Consistent analysis of fusion data without adjustable parameters for systems involving odd nuclei

In a recent article, we calculated fusion cross sections for systems involving even-even nuclei based on a method that takes into account the couplings to a complete set of states for surface vibrations of the nuclear densities. The predictions were obtained without using any adjustable parameter, and are in good agreement with the experimental results for most of the systems, even at sub-barrier energies. In the present work, we extend these calculations to systems involving nuclei with odd number of protons and/or of neutrons.

Figure 1

The values of the experimental excitation energies of the $2^{+}$ state 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 experimental excitation energies of the $3^{-}$ state 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 $2^{+}$ excitation energies as a function of the number of protons (or neutrons) of the nuclei.

Figure 4

The average values for isotopes (or isotones) of the $3^{-}$ excitation energies as a function of the number of protons (or neutrons) of the nuclei.

Figure 5

The functions $E_2$ and $E_3$ that approximately describe the behavior of the excitation energies. The dotted lines represent the magic numbers.

Figure 6

Schematic representation of the collision between two deformed nuclei. The $s$ coordinate represents the variation of the nuclear radius relative to the spherical shape in the direction of $\overrightarrow{R}$.

Figure 7

Effective (sum of nuclear and Coulomb) potentials obtained considering spherical or deformed densities of the nuclei and parabolic representation of the spherical potential. The shift in the barrier height ($\Delta V_B$) is indicated in the figure.

Figure 8

Fusion data (from Refs. [32, 33, 34]) and corresponding undeformed BPM (dashed lines) and ZPM (solid lines) theoretical cross sections for the ${}^{10}\mathrm{B}+{}^{159}\mathrm{Tb}$, ${}^{11}\mathrm{B}+{}^{14}\mathrm{N}$, ${}^{159}\mathrm{Tb}$, and ${}^{232}\mathrm{Th}$ systems. The $s$-wave barrier heights of the undeformed potential are presented in the figure.

Figure 9

The same as in Fig. 8 for the systems indicated in the figure. The data are from Refs. [33, 35, 36, 37].

Figure 10

The same as in Fig. 8 for the systems indicated in the figure. The data are from Refs. [35, 36, 38, 39].

Figure 11

The same as in Fig. 8 for the systems indicated in the figure. The data are from Ref. [40].

Figure 12

The same as in Fig. 8 for the systems indicated in the figure. The data are from Refs. [41, 42].

Figure 13

The same as in Fig. 8 for the systems indicated in the figure. The data are from Refs. [41, 42].

Figure 14

The same as in Fig. 8 for the systems indicated in the figure. The data are from Refs. [43, 44, 45]. There are different data sets (indicated by different symbols) with measured cross sections that can significantly differ from each other.

Figure 15

The same as in Fig. 8 for the systems indicated in the figure. The data are from Refs. [43, 44, 45]. There are different data sets (indicated by different symbols) with measured cross sections that can significantly differ from each other.

Figure 16

The same as in Fig. 8 for the systems indicated in the figure. The data are from Refs. [43, 46, 47, 48].

Figure 17

The same as in Fig. 8 for the systems indicated in the figure. The data are from Refs. [41, 46, 49].

Figure 18

Ratio between fusion data and BPM (left side) or ZPM (right side) theoretical cross sections for systems involving odd (bottom panels) or even-even (top panels) nuclei.

Figure 19

(Top) Ratio between fusion data and ZPM cross sections for systems involving odd and even-even nuclei. (Bottom) Average values of the order of magnitude of the deviations between fusion data and ZPM cross sections for systems involving odd and even-even nuclei and also for systems involving the weakly bound ${}^{6,7}\mathrm{Li}$ and ${}^9\mathrm{Be}$ nuclei.

Figure 20

The same as in Fig. 8 for the systems indicated in the figure. The data are from Refs. [50, 51, 52].

Figure 21

The same as in Fig. 8 for the systems indicated in the figure. The data are from Refs. [23, 50, 51, 53].