in the preformation energy when nuclei interacted via pole to pole, which on the other hand remains the unchanged for the case of non-polar interactions. This result is independent of the skyrme force used. The addressal of experimental data within the non-polar orientation demands higher value of neck-length parameter ΔR and consequently larger barrier modification as compared to the polar configuration.