Optical-model potential in nuclear matter from Reid's hard core interaction

We describe a method for the calculation of the leading term of a previously proposed low-density expansion for the self-energy of nucleons in nuclear matter. We compute the single-particle complex potential energy, the average binding energy per nucleon, the complex symmetry potential, and the symmetry energy. We use Reid's hard core nucleon-nucleon interaction and take a Fermi momentum $k_F=1.4\mathrm{}$ ${\mathrm{fm}}^{-1\mathrm{}}$. The calculated single-particle potential energy is compared with the phenomenological values of the optical-model potential in the inner region of a nucleus. The real part of our theoretical value is given by $56-0.3E$ (MeV) below $E=150\mathrm{}$ MeV, and changes sign at 200 MeV. The imaginary part rises from 2 MeV at low energy to about 20 MeV at $E=200\mathrm{}$ MeV. These features are in good agreement with experimental evidence. The average binding energy $B$ per nucleon calculated with a self-consistent potential energy for the particle states above $k_F$ is equal to -11 MeV. In the standard approach, with no potential energy for intermediate particle states above $k_F$, one finds -8.65 MeV. We also calculate the symmetry potential. At low energy, its real part is equal to $14 \frac{(N-Z)}{A}$ (MeV); it changes sign at 110 MeV. Its imaginary part is equal to $3.5\frac{(N-Z)}{A}$ (MeV) at low energy, and rises to $8.5\frac{(N-Z)}{A}$ (MeV) at 200 MeV. The symmetry energy is equal to 27.8 MeV.