Effect of the splitting of the neutron and proton effective masses on the nuclear symmetry energy at finite temperatures

We present the temperature and density dependence of symmetry energy for nuclear matter at finite temperatures based on the thermodynamics approach with the Skyrme energy density functional. We first classify the Skyrme interactions into 7 groups according to their ranges of the variation of neutron and proton effective masses with densities in neutron matter limit (99.99 percent neutron in the matter). We find that there is an obvious correlation between the temperature dependence of the symmetry energy and the splitting of neutron and proton effective mass. For the Skyrme interactions with mn⁎>mp⁎ and the strong neutron and proton effective mass splitting in asymmetric nuclear matter, a transition of symmetry energy from decreasing with temperature at low densities to increasing with temperature at high densities appears. For other Skyrme interactions, we do not observe this phenomenon and the symmetry energy decreases with temperature in the entire density region.