Calculation of compound-nucleus cross sections for symmetric very-heavy-ion reactions

Abstract

Cross sections for forming compound nuclei in symmetric very-heavy-ion reactions are calculated by use of the criterion that the dynamical trajectory for the fusing system must pass inside the fission saddle point in a multidimensional space in order to form a compound nucleus. The dynamical trajectory is obtained by solving numerically the modified classical Lagrange equations of motion for a system whose shape is specified in terms of smoothly joined portions of three quadratic surfaces of revolution. This restriction to nuclear shapes that are axially symmetric (about an axis that is in general rotating in space) is a good approximation for systems with small angular momentum, but it may be seriously deficient for systems with large angular momentum. The nuclear potential energy of deformation is determined by means of a macroscopic model that includes the Coulomb energy and the nuclear macroscopic energy. This latter quantity is calculated in terms of a double volume integral of a Yukawa two-body interaction potential, which includes the surface energy of the liquid-drop model but also takes into account the lowering in energy due to the finite range of the nuclear force. For systems with angular momentum, we include a centrifugal pseudopotential calculated for rigid-body rotation. The kinetic energy of collective motion is calculated for nuclear flow that is a superposition of incompressible, nearly irrotational collective-shape motion and rigid-body rotation. Nuclear dissipation (the transfer of energy of collective motion into internal single-particle excitation energy) is included in the formulation by means of Rayleigh's dissipation function. However, to emphasize that compound-nucleus cross sections become small for very heavy systems even in the absence of dissipation, our current results are presented for the case of zero dissipation. For nuclear systems lighter than about ${}_{}{}^{110}{}_{}{}^{}\mathrm{Pd}$ + ${}_{}{}^{110}{}_{}{}^{}\mathrm{Pd}$ → ${}_{}{}^{220}{}_{}{}^{}\mathrm{U}$ and for relatively low angular momentum, the fission saddle point lies outside the point of hard contact in heavy-ion reactions, which permits the compound-nucleus cross section at relatively low bombarding energy to be calculated in terms of a one-dimensional interaction barrier, as is customarily done. For heavier nuclear systems and/or for high angular momentum, the fission saddle point lies inside the contact point, which reduces the compound-nucleus cross section compared with that calculated for a one-dimensional interaction barrier.