Fusion hindrance for ${}^{27}\mathrm{Al}+{}^{45}\mathrm{Sc}$ and other systems with a positive $Q$ value

Fusion evaporation cross sections for the ${}^{27}\mathrm{Al}+{}^{45}\mathrm{Sc}$ ($Q=9.63$ MeV) system are measured down to about 300 nb. Deviations from standard coupled-channels calculations were observed in this system at the lowest energies. The steep fall-off of the fusion cross sections can be reproduced by calculations using a shallow potential model, which was originally developed to explain the hindrance behavior of heavy-ion fusion in medium-mass systems with negative $Q$ values. Comparisons of the hindrance behavior between the present experiment and other systems, for example, ${}^{28}\mathrm{Si}+{}^{30}\mathrm{Si}$ ($Q=14.3$ MeV) and ${}^{36}\mathrm{S}+{}^{48}\mathrm{Ca}$ ($Q=7.55$ MeV) are presented.

Figure 1

$M/q$ spectra at the focal plane of the Fragment Mass Analyzer for three incident beam energies. Here $M$ and $q$ are the mass number and charge state of the evaporation residues measured at the focal-plane detector. The counts at the lowest energy $E=31.74$ MeV, marked B, are background events.

Figure 2

Comparisons of experimental evaporation residue cross sections with calculations for the system ${}^{27}\mathrm{Al}+{}^{45}\mathrm{Sc}$. Green long-dashed line is the optical model; solid magenta line is an extrapolation based on the equation $L(E)=A_0+B_0/E^{1.5}$; black dashed-dotted line is the standard CC calculation using a Woods-Saxon potential.

Figure 3

(a) The logarithmic derivative $L(E)=d(\ln \sigma E)/\mathit{dE}$ plotted as a function of the center-of-mass energy $E$. The stars were derived from the data by least-squares fits to three consecutive data points while the solid circles were obtained with the two-points method. The black dashed line is the constant $S$-factor function $L_{\mathit{cs}}(E)$. The solid magenta curve is a fit to the low-energy part of the data with a formula $L(E)=A_0+B_0/E^{1.5}$. (b) The $S$ factor versus $E$ plot. The solid magenta curve is an extrapolated one (see text for details). The green long-dashed and the black dash-dotted curves are calculations from optical model and standard CC, respectively.

Figure 4

Comparison between experimental $\sigma (E)$ data for system ${}^{14}\mathrm{N}+{}^{14}\mathrm{N}$. Black triangles are for the fusion channel red-open and blue-solid circles are for neutron and proton transfer channels, respectively.

Figure 5

Comparison of potentials used in the CC calculations. Different stiffnesses $a_{\mathrm{rep}}$ of the hard core density are used ($a_{\mathrm{rep}}=0.40$, $0.425$, and $0.44$ fm, respectively), see Ref. [25]. The black long-dashed curve is the pure M3Y potential and the red dotted curve is the Woods-Saxon potential from Ref. [29].

Figure 6

Comparison between the experimental $S(E)$ curve for the fusion reaction ${}^{27}\mathrm{Al}+{}^{45}\mathrm{Sc}$ and various calculations. The black dash-dotted and green long-dashed curves are calculations from standard CC (with a Woods-Saxon potential) and optical model. The blue dashed and light-blue dotted curves are CC calculations from a shallow potential and with $a_{\mathrm{rep}}=0.44$ and $0.40$ fm, respectively. The red solid line is the extrapolation with the hindrance recipe.

Figure 7

$L(E)$ [(a), (c), and (e)] and $S(E)$ [(b), (d), and (f)] plots for the reactions ${}^{28}\mathrm{Si}+{}^{30}\mathrm{Si}$, ${}^{27}\mathrm{Al}+{}^{64}\mathrm{Ni}$ and ${}^{28}\mathrm{Si}+{}^{64}\mathrm{Ni}$. The black dashed curves display the constant $S$-factor functions and the red arrows are energy locations of $S$-factor maxima predicted by the hindrance systematics [from Eqs. (3) and (4)]. The blue dash-dotted curves are from standard CC calculations with a standard Woods-Saxon potential and the magenta curves represent fits to $L(E)$ data at the low-energy region [(a), (c), and (e), with Eq. (1)] and to the extrapolations in the $S$-factor representation [(b), (d), and (f)]. In (a), (c), and (e), solid and open circles correspond to slope determinations from consecutive data points and least-squares fits to three data points, respectively. The lowest cross sections (${\sigma }_{\min }$) measured in these experiments are included in (b), (d), and (f), respectively.

Figure 8

$L(E)$ [(a), (c), and (e)] and $S(E)$ [(b), (d), and (f)] plots for the reactions ${}^{40}\mathrm{Ca}+{}^{48}\mathrm{Ca}$, ${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$, and ${}^{36}\mathrm{S}+{}^{48}\mathrm{Ca}$. The black dashed curves, the red arrows, the blue dash-dotted curves, and the magenta curves have the same meanings as in Fig. 7. The lowest cross sections (${\sigma }_{\min }$) measured in these experiments are included in (b), (d), and (f), respectively.