Symmetry Energy and the Isotopic Spin Dependence of the Single-Particle Potential in Nuclear Matter

Nuclear matter with a given neutron excess is treated within the frame of the $K$-matrix theory. General expressions for the symmetry energy ${\epsilon }_{\mathrm{sym}}$ and the single-particle potential are obtained with the help of the reaction matrix which depends on two different Fermi momenta for neutrons and protons. In particular, an expression for the isotopic spin-dependent part $U_1$ of the single-particle potential is obtained, and then specialized for single particles at the Fermi surface. With suitable approximations numerical values of ${\epsilon }_{\mathrm{sym}}$ and $U_1$ at the Fermi surface are obtained with the help of the Bruckner-Gammel solution for nuclear matter. The results are: ${\epsilon }_{\mathrm{sym}}=64\mathrm{}$ MeV, $U_1\left(k_F\right)=126\mathrm{}$ MeV.