Isotopic effects in sub-barrier fusion of Si + Si systems

Background: Recent measurements of fusion cross sections for the ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ system revealed a rather unsystematic behavior; i.e., they drop faster near the barrier than at lower energies. This was tentatively attributed to the large oblate deformation of ${}^{28}\mathrm{Si}$ because coupled-channels (CC) calculations largely underestimate the ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ cross sections at low energies, unless a weak imaginary potential is applied, probably simulating the deformation. ${}^{30}\mathrm{Si}$ has no permanent deformation and its low-energy excitations are of a vibrational nature. Previous measurements of this system reached only 4 mb, which is not sufficient to obtain information on effects that should show up at lower energies.

Purpose: The aim of the present experiment was twofold: (i) to clarify the underlying fusion dynamics by measuring the symmetric case ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$ in an energy range from around the Coulomb barrier to deep sub-barrier energies, and (ii) to compare the results with the behavior of ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ involving two deformed nuclei.

Methods: ${}^{30}\mathrm{Si}$ beams from the XTU tandem accelerator of the Laboratori Nazionali di Legnaro of the Istituto Nazionale di Fisica Nucleare were used, bombarding thin metallic ${}^{30}\mathrm{Si}$ targets (50 $\mu \mathrm{g}$/${\mathrm{cm}}^2$) enriched to $99.64\%$ in mass 30. An electrostatic beam deflector allowed the detection of fusion evaporation residues (ERs) at very forward angles, and angular distributions of ERs were measured.

Results: The excitation function of ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$ was measured down to the level of a few microbarns. It has a regular shape, at variance with the unusual trend of ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$. The extracted logarithmic derivative does not reach the $L_{\mathrm{CS}}$ limit at low energies, so that no maximum of the $S$ factor shows up. CC calculations were performed including the low-lying $2^{+}$ and $3^{-}$ excitations.

Conclusions: Using a Woods-Saxon potential the experimental cross sections at low energies are overpredicted, and this is a clear sign of hindrance, while the calculations performed with a M3Y + repulsion potential nicely fit the data at low energies, without the need of an imaginary potential. The comparison with the results for ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ strengthens the explanation of the oblate shape of ${}^{28}\mathrm{Si}$ being the reason for the irregular behavior of that system.

Figure 1

Two-dimensional spectra of time of flight vs energy loss measured at different beam energies around (a) and far below the Coulomb barrier (b), for ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$. The nominal beam energies and the corresponding fusion cross sections are reported. The detected events within the oval shapes are identified as fusion evaporation residues (ER) and are indicated by the red arrows. They are well separated from the other kinds of events due to degraded beam and to fusion on carbon and oxygen present in the target.

Figure 2

Fusion excitation function of ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$ measured in this work, compared to the system ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ [1]. The energy scale is normalized to the Akyüz-Winther Coulomb barrier [11]. The reported errors are purely statistical uncertainties. The insert shows the logarithmic derivatives (slopes) of the excitation functions for the two systems.

Figure 3

Logarithmic (a) and linear plot (b) of the present and previously measured (Bozek et al. [5]) fusion cross sections for ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$. The data are compared to Ch-1 and Ch-15 CC calculations that are based on the M3Y + repulsion potential with parameters $a_r=0.38$ fm and radius $R=3.30$ fm of the neutron density. Also shown are the results of Ch-15 calculations that use a Woods-Saxon (WS) potential with parameters $R=7.177$ fm, $U_0=-55.37$ MeV, and $a=0.637$ fm. The radius $R$ was adjusted to reproduce the data at energies above the Coulomb barrier. The calculation Ch-15w0 shown in panel (b) illustrates the importance of an imaginary potential at high energies.

Figure 4

${\chi }^2$ analysis of the present data by Ch-10w5 and Ch-15w5 calculations, respectively, as functions of the radius $R$ of the neutron density in ${}^{30}\mathrm{Si}$. The data have been normalized to the data of Bozek et al. [5] at 34.9 MeV.

Figure 5

Logarithmic (a) and linear (b) plots of the best fits to the ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ fusion data [1] and the present ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$ fusion data obtained with Ch-10w5 and Ch-15w5 calculations, respectively.

Figure 6

$S$ factors (upper panels) and logarithmic slopes of the excitation functions (lower panels) for the fusion of ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ [1] and ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$ (see also Ref. [5]) are compared to the best Ch-10 and Ch-15 calculations, respectively, with (w5) and without (w0) an imaginary potential. Ch-1w0 calculations (no coupling results) are reported for reference. Results that are based on a WS potential (adjusted to reproduce the high-energy data) are also shown.

Figure 7

The first derivative of the energy-weighted cross section for the fusion of ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ (Montagnoli et al. [12]) (a) and ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$ (b) (present data and Ref. [5]) are compared to the best Ch-10aw3w5 and Ch-15w5 calculations, respectively. The results of Ch-1 calculations are also shown.

Figure 8

Ratio of the measured and the calculated cross sections shown in Fig. 5 for ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$ (a) and ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$ (b). The ratio to the WS calculations indicates a fusion hindrance just below the Coulomb barrier for ${}^{28}\mathrm{Si}+{}^{28}\mathrm{Si}$, but that effect is seen to disappear at lower energies (the older data of Gary and Volant are from Ref. [16]). For ${}^{30}\mathrm{Si}+{}^{30}\mathrm{Si}$ a fusion hindrance appears at the lowest measured energies.