Examination of fusion cross sections and fusion oscillations with a generalized Wong formula

We re-examine the well-known Wong formula for heavy-ion fusion cross sections. Although this celebrated formula yields almost-exact results for single-channel calculations for relatively heavy systems such as ${}^{16}\mathrm{O}+{}^{144}\mathrm{Sm}$, it tends to overestimate the cross section for light systems such as ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$. We generalize the formula to take account of the energy dependence of the barrier parameters and show that the energy-dependent version gives results practically indistinguishable from a full quantal calculation. We then examine the deviations arising from the discrete nature of the intervening angular momenta, whose effect can lead to an oscillatory contribution to the excitation function. We recall some compact, analytic expressions for these oscillations and highlight the important physical parameters that give rise to them. Oscillations in symmetric systems are discussed, as are systems where the target and projectile identities can be exchanged via a strong transfer channel.

Figure 1

For an incident energy $E$ above the Coulomb barrier $B$, the radius for the barrier with grazing angular momentum $l_g$ occurs at a separation $R_E<R_B$. The sum of the Coulomb and nuclear potentials at this point is $V_E<B$. Furthermore, the curvature $\hslash {\omega }_E$ of the barrier in the total potential is larger than its value for $l=0$. Curves are calculated for an exponential potential with $a=0.8$ fm and a depth that yields $B=6.22$ MeV for the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system. For $l=0$ this potential has $R_B=7.44$ fm and $\hslash \omega =2.52$ MeV.

Figure 2

(a) Fusion cross sections for the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system obtained with the Wong formula with $l=0$ values of $[B,R_B,\hslash \omega ]$ (dashed curve) and with their energy-dependent values (dot-dashed curve). The Bose symmetry of the identical spin-0 system is ignored, and both even and odd partial waves are summed up in the cross sections. (b) The energy-dependent Wong result agrees extremely well with a full quantal calculation shown by the solid line. However, the latter shows very weak oscillations $(\approx 1$ mb) coming from the partial-wave sum, (6), even though all $l$'s are summed.

Figure 3

(a) Barrier position, $R_B$, and (b) barrier curvature, $\hslash \omega$, as a function of angular momentum, $l$, for the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system (solid line) and the ${}^{16}\mathrm{O}+{}^{144}\mathrm{Sm}$ system (dashed line).

Figure 4

Symmetrized cross section (even $l$ only) for ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ is compared with the cross section for all $l$. One sees an order-of-magnitude difference in the oscillatory terms (see text). It is unlikely that the oscillations shown in Fig. 2 (b) $(\approx 1$ mb) could be observed in a system without some asymmetry between odd and even $l$ values.

Figure 5

(a) The $\xi$ parameter defined by Eq. (17) in the oscillatory part of fusion cross sections. Dot-dashed and solid lines show the $\xi$ parameters for the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system and the ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ system, respectively, as a function of the energy measured from the barrier height $B$ of each system. Both lines are obtained with an exponential potential with the diffuseness parameter of $a=0.8$ fm. (b) The oscillatory contribution of fusion cross sections given by Eq. (19). (c) Total fusion cross sections for the two systems given as the sum of the smooth part and the oscillatory part.

Figure 6

First energy derivative of the penetrability for each angular momentum $l,d{T}_l/dE$, for (a) the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system and (b) the ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ system. Numbers above the peaks are the corresponding values of $l$.

Figure 7

Two sets of measurements [31, 33] for fusion in the ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ system are shown as $E\sigma$ in the upper panel and as $d\left(E\sigma \right)/dE$ in the lower one. Data are compared with uncoupled calculations with two values of surface diffuseness in the nuclear potential. See text.

Figure 8

Parametrized ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ cross section [9, 21] with $[B,R_B,\hslash \omega ]=[5.6,6.3,3.0]$. The smooth-only cross section is shown by the dashed curve. Experimental data are taken from Refs. [1, 35].

Figure 9

(a) Potential-model fit to the same data as in Fig. 8 with $[B,a]=[6.22,0.8]$ (even $l$ only). (b) Same as (a), but with the transmission of $l=14$ reduced by a factor of 2 and all higher waves absent.

Figure 10

Fits to the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ cross section with different potential parameters. The surface diffuseness $a$ is constrained both by the magnitude of the smooth cross section and by the presence of oscillations. Results with (a) $a=0.6$ fm and $B=6.54$ and (b) $a=0.9$ fm and $B=6.06$ MeV. The intermediate value of $a=0.8$ fm appears to be best (see Fig. 9).

Figure 11

(a) Effect of channel coupling on the fusion excitation function for the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system. The dashed line is the same as the solid line in Fig. 9 obtained with the potential model. The solid line, on the other hand, was obtained by including the rotational coupling to the first $2^{+}$ state in both the projectile and the target nuclei. The depth parameter of the nuclear potential is slightly adjusted in order to remove the trivial barrier renormalization due to the couplings. The penetrability $T_l$ and its first derivative $d{T}_l/dE$ are shown in the middle and the bottom panels, respectively, for $l=8$, 10, and 12.

Figure 12

(a) Comparison between the experimental fusion excitation function and theoretical curves for the ${}^{12}\mathrm{C}+{}^{13}\mathrm{C}$ system. The dashed line was obtained with the Wong formula, while the solid line shows a fit to the data using Eq. (19). Experimental data were taken from Ref. [35]. (b) Same as (a), but using Eq. (23) and $\alpha =-0.5$.

Figure 13

Fusion excitation function for ${}^{12}\mathrm{C}+{}^{13}\mathrm{C}$ obtained with a parity-dependent potential. The parity dependence is introduced by replacing $V_0$ with $V_0(1+{(-1)}^l\epsilon )$ in the exponential potential $V_N\left(r\right)=V_0{e}^{-r/a}$, where $V_0$ takes a negative value. The solid and the dashed lines were obtained with $\epsilon =-0.15$ and 0, respectively. The angular momentum sum is truncated at $l=14$ as in Fig. 9.

Figure 14

(a) Exact and approximate barrier positions $R_E$ for the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system as a function of the incident energy $E$. An exponential nuclear potential with the diffuseness parameter $a=0.8$ fm was used. The solid line shows the exact value obtained with Eq. (A3), while the dotted and dashed lines were obtained from Eq. (A5) at first and second order, respectively. (b) Same as (a), but for the value of the $s$-wave potential at $R_E$. (c) Same as (b), but for the curvature of the barrier for the grazing angular momentum $l_g$.