Journal of Nuclear Physics, Material Sciences, Radiation and Applications Investigation for Suitable Target-Projectile Combination for Fusion from the Isotopes of Ti and Nd using Intrinsic Fusion and Fission Barriers Analysis

Background: A configuration is most suitable for the fusion if it corresponds to a minimum intrinsic fusion barrier and maximum fission barrier. Purpose: To find a suitable target-projectile combination from the isotopes of Ti and Nd by analyzing the intrinsic fusion and fission barriers theoretically by including the deformations up to hexadecapole order. Methods: The fragmentation theory has been used for the calculations. Results: The intrinsic fusion barrier is minimum and fission barrier is maximum for the target- projectile combination: 43 Ti+ 150 Nd in belly-belly configuration, and the inclusion of deformation of higher order leads to the decrease of fission barrier for the prolate shaped cases and compactness for most of the cases. Conclusions: The most suitable target-projectile combination from the isotopes of Ti and Nd for the fusion is 43 Ti+ 150 Nd.


Introduction
In deformed heavy-ion collisions the final-state observables as well as properties of the dynamics of the fusion or fusion-fission process is expected to be different from the spherical cases. This is due to the distinct differences in the overlap region of aligned deformed nuclei in the process of fusion and hence an investigation of the shape effects during collisions is important to understand the mechanism and dynamics of the process. The interaction potential is not only the function of the deformations of the interacting nuclei but it also depends on their orientations. The orientation which corresponds to the maximum height of fission barrier V B and minimum interaction radius R B is known as hot compact and which gives minimum V B and maximum R B is cold elongated [1]. In a recent work of ref. [2] hot compact and cold elongated configurations have been obtained for the isotopes of Ti and Nd, with nuclei oriented at bellybelly (b-b) and tip-tip (t-t) configurations, respectively, with quadrupole deformation only. As the nuclei are not quadrupole deformed only so investigations with respect to the higher deformations will be interesting.
In this paper, we have obtained a suitable targetprojectile for the fusion from the various combinations of the deformed and oriented isotopes of Ti and Nd: ⁴⁴Ti (β 2 =β 4 =0), 43 Ti (β 2 =-0.042, β 4 =0.012), 48 Ti (β 2 =0.011; β 4 =0) and 142 Nd ( β 2 =β 4 =0), 181 Nd (β 2 =-0.125; β 4 =-0.006) and 150 Nd (β 2 =0.237; β 4 =0.110) by investigating the V B and R B for the b-b and t-t configurations, and intrinsic fusion barrier B fus, in mass asymmetric coordinates, for b-b configuration only due to lesser values of V B for t-t configuration. It may be noted that β 3 =0 for these nuclei and the deformation values are of ref. [3]. These barriers play an important role in understanding the competition between quasi-fission and complete fusion. Following the definition of B fus in charge asymmetry coordinate of ref. [4], B fus in mass asymmetry coordinate is defined as the height of the saddle point (the maximum fragmentation potential) from the potential which corresponds to the incoming channel mass asymmetry or mass number of the projectile. The smaller value of B fus favours complete fusion while the larger value of it is a hindrance to the fusion process.
In the following we discussed the methodology, calculations and results, and conclusion of the study.

Methodology
The fragmentation theory [5,6] used to obtain the fragmentation potentials is worked out in terms of mass asymmetry coordinate η = (A 1 -A 2 )/ (A 1 + A 2 ) or charge asymmetry coordinate η Z = (Z 1 -Z 2 )/(Z 1 + Z 2 ), relative separation R, the neck-length parameter ε and the deformations of the interacting nuclei β λi (i = 1, 2 and λ = 2, 3, 4 for the quadrupole, octupole, and hexadecapole deformations). According to the fragmentation theory the fragmentation potential between two deformed and oriented nuclei colliding in a plane (φ = 0) at fixed inter-nuclear separation R a is where V LDM (A i , Z i ) , δU i ,V P, V C and V  respectively are the liquid drop energies, shell corrections, proximity potential, Coulomb potential and centrifugal potential between the fragments. For a given nucleus V LDM (A, Z) [7] is where α, β, γ, η and a a are Seeger's constants [8], t ζ = a a (Z -N) is the asymmetry term and a a is the asymmetry constant. The shell correction δU for a given nucleus is taken from [9]. The bulk α and asymmetry a a constants are obtained by equating the ground state mass excess of AME2016 [10] or of FRDM(2012) [3] by with V A Z U LDM , ( )+δ [11].
The Coulomb and proximity potentials for deformed and oriented nuclei for φ = 0 are given as:   where s 0 is the minimum separation between the surfaces of the interacting nuclei per unit surface thickness. The mean curvature radius R for axially symmetric deformed and oriented nuclei is of ref. [12]. The separation between the surfaces of two interacting nuclei is, s 0 = R − R 1 (α 1 ) cos(θ 1 -α 1 )− R 2 (α 2 ) cos(180 + θ 2 − α 2 ) and become minimum when ds 0 /dα 1 = ds 0 /dα 2 = 0 (for detail see [1,13] and references therein). The scattering/interaction potential between the two interacting nuclei V(R) is the sum of the proximity potential V P and Coulomb potential V C , i.e., V T (R) = V P (R) + V C (R) for =0 case. Fig. 1 shows the interaction potential for the oblate shaped projectile-target combination 43 Ti+ 181 Nd oriented at (0°, 0°) for hot and (90°, 90°) for cold configurations called bellybelly and tip-tip configurations, respectively. The barrier height (V B ) is maximum and interaction radius (R B ) is minimum for the belly-belly configuration while for the tiptip configuration corresponds to the minimum of V B and maximum of R B , whatever may be the sign of the deformation.

Calculations and Results
This has been explored for various combinations of Ti and Nd: oblate-oblate, oblate-prolate, oblate-spherical, prolateoblate, prolate-prolate, prolate-spherical, spherical-oblate and spherical-prolate, as tabulated in Table: 1 and 2 below along with the inclusion of deformation of higher order (±). Table 1: P-T oriented for belly-belly (b-b) configuration (Hot fusion).  Table 2: P-T oriented for tip-tip (t-t) configuration (Cold fusion). It can be seen from Table 1 and 2 that the V B is maximum and R B is minimum for the reaction 43 Ti+ 150 Nd in bellybelly configuration, and hence is predicted to be the best target-projectile for fusion. The addition of higher order deformation leads to the decrease of height of fission barrier (V B ) in p + (prolate with β 4 =+ve) cases and increase of the interaction radius in most of the cases. This means that the systems become less compact with the inclusion of deformation of higher order and compact configuration is expected to be at some other orientation, as can be seen in ref. [14]. Thus a configuration of compactness is most suitable for the fusion process. The suitability is explored further in terms of B fus . It may be noted that the inclusion of the deformations of higher order (±) does not change the optimal orientation, but leads to a decrease in the compactness. Fig. 2 illustrate the intrinsic fusion barrier B fus for 43 Ti+ 181 Nd reaction when target and projectile considered are: (i) spherical, (ii) quadrupole deformed (β 2i ) and (iii) with deformation of higher order such as octupole (β 3i ) and hexadecapole (β 4i ) i.e., p ± , o ± . The intrinsic fusion barrier B fus increases with the inclusion of the deformation of higher order, i.e., B fus (β 2i ) < B fus (β 2i + β 3i + β 4i ), which seems to be due to the decrease of the compactness with the higher deformations [14]. Table 3 clearly shows that for 43 Ti+ 150 Nd→ 193 Pb * the intrinsic fusion barrier B fus is zero when target and projectiles are considered quadrupole deformed (β 2i ) and is minimum (=0.84 MeV) when the deformations of higher orders (β 3i and β 4i ) are included. Thus 43 Ti+ 150 Nd reaction or target-projectile combination seems to be the best for the fusion when oriented at belly-belly configuration.

Conclusions
An analysis of intrinsic fusion barrier, fission barrier height and position reveals that the most compact orientation (belly-belly) of the colliding nuclei seems to be the best for the fusion of heavy-ions and can be obtained using the barrier analysis, here predicting 43 Ti+ 150 Nd as the best target-projectile combination for the synthesis of Pb.