Effects of mutual excitations in the fusion of carbon isotopes

Fusion data for ${}^{13}$C+${}^{13}$C, ${}^{12}$C+${}^{13}$C, and ${}^{12}$C+${}^{12}$C are analyzed by coupled-channels calculations that are based on the M3Y+repulsion, double-folding potential. The fusion is determined by ingoing-wave-boundary conditions (IWBC) that are imposed at the minimum of the pocket in the entrance channel potential. Quadrupole and octupole transitions to low-lying states in projectile and target are included in the calculations, as well as mutual excitations of these states. The effect of one-neutron transfer is also considered but the effect is small in the measured energy regime. It is shown that mutual excitations to high-lying states play a very important role in developing a comprehensive and consistent description of the measurements. Thus, the shapes of the calculated cross sections for ${}^{12}$C+${}^{13}$C and ${}^{13}$C+${}^{13}$C are in good agreement with the data. The fusion cross sections for ${}^{12}$C+${}^{12}$C determined by the IWBC are generally larger than the measured cross sections but they are consistent with the maxima of some of the observed peak cross sections. They are therefore expected to provide an upper limit for the extrapolation into the low-energy regime of interest to astrophysics.

Figure 1

Experimental point-proton density of ${}^{12}$C [18] is compared to the symmetrized Fermi function distribution with diffuseness $a$ $=$ 0.44 fm and radius $R$ $=$ 2.155 fm.

Figure 2

The M3Y+repulsion double-folding potentials for different combinations of carbon isotopes. They are based on the parameter $a_r$ $=$ 0.31 fm and the radius $R$ $=$ 2.28 fm for ${}^{13}$C and $R=2.155$ fm for ${}^{12}$C. The black dashed cures are the pure M3Y potentials. Also shown are the WS potentials discussed in the text and the ground-state energies of the compound nuclei.

Figure 3

Excitation energies of the channels considered in the calculations of the fusion of ${}^{13}$C+${}^{13}$C. The (red) solid lines are the excitations in projectile and target of the ${}^{13}$C states shown in Table . The (blue) dashed lines are the six mutual excitations that involve the 3/2${}^{-}$ and 5/2${}^{+}$ states in projectile and target.

Figure 4

Enhancement of the fusion of ${}^{13}$C+${}^{13}$C relative to the cross section of the no-coupling calculation. Shown are the enhancement of the one-phonon calculation (ph1), the mutual excitation (Ch13) calculation, and the full calculation (Ch26) which also includes one-neutron transfer. The calculations are based on a standard Woods-Saxon potential with radius $R_{\mathrm{ws}}=5.268$ fm. Also shown are the ratios of the measured cross sections [2, 3] relative to the no-coupling limit.

Figure 5

Results of the ${\chi }^2$ analysis of the ${}^{13}$C+${}^{13}$C fusion data [2] as function of the radius of ${}^{13}$C. The dashed curves show the ${\chi }^2/N$ for the indicated fixed values of $a_r$ and minimized with respect to the scaling factor $S_c$. The parameters of the best solution for $a_r=0.31$ fm are shown in Table .

Figure 6

Ratios of the measured [2] and calculated fusion cross sections for ${}^{13}$C+${}^{13}$C. The coupled-channels calculations include 26 channels and are based on the M3Y+repulsion potential using the ${}^{13}$C radius $R=2.17$ fm (BF, $S_c=1$) and $R=2.28$ fm (ABF, $S_c=0.843$), respectively. Also shown is the ratio of Charvet's data [3] and the calculation that uses the radius $R=2.28$ fm.

Figure 7

Enhancement of the fusion of ${}^{13}$C+${}^{13}$C [2] relative to the cross sections obtained in the no-coupling limit. The curves show the enhancement of calculations that include one-phonon (ph1) and mutual excitations (Ch13) and the full calculation (Ch26) which also includes one-neutron transfer. The enhancement factors for Trentalange's data [2] have been divided by the optimum scaling factor, $S_c=0.843$.

Figure 8

$S$ factors for the fusion of ${}^{13}$C+${}^{13}$C calculated using the M3Y+repulsion potential. They are compared to the data of Ref. [2], which have been divided by the scaling factor $S_c=0.843$.

Figure 9

Measured fusion cross sections for ${}^{13}$C+${}^{13}$C [3] are compared to the full coupled-channels calculation (Ch26, the solid curve). The thin (green) dashed curves are the calculated cross sections for a maximum angular momentum of $L_{\max }=2,4,6,8,10$, and 12. The (black) dashed curve shows the no-coupling limit.

Figure 10

Excitation energies of the channels considered in the calculations of the ${}^{12}$C+${}^{13}$C fusion cross section. The (red) solid lines are the states given in Table . The blue lines are mutual excitations considered in the calculations.

Figure 11

Enhancement factors of the measured ${}^{12}$C+${}^{13}$C fusion cross sections [4, 5] relative to the cross sections obtained in the no-coupling limit. The curves show the enhancement of calculations that include the 24 (Ch24) and 26 (Ch26) channels described in the text. The associated calculations without transfer, Ch12 and Ch13, are shown by the thin curves.

Figure 12

$S$ factors for the fusion of ${}^{12}$C+${}^{13}$C [4, 5] are compared to calculations that are based on a Woods-Saxon potential and 26 channels (WS Ch26) and on the M3Y+repulsion potential with 24 and 26 channels, respectively. The no-coupling limit is also shown.

Figure 13

Excitation energies of the channels considered in the calculations of the ${}^{12}$C+${}^{12}$C fusion cross section. The (red) solid lines are the states given in Table . The (blue) dashed lines are the mutual excitations considered in the calculations. The $(3^{-},3^{-})$ mutual excitation of projectile and target is indicated but it is ignored in the calculations.

Figure 14

Measured fusion cross sections for ${}^{12}$C+${}^{12}$C [27, 28, 29, 30] are compared to the Ch10 and Ch12 coupled-channels calculations discussed in the text. The data of Refs. [27, 28] have been shifted in energy ($\Delta E=+$100 keV [27] and $\Delta E=+$75 keV [28], respectively) following the suggestions of Ref. [29].

Figure 15

Measured $S$ factors for the fusion of ${}^{12}$C+${}^{12}$C [27, 28, 29, 30, 31] are compared to the coupled-channels calculations Ch10 and Ch12 and to the no-coupling limit. The thin solid curves show the error band in Ch10 calculations due to the uncertainty in the octupole strength, ${\beta }_3=0.90\pm 0.07$.

Figure 16

Ratios of measured fusion cross sections relative to the calculation Ch26 of the fusion of ${}^{13}$C+${}^{13}$C. The error bars include a 15$\%$ systematic error. The average ratio for the ${}^{12}$C+${}^{13}$C data [4] is 1.02, whereas the average ratio for the ${}^{13}$C+${}^{13}$C data [2] is 0.843.

Figure 17

$S$ factors for the fusion of ${}^{13}$C+${}^{13}$C [2], ${}^{12}$C+${}^{13}$C [4, 5], and ${}^{12}$C+${}^{12}$C [27, 28, 29, 30, 31] are compared to the coupled-channels calculations described in the text. The ${}^{13}$C+${}^{13}$C data have been divided by $S_c=0.843$, which optimizes the fit to the data. The thin solid curves show the error band due to the uncertainty in the octupole strength in ${}^{12}$C, ${\beta }_3=0.90\pm 0.07$. The black dashed curve is the no-coupling limit for ${}^{12}$C+${}^{12}$C.