Dynamical thresholds for compound-nucleus formation in symmetric heavy-ion reactions

We study the effect of the nuclear macroscopic energy, nuclear dissipation, and shape parametrization on dynamical thresholds for compound-nucleus formation in symmetric heavy-ion reactions. This is done by solving numerically classical equations of motion for head-on collisions to determine whether the dynamical trajectory in a multidimensional deformation space passes inside the fission saddle point and forms a compound nucleus, or whether it passes outside the fission saddle point and reseparates. Specifying the nuclear shape in terms of smoothly joined portions of three quadratic surfaces of revolution, we take into account three symmetric deformation coordinates. However, in some cases we reduce the number of coordinates to two by requiring the ends of the fusing system to be spherical in shape. The nuclear potential energy of deformation is determined in terms of a Coulomb energy and a nuclear macroscopic energy that is usually taken to be a double volume energy of a Yukawa-plus-exponential folding function, although a double volume integral of a single-Yukawa folding function and ordinary surface energy are also considered. The collective kinetic energy is calculated for incompressible, nearly irrotational flow by means of the Werner-Wheeler approximation. Four possibilities are studied for the transfer of collective kinetic energy into internal single-particle excitation energy: (1) zero dissipation, (2) ordinary two-body viscosity, (3) one-body wall-formula dissipation, and (4) one-body wall-and-window dissipation. For systems with $\frac{Z^2}{A}$ larger than a threshold value ${\left(\frac{Z^2}{A}\right)}_{\mathrm{thr}}$ which depends somewhat upon dissipation, the center-of-mass bombarding energy must exceed the maximum in the one-dimensional interaction barrier by an amount $\Delta E$ in order to form a compound nucleus. For all four possibilities considered, we find that the dependence of $\Delta E$ on $\frac{Z^2}{A}-{\left(\frac{Z^2}{A}\right)}_{\mathrm{thr}}$ is more complicated than the lowest-order quadratic dependence found in some previous approximate solutions. For both types of one-body dissipation, our calculated values of $\Delta E$ are an order of magnitude larger than those for zero dissipation and ordinary two-body viscosity. We compare our results calculated for symmetric systems with experimental values for asymmetric systems by use of a tentative scaling involving ${\left(\frac{Z^2}{A}\right)}_{\mathrm{mean}}$, defined as the geometric mean of $\frac{Z^2}{A}$ for the combined system and an effective value ${\left(\frac{Z^2}{A}\right)}_{\mathrm{eff}}$, for the projectile and target. When compared in this way, the experimental values of $\Delta E$ agree better with results calculated for two-body viscosity than with results calculated for either type of one-body dissipation.