Relativistic two-fluid model of nucleus-nucleus collisions

Abstract

To take into account the expected partial transparency of nuclei during collisions at high energy, we introduce a two-fluid dynamical model, in which coupled relativistic equations of motion are solved for separate target and projectile nuclear fluids. The terms in the equations that couple the two nuclear fluids are obtained from the cross section and momentum transfer for individual nucleon-nucleon collisions. At low relative velocities the target and projectile fluids merge, in which case the conventional one-fluid dynamical model is recovered. For a given nuclear equation of state and for given initial conditions, the equations of motion are solved as functions of time for the nucleon number density, momentum density, energy density, pressure, and velocity for each fluid. This is done in three spatial dimensions by means of a relativistic generalization of a standard particle-in-cell finite-difference computing method for multiphase fluid-dynamics problems. For each of several impact parameters, the velocity distribution at some large time is converted to an energy and angular distribution for the expanding matter. Integration of these results over impact parameter then gives the double differential cross section $\frac{{d}^{2}\ensuremath{\sigma}}{\mathrm{dEd}\ensuremath{\Omega}}$. For $^{20}\mathrm{Ne}$ + $^{238}\mathrm{U}$ at laboratory bombarding energies per nucleon of 250 MeV, 400 MeV, and 2.1 GeV, as well as for $^{4}\mathrm{He}$ + $^{238}\mathrm{U}$ at a laboratory bombarding energy per nucleon of 400 MeV, we compare calculated and experimental energy spectra for outgoing charged particles at several laboratory angles. The calculations reproduce correctly the experimental slopes at each angle, as well as the overall decrease in the experimental cross section when going from forward to backward angles. However, for $^{20}\mathrm{Ne}$ + $^{238}\mathrm{U}$ at laboratory bombarding energies per nucleon of 250 and 400 MeV, the calculated values at 30\ifmmode^\circ\else\textdegree\fi{} are only one-third the experimental values. Also, for $^{4}\mathrm{He}$ + $^{238}\mathrm{U}$ at a laboratory bombarding energy per nucleon of 400 MeV, the calculated values at all angles are substantially smaller than the experimental values. As a prediction for future experiments, we also use the theory in a context where our numerical approximations and simplifications (such as neglecting the diffuseness of the nuclear surface) should be more reliable, to deduce results for $^{238}\mathrm{U}$ + $^{238}\mathrm{U}$ at laboratory bombarding energies per nucleon of 250 MeV and 2.1 GeV. By means of a one-dimensional calculation we also study the maximum compression and excitation that can be achieved in such collisions. The maximum rest-frame compression is about 2.4 to 2.6 at a bombarding energy per nucleon of 250 MeV, and is about 6 to 8 at a bombarding energy per nucleon of 2.1 GeV. The available energy is primarily in the form of compressional energy, with little in the form of internal excitation energy, Significant rest-frame compressions (greater than twice normal density for 250 MeV and greater than 3 times normal density for 2.1 GeV) are achieved for substantial numbers of nucleons and substantial time periods.NUCLEAR REACTIONS $^{20}\mathrm{Ne}$ + $^{238}\mathrm{U}$, $\frac{{E}_{\mathrm{bom}}}{20}=250 \mathrm{MeV}, 400 \mathrm{MeV}, 2.1 \mathrm{GeV}$; $^{4}\mathrm{He}$ + $^{238}\mathrm{U}$, $\frac{{E}_{\mathrm{bom}}}{4}=400 \mathrm{MeV}$; $^{238}\mathrm{U}$ + $^{238}\mathrm{U}$, $\frac{{E}_{\mathrm{bom}}}{238}=250 \mathrm{MeV}, 2.1 \mathrm{GeV}$. Calculated $\frac{{d}^{2}\ensuremath{\sigma}}{\mathrm{dEd}\ensuremath{\Omega}}$ for outgoing charged particles. Relativistic two-fluid dynamical model, nuclear equation of state, nuclear matter at high density and high excitation energy.