Proton differential elliptic flow and the isospin dependence of the nuclear equation of state

Within an isospin-dependent transport model for nuclear reactions involving neutron-rich nuclei, we study the first-order direct transverse flow of protons and their second-order differential elliptic flow as a function of transverse momentum. It is found that the differential elliptic flow of midrapidity protons, especially at high transverse momenta, is much more sensitive to the isospin dependence of the nuclear equation of state than the direct flow. Origins of these different sensitivities and their implications to the experimental determination of the isospin dependence of the nuclear equation of state by using neutron-rich heavy-ion collisions at intermediate energies are discussed. The isospin dependence of the nuclear equation of state ~EOS! is one of the most important but very poorly known properties of neutron-rich matter @1#. Its determination in laboratory-controlled experiments has profound implications for the study of the structure and evolution of many astrophysical objects @2#. Nuclear reactions induced by stable neutron-rich nuclei and/or radioactive beams provide a means to extract useful information about the isospin dependence of the nuclear EOS and to explore novel phenomena in nuclear matter at extreme isospin asymmetries. A number of dedicated experiments to study the isospin dependence of the nuclear EOS have been performed/planned at several available radioactive beam facilities and the future Rare Isotope Accelerator @3#. For these experiments to be fruitful it is important to first understand well the role of the isospin degree of freedom in nuclear reaction dynamics. Moreover, theoretical predictions on the sensitivity of experimental observables to the isospin dependence of the nuclear EOS are useful for explaining available data and planning new experiments. Recently, several useful observables have been identified in neutron-rich heavy-ion collisions at intermediate energies. These include the isospin fractionation @ 4‐1 0#, neutron to proton ratio of heavy residues @11# or projectilelike fragments @7,12#, and the neutron-proton differential flow @13#. Most of these observables make use of the relative multiplicities and kinetic energies of mirror nuclei. Information about the isospin dependence of the nuclear EOS can also be obtained from studying free nucleons as neutrons and protons have the opposite symmetry potentials in nuclear media. In a recent work, one of the present authors has shown that the neutron-proton differential flow is a rather useful probe of the isospin dependence of the nuclear EOS. The method utilizes constructively both the isospin fractionation and the nuclear collective flow as well as their sensitivities to the nuclear EOS. It, however, requires measuring the neutron and proton collective flows simultaneously. Experimentally, it is easier to identify charged particles and measure their momenta accurately, thus observables using charged particles only are more useful. In this work, within an isospin-dependent transport model for nuclear reactions involving neutron-rich nuclei, we explore the sensitivity of proton collective flow to the isospin dependence of the nuclear EOS in heavy-ion collisions at intermediate energies. We examine both the first-order transverse flow and the second-order differential elliptic flow as functions of transverse momenta. It is found that the differential elliptic flow of midrapidity protons, especially at high transverse momenta, is very sensitive to the isospin dependence of the nuclear EOS. This sensitivity is much stronger than that found in the first-order transverse collective flow around the projectile and target rapidity. At present, nuclear many-body theories predict vastly different isospin dependence of the nuclear EOS depending on both the calculation techniques and the bare two-body and/or three-body interactions employed, see, e.g., Refs. @14 ‐17#. Various theoretical studies ~e.g., Refs. @18,19#! have shown that the energy per nucleon e(r,d) in nuclear matter of density r and isospin-asymmetry parameter d defined as d[~r n2r p!/~r n1r p! ~1!