Microscopic Theory of the Optical-Model Potential and the Hole-Particle Model in Nuclear Spectroscopy

Approximate expressions for the optical-model potential ${\mathcal{U}}_{\mathrm{opt}}$ in terms of the nucleon-nucleon interaction are discussed. The identity of all the $A+1\mathrm{}$ nucleons of the scattering problem is approximately taken into account in our final formulas. The imaginary part of ${\mathcal{U}}_{\mathrm{opt}}$ is calculated for the case of ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ and the incident-nucleon energy of 20 MeV. The spherical-model random-phase-approximation (RPA) eigenvalues and eigenvectors are assumed for the complete set of intermediate nuclear states involved. The results are rather insensitive to the exchange-force mixture assumed for our zero-range nucleon-nucleon potential. The anti-symmetrization of our $V$-matrix elements is extremely important as it reduces our $\mathrm{Im}{\mathcal{U}}_{\mathrm{opt}}$ by a factor of 2-3. For a reasonable set of our RPA intermediate eigenvalues and eigenvectors we obtain a semiquantitative agreement with the best phenomenological $\mathrm{Im}{\mathcal{U}}_{\mathrm{opt}}$ available for our case. The most important contribution to $\mathrm{Im}{\mathcal{U}}_{\mathrm{opt}}$ corresponds to the first excited $T=0, 2^{+}$ state in ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$.