Approximate Independence of Optical-Model Elastic Scattering Calculations on the Potential at Small Distances

In optical-model calculations of the elastic scattering of alpha particles or other heavier nuclei on nuclei, it has been observed that the cross section is rather insensitive to variation of the optical potential at small values of the interaction distance. In the present note it is pointed out that in many cases this effect can be explained with the aid of the JWKB approximation. If the real part $V$ of the optical potential $V+iW$ is sufficiently deep and if $W$ produces sufficient absorption, then at some point $R_b$ contained inside the interaction region, the outgoing branch of the JWKB expression for the radial wave function becomes negligible compared to the ingoing branch. In this case the choice of the potentials for $r<\mathrm{}{R}_b$ does not affect the phase shifts, and the use of an ingoing wave boundary condition applied at $R_b$ to the radial wave function is nearly equivalent to the use of conventional optical-model procedures. Under such circumstances the phase shifts are in essence determined by the penetration of the radial wave function through the barrier formed by the combined action of the Coulomb, nuclear, and centrifugal potentials.