Approximate treatment of particle collisions in the time-dependent mean-field theory

In order to study the gross effects of particle collisions, we formulate a model to include particle collisions in the time-dependent mean-field theory. The model consists of the time-dependent Hartree-Fock equation for the single-particle states and a linearized but truncated equation for the occupation numbers, with the $H$-theorem and the conservation laws properly satisfied. Numerical calculations are performed with a "conventional" set of basis states in which the occupied and the unoccupied static Hartree-Fock states are boosted in the same way. They are carried out for the head-on collisions of ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ on ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ and ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$ on ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$ at 100 MeV per nucleon. It is found that the dynamics of the reaction is affected in only a minor way when particle collisions are included. The projectile and the target nuclear matter appear to interpenetrate each other, with only a 20% reduction in the center-of-mass kinetic energies, just as in mean-field calculations with no particle collisions. This result reinforces the idea that the conventional basis states do not include some important degrees of freedom. It is suggested that in future work one should modify the basis in order to allow for ample wave propagation in the direction normal to the collision axis.