Fusion of ${}^{28}\mathrm{Si}+{}^{28,30}\mathrm{Si}$: Different trends at sub-barrier energies

Background: The fusion excitation function of the system ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ at energies near and below the Coulomb barrier is known only down to $\simeq 15$ mb. This precludes any information on both coupling effects on sub-barrier cross sections and the possible appearance of hindrance. For ${}^{28}\mathrm{Si}+{}^{30}\mathrm{Si}$ even if the fusion cross section is measured down to $\simeq 50$ $\mu \mathrm{b}$, the evidence of hindrance is marginal. Both systems have positive fusion $Q$ values. While ${}^{28}\mathrm{Si}$ has a deformed oblate shape, ${}^{30}\mathrm{Si}$ is spherical.

Purpose: We investigate 1. the possible influence of the different structure of the two Si isotopes on the fusion excitation functions in the deep sub-barrier region and 2. whether hindrance exists in the $\text{Si}+\text{Si}$ systems and whether it is strong enough to generate an $S$-factor maximum, thus allowing a comparison with lighter heavy-ion systems of astrophysical interest.

Methods: ${}^{28}\mathrm{Si}$ beams from the XTU Tandem accelerator of the INFN Laboratori Nazionali di Legnaro were used. The setup was based on an electrostatic beam separator, and fusion evaporation residues (ER) were detected at very forward angles. Angular distributions of ER were measured.

Results: Fusion cross sections of ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ have been obtained down to $\simeq 600$ nb. The slope of the excitation function has a clear irregularity below the barrier, but no indication of a $S$-factor maximum is found. For ${}^{28}\mathrm{Si}+{}^{30}\mathrm{Si}$ the previous data have been confirmed and two smaller cross sections have been measured down to $\simeq 4$ $\mu \mathrm{b}$. The trend of the $S$-factor reinforces the previous weak evidence of hindrance.

Conclusions: The sub-barrier cross sections for ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ are overestimated by coupled-channels calculations based on a standard Woods-Saxon potential, except for the lowest energies. Calculations using the M3Y+repulsion potential are adjusted to fit the ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ and the existing ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$ data. An additional weak imaginary potential (probably simulating the effect of the oblate ${}^{28}\mathrm{Si}$ deformation) is required to fit the low-energy trend of ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$. The parameters of these calculations are applied to predict the ion-ion potential for ${}^{28}\mathrm{Si}+{}^{30}\mathrm{Si}$. Its cross sections are well reproduced by also including one- and successive two-neutron transfer channels, besides the low-lying surface excitations.

Figure 1

Excitation functions measured in this work for ${}^{28}\mathrm{Si}+{}^{28,30}\mathrm{Si}$, together with the previous data on ${}^{28}\mathrm{Si}+{}^{30}\mathrm{Si}$ [8], obtained at Argonne (ANL). The arrows mark the positions of the Akyüz-Winther barriers [11] for the two systems.

Figure 2

Cross sections for the fusion of ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ (this work and Ref. [6]) (a) and ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$ [7] (b) compared to the no-coupling calculation Ch1 and to Ch10 coupled-channels calculations that are based on the M3Y+rep potential. A weak, short-ranged imaginary potential had to be applied (Ch10 w5, solid curve) in order to reproduce the low-energy data of ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$. Also shown in (a) is a Ch10 calculation that is based of a Woods-Saxon (WS) potential [11]. No imaginary potential was used in this case.

Figure 3

Logarithmic derivative (slope) of the excitation function of ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ as measured in this work. It is compared with the results of CC calculations [see also Fig. 2] and with the slope expected for a constant astrophysical $S$ factor (“Constant S”). The need for a weak imaginary potential is emphasized in this representation.

Figure 4

Channel potentials for ${}^{28}\mathrm{Si}+{}^{30}\mathrm{Si}$. The solid curve is the M3Y+rep entrance channel potential. Also shown are channel potentials for the $2^{+}$ and $3^{-}$ excited states in ${}^{28}\mathrm{Si}$ (a) and in ${}^{30}\mathrm{Si}$ (b). The channel potentials were calculated for $L$ = 0 with the parameters shown in Tables 2 and 3; they have been displaced by the excitation energies $E_x$. The entrance channel potential for the adjusted Woods-Saxon (WS) potential mentioned in Table 2 is also shown.

Figure 5

Fusion cross sections for ${}^{28}\mathrm{Si}+{}^{30}\mathrm{Si}$ (with previous data from [6, 8]) compared to the no-coupling Ch1 calculation and to Ch10 and Ch30 coupled-channels calculations that are based on a Woods-Saxon (WS) potential and the predicted M3Y+rep potential. No imaginary potential was used. The one- and successive two-neutron transfer cross sections, ${\sigma }_{1n}$ and ${\sigma }_{2n}$, obtained in Ch30 calculations are also shown.

Figure 6

Measured fusion cross sections for ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ and ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$ [7] compared to Ch10 calculations, and the data for ${}^{28}\mathrm{Si}+{}^{30}\mathrm{Si}$ compared to Ch30 calculations. All calculations are based on M3Y+rep potentials. Note that it is necessary to use an imaginary potential with parameters $a_w=0.2$ fm and $W_0=5$ MeV in order to reproduce the ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ data at low energy.

Figure 7

$S$ factors for the fusion of ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ (a) and ${}^{28}\mathrm{Si}+{}^{30}\mathrm{Si}$ (b). A weak imaginary potential ($a_w$ = 0.2 fm $W_0=5$ MeV) was used in the Ch10 w5 calculation of the fusion of ${}^{28}\mathrm{Si}$ + ${}^{28}\mathrm{Si}$.