Proximity potential for deformed, oriented nuclei: “Gentle” fusion and “hugging” fusion

The proximity potential is obtained in the form of the generalized “pocket formula” for a collision between any two symmetric or asymmetric mass, deformed and noncoplanar (including also the case of coplanar) nuclei, having the fixed orientations ${\theta }_1$ and ${\theta }_2$ and any azimuthal angle $\varphi (=0^{°}-{90}^{°}$). The method is applied first to some illustrative axially symmetric noncoplanar nuclei with the known “gentle”- and “hugging”-fusion configurations (${\theta }_1={\theta }_2={90}^{°},\varphi ={90}^{°}$). The very general case of noncoplanar nuclei having any orientation and azimuthal angles is also discussed. Application of the method to a specific reaction that has been used in experiments for synthesizing a superheavy nucleus is also made.

Figure 1

Two unequal nuclei, oriented at angles ${\theta }_1$ and ${\theta }_2$, with their principal planes $X^{\text{'}}{Z}^{\text{'}}$ and XZ making an azimuthal angle ϕ. The angle ϕ is shown by a dashed line, since it is meant to be an angle coming out of the plane XZ. Nucleus 2 is in XZ plane and for the out-of-plane nucleus 1, another principal plane $Y^{\text{'}}{Z}^{\text{'}}$, perpendicular to $X^{\text{'}}{Z}^{\text{'}}$, is also shown. The angles ${\theta }_i$ are measured counter-clockwise from the collision Z axis, and the angles ${\alpha }_i$ (and ${\delta }_i$) are measured in the clockwise direction from the nuclear symmetry axis. Only lower halves of the two nuclei are depicted.

Figure 2

The nuclear proximity potential $V_P(R)$ for the fixed ${\theta }_1={\theta }_2={90}^{°}$ orientations of the identical, noncoplanar ($\varphi \ne 0^{°}$) ${}^{150}\mathrm{Nd}$+${}^{150}\mathrm{Nd}$ nuclei, compared with the coplanar ($\varphi =0^{°}$) case, for the nuclei considered with (a) quadrupole deformations only, and (b) the octupole and hexadecupole deformations also included. The data used here for deformations are taken from Möller et al. [12]. ${}^{150}\mathrm{Nd}$ is a prolate deformed nucleus with positive hexadecupole deformation.

Figure 3

Same as for Fig. 2, but for diffrent mass nuclei ${}^{180}\mathrm{Hf}$+${}^{86}\mathrm{Kr}$. Both the nuclei are prolate deformed nuclei with negative hexadecupole deformations.

Figure 4

Same as for Fig. 3, but for arbitrary orientations ${\theta }_1={30}^{°}$ and ${\theta }_2={110}^{°}$.

Figure 5

The scattering potentials $V(R)[=V_P(R)+V_C(R)]$ for the same two reactions and using the same ${\theta }_i$ and ϕ values, as in Figs. 2a and 3a, i.e., for nuclei considered with quadrupole deformations alone.

Figure 6

Same as for Fig. 5, but for $p^{+}{p}^{+}$ collisions, i.e., with positive hexadecupole deformations also included. Here, the hexadecupole deformations (${\beta }_4$ values) for both ${}^{180}\mathrm{Hf}$ and ${}^{86}\mathrm{Kr}$ nuclei are taken to be with plus signs, though their listed values [12] are both with minus signs.

Figure 7

Same as for Fig. 6, but for $p^{-}{p}^{-}$ collisions, i.e., with negative hexadecupole deformations included. Here, the ${\beta }_4$ value for ${}^{150}\mathrm{Nd}$ is taken to be with minus sign, though its listed value [12] is with plus sign.

Figure 8

Same as for Fig. 7b, but for arbitrary orientations ${\theta }_1={30}^{°}$ and ${\theta }_2={110}^{°}$.

Figure 9

The scattering potentials for $p^{+}{p}^{+},p^{-}{p}^{-}$, and $p^{+}{p}^{-}$ configurations of ${}^{150}\mathrm{Nd}$+${}^{150}\mathrm{Nd}$ reaction oriented at $\varphi ={90}^{°}$ with ${\theta }_1={\theta }_2={90}^{°}$ (solid lines) and ${\theta }_1={30}^{°},{\theta }_2={110}^{°}$ (dashed lines). The listed ${\beta }_4$ value for ${}^{150}\mathrm{Nd}$ is with plus sign.

Figure 10

Same as for Fig. 2, but for ${}^{48}\mathrm{Ca}$+${}^{238}\mathrm{U}$ collision with ${}^{238}\mathrm{U}$, a prolate deformed nucleus with positive hexadecupole deformation, oriented at various different ${\theta }_2$ values and $\varphi (=0^{°}$) fixed. ${}^{48}\mathrm{Ca}$ is a spherical nucleus.

Figure 11

Same as for Fig. 6, but for ${\mathrm{sp}}^{+}$ collision ${}^{48}\mathrm{Ca}$+${}^{238}\mathrm{U}$ with ${}^{238}\mathrm{U}$ oriented at various ${\theta }_2$ values and $\varphi (=0^{°}$) fixed. ${}^{48}\mathrm{Ca}$ is a spherical nucleus.