On the theory of the optical potential

We consider elastic scattering of a projectile by a complex many-body system. The incident energy is assumed high enough so that closure may be used on the target, and the problem is reduced to computing the target ground-state expectation value of the scattering amplitude from A fixed scattering centers. We solve for this A-body scattering amplitude exactly, using a model of a sum of separable potentials (the approach is extended to a more general class of potentials in an appendix). The exact amplitude exhibits many interesting properties. We then make a multiple scattering expansion of this amplitude, and simultaneously, an expansion of the ground-state density in terms of single-particle densities, two-particle densities, and so on. By comparing the resulting multiple scattering amplitude term by term with the scattering series from a one-particle Schrödinger equation for the projectile, we can identify an equivalent, or optical potential. We identify a lowest-order optical potential, stating precisely under what conditions it will generate the correct scattering amplitude. By summing additional classes of multiple scattering graphs, we include modifications due to two-particle correlations and multiply-struck target particles (“local field” corrections). The limiting cases of S-wave scattering and very high energy projectiles are discussed in detail. Some calculations of the high-energy optical potential based on more realistic elementary-particle scattering amplitudes are presented. The effects of spin and isospin are discussed, and an appendix is included on center-of-mass motion.