Spin and Spin-Isospin Symmetry Energy of Nuclear Matter

The expression, $\frac{E}{A}={\epsilon }_{\mathrm{vol}}+\frac{1}{2}{\epsilon }_{\tau }\alpha _{\tau }^{}{}_{}{}^2+\frac{1}{2}{\epsilon }_{\sigma }{({\alpha }_n+{\alpha }_p)}^2+\frac{1}{2}{\epsilon }_{\sigma \tau }{({\alpha }_n-{\alpha }_p)}^2$, for the ground-state energy of nuclear matter with an excess of neutrons, of spin-up neutrons, and of spin-up protons (characterized by the corresponding parameters, ${\alpha }_{\tau }=\frac{\mathrm{}(N-Z)\mathrm{}}{A}$, ${\alpha }_n=\frac{(N_{\uparrow }-N_{\downarrow })}{A}$, and ${\alpha }_p=\frac{(Z_{\uparrow }-Z_{\downarrow })}{A}$), contains three symmetry energies: the isospin symmetry energy ${\epsilon }_{\tau }$, the spin symmetry energy ${\epsilon }_{\sigma }$, and the spin-isospin symmetry energy ${\epsilon }_{\sigma \tau }$. General expressions for ${\epsilon }_{\sigma }$ and ${\epsilon }_{\sigma \tau }$ are obtained in terms of the $K$ matrix which depends on four different Fermi momenta. With suitable approximations, numerical values of ${\epsilon }_{\sigma }$ and ${\epsilon }_{\sigma \tau }$ (and also of ${\epsilon }_{\tau }$) are derived using the Brueckner-Gammel-Thaler, the Hamada-Johnston, and the Reid softcore nucleon-nucleon potentials. The most reliable results, obtained with the Reid soft-core potential, are: ${\epsilon }_{\tau }=61\mathrm{}$ MeV, ${\epsilon }_{\sigma }=74\mathrm{}$ MeV, and ${\epsilon }_{\sigma \tau }=73\mathrm{}$ MeV. The possibility of estimating the energies of the spin and spin-isospin modes of collective nuclear excitations is discussed.