Moments of fusion-barrier distributions

A study of fusion-barrier distributions through an analysis of their moments is presented. The moments can be obtained from least-squares fits of the energy-weighted fusion cross sections without the need of calculating second derivatives. The zeroth and first moments determine the fusion radius $R$ and the Coulomb barrier $V_C$. These two quantities are the same as the parameters $R$ and $V_C$ that are used in the well-known expression, $E\sigma =\pi R^2(E-V_C)$, for the fusion cross section at high energies. The second and third moments, $M_2$ and $M_3$, determine the width and skewness of the barrier distribution, respectively. From these global parameters new correlations for the study of heavy-ion-induced fusion reactions can be obtained. Systems exhibiting a large coupling to transfer reactions show a small fusion radius as well as a large second moment. A negative third moment is correlated with a prolate deformation of the target nucleus.

Figure 1

Plot of $E\sigma$ vs $E$ for the fusion reaction ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$. The black line is given by the equation $\pi R^2(E-V_C)$ with the parameters $R$ and $V_C$ obtained from a least-squares fit to the data at $E>1.04V_c$ (red solid circles). The second and third moments $M_2$ and $M_3$ are calculated from the green-shaded area. See text for details.

Figure 2

Plots of $E\sigma$ vs $E$ for the fusion systems $\mathrm{Ca}+\mathrm{Ca}$ and $\mathrm{Ca}+\mathrm{Zr}$. The straight lines are least-squares fits of Eq. (4) to the data at high energies (solid circles). The slopes of these lines determine the $\pi R^2$ values for each system.

Figure 3

Plots of $R-R_{\mathrm{ref}}$ vs $\sqrt{M_2}$ for S-induced (a) and Ca- and Ni-induced (b) fusion reactions listed in Table 2 (the results for O, $\mathrm{Si}+\mathrm{Sm}$, and $\mathrm{O}+\mathrm{W}$ are not included). Systems with the same $Z_1$ and $Z_2$ are connected by dashed lines. Systems that exhibit strong transfer reactions are shown by solid points (see text for details).

Figure 4

Plot of $M_3/M_2$, for systems where the uncertainties allow the calculation of a meaningful value of $(M_3/M_2)$. The $x$ axis is the order in which the systems are quoted in Table 2.

Figure 5

Plots of $E\sigma$ vs $E$ for two systems with radioactive beams, (a) ${}^{58}\mathrm{Ni}+{}^{132}\mathrm{Sn}$ and (b) ${}^{15}\mathrm{C}+{}^{232}\mathrm{Th}$. The red line and symbols in panel (a) show the data plotted on a reduced scale. The solid circles are data used in the least-squares fits using Eq. (4).

Figure 6

(a) Barrier distributions for three $\mathrm{Ca}+\mathrm{Ca}$ fusion reactions. The ${}^{40}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ system (black) exhibits the widest distribution (large $M_2$) and the lowest maximum (small $R$). The symbols are obtained from calculating the second derivative of the energy-weighted, high-statistics cross sections. The curves are the barrier distributions $B_w\left(E\right)$ of the Wong model [Eq. (11)]. (b) Plot of $E\sigma$ vs $E$ for energies close to the barrier. The second moment $M_2$ is calculated from the cross-hatched area, which is largest for the ${}^{40}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ system shown in black. (c) Plot of $R, M_2$, and $M_3$ vs $A_1+A_2$.