Orientation dependence of the heavy-ion potential between two deformed nuclei

The deformation and orientation dependence of the real part of the interaction potential is studied for two heavy deformed nuclei using the Hamiltonian energy density approach derived from the well-known Skyrme NN interaction with two parameter sets SIII and SkM${}^{*}$. We studied the real part of the heavy ion (HI) potential for ${}^{238}\mathrm{U}$+${}^{238}\mathrm{U}$ pair considering quadrupole and hexadecapole deformations in both nuclei and taking into consideration all the possible orientations, coplanar and noncoplanar, of the two interacting nuclei. We found strong orientation dependence for the physically significant region of the HI potential. The orientation dependence increases with adding the hexadecapole deformation. The azimuthal angle dependence is found to be strong for some orientations. This shows that taking the two system axes in one plane produces a large error in calculating the physical quantities.

Figure 1

Schematic representation of the two interacting axially symmetric deformed nuclei. The unit vectors in the direction of the symmetry axes of target and projectile are ${\stackrel{\wedge }{\Omega }}_T({\theta }_T,{\varphi }_T)$ and ${\stackrel{\wedge }{\Omega }}_P({\theta }_P,{\varphi }_P)$, respectively.

Figure 2

The variation of the real part of the nuclear potential at R = 0 with the angle ${\varphi }_T$ for different symmetry axes orientations for interacting pair ${}^{238}\mathrm{U}$ + ${}^{238}\mathrm{U}.$

Figure 3

(a) The real part of the nuclear potential for ${}^{238}\mathrm{U}$+${}^{238}\mathrm{U}$ calculated using Skyrme force with parameter set SIII for various orientations with ${\theta }_T={\theta }_P$ of the two deformed nuclei. (b) Same as (a) but for ${\theta }_T\ne {\theta }_P$.

Figure 4

(a) The ϕ-dependence of HI potential for ${}^{238}\mathrm{U}$+${}^{238}\mathrm{U}$ calculated using SIII parameters of Skyrme force when the ${\theta }_P$-orientation is 0${}^{°}$ or 180${}^{°}$. (b) Same as (a) but when the ${\theta }_P$-orientation is 90${}^{°}$.

Figure 5

(a) The ϕ-dependence of HI potential for ${}^{238}\mathrm{U}$+${}^{238}\mathrm{U}$ calculated using SIII parameters of Skyrme force for the orientation angles $({\theta }_T,{\theta }_P)=({30}^{°},{30}^{°})$. (b) For the orientation angles $({\theta }_T,{\theta }_P)=({30}^{°},{60}^{°})$. (c) For the orientation angles $({\theta }_T,{\theta }_P)=({30}^{°},{120}^{°})$. (d) For the orientation angles $({\theta }_T,{\theta }_P)=({30}^{°},{150}^{°})$. (e) For the orientation angles $({\theta }_T,{\theta }_P)=({60}^{°},{60}^{°})$.

Figure 6

Fusion cross sections for ${}^{238}\mathrm{U}$+${}^{238}\mathrm{U}$ calculated at the three orientations $({\theta }_T,{\theta }_P)=(0^{°},0^{°}),({90}^{°},{90}^{°})$, and $(0^{°},{90}^{°})$ of the symmetry axes of the two interacting nuclei. The quadrupole deformations of both target and projectile are considered with the value ${\beta }_2\hspace{1em}=\hspace{1em}$0.331. The thick solid curve shows the average fusion cross section over all possible orientations. The thin solid curve is the fusion cross section calculated assuming two spherical nuclei.