Nuclear matter symmetry energy from generalized polarizabilities: Dependences on momentum, isospin, density, and temperature

Symmetry energy terms from macroscopic mass formulas are investigated as generalized polarizabilities of nuclear matter. Besides the neutron-proton (n-p) symmetry energy, the spin-dependent symmetry energies and a scalar one are also defined. They depend on the nuclear densities (ρ), neutron-proton asymmetry (b), temperature (T), and exchanged energy and momentum (q). Based on a standard expression for the generalized polarizabilities, a differential equation is proposed to constrain the dependence of the symmetry energy on the n-p asymmetry and on the density. Some solutions are discussed. The q dependence (zero frequency) of the symmetry energy coefficients with Skyrme-type forces is investigated in the four channels of the particle-hole interaction. Spin-dependent symmetry energies are also investigated indicating much stronger differences in behavior with q for each Skyrme force than the results for the neutron-proton one.