Coulomb and nuclear potentials between deformed nuclei

We present a systematics for the octupole and quadrupole deformation parameters of heavy nuclei. We also provide useful procedures to calculate the Coulomb and nuclear potentials for systems involving deformed nuclei.

Figure 1

The values of the quadrupole deformation lengths as a function of the number of protons (bottom) or neutrons (top) of the nuclei. The dotted lines represent the magic numbers.

Figure 2

The values of the octupole deformation lengths as a function of the number of protons (bottom) or neutrons (top) of the nuclei. The dotted lines represent the magic numbers.

Figure 3

The average values for isotopes (or isotones) of the quadrupole deformation lengths as a function of the number of protons (or neutrons) of the nuclei.

Figure 4

The average values for isotopes (or isotones) of the octupole deformation lengths as a function of the number of protons (or neutrons) of the nuclei. The solid line represents a function proportional to $X^{-1∕2}$, where $X$ could be $Z$ or $N$.

Figure 5

The functions $D_2$ and $D_3$ that describe the behavior of the deformation lengths. The dotted lines represent the magic numbers.

Figure 6

Schematic picture of the collision of two deformed nuclei. The dashed circle represents the nucleus 1 if it did not have deformation.

Figure 7

The Coulomb potential calculated through different models for the ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ system. The arrow represents the approximate position of the corresponding s-wave barrier radius.

Figure 8

The figure presents the total Coulomb potential $\left(V_C\right)$, the correction $\left(V_C^{\mathrm{Cor}}\right)$ and the corresponding relative values $(100V_C^{\mathrm{Cor}}∕V_C)$ for the ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ system. The arrow indicates the approximate position of the barrier radius.

Figure 9

The figure presents the correction of the Coulomb potential for the ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ system. The nucleus 2 is assumed undeformed and the diffuseness of both distributions is $a=0$. The deformation parameters of nucleus 1 are indicated in the figure. The solid lines represent the results of Eqs. (15, 16, 17, 18, 19, 20, 21, 22), and the dashed lines represent the asymptotic expression (23). The arrows indicate the approximate position of the barrier radius.

Figure 10

The figure presents the correction of the Coulomb potential for the ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ system, where ${\beta }_2=0.205$, ${\beta }_3=0.235$, and $a=0$ were assumed for both nuclei. The angles of the deformation axes are indicated in the figure. The arrows indicate the approximate position of the barrier radius.

Figure 11

The same as Fig. 10, but for $a=0.53\mathrm{fm}$.

Figure 12

The deformed nuclear potential (solid lines) compared with the nondeformed one (dashed lines) for the ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ system. The figure also presents the results obtained with the usual approximations, Eqs. (35, 36). ${\beta }_2=0.205$, ${\beta }_3=0.235$, and $a_M=0.56\mathrm{fm}$ were assumed for both nuclei. The angles of the deformation axes are indicated in the figure.

Figure 13

The absolute value for the correction $V_F^{\mathrm{Cor}}=V_F-V_F^{\left(0\right)}$, where the folding potential was obtained from Eqs. (32, 35, 36). The deformation parameters are the same as those in Fig. 12.