Optical phase shifts from low-energy neutron cross sections

The Lorentz-weighted average of the S-matrix introduced by G.E. Brown is used in the Feshbach theory of the generalized optical potential to show that the average many-body S-matrix for elastic scattering is exactly equal to the two-body S-matrix of an optical potential. However, the optical potential S-matrix must be evaluated at the complex energy $\text{E}\text{= E + iI,}\text{where}\text{I}$ is the half-width of the lorentzian. The resulting equation for the optical phase shifts (OPS) δ_c, exp $\text{2iδ}{}_{\mathrm{c}}\text{(}\text{E}\text{)] = 〈S}{}_{\mathrm{cc}}\text{(E)〉}$, holds even when the level spacing D forces the use of an averaging half-width I > D which is comparable to the energy E, providing that the OPS are also evaluated at the complex $\text{E}$ instead of being approximated by their values on the real energy axis at E. An appendix discusses briefly the conditions on a potential necessary for the result obtained by Brown that $\text{〈S}{}_{\mathrm{cc}}\text{(E)〉 = S}{}_{\mathrm{cc}}\text{(}\text{E}\text{)}$ when Lorentz-weighted averaging is used.