Coulombic Modified Effective-Range Theory for Long-Range Effective Potentials

A modified effective-range theory (MERT) was introduced previously to describe the scattering of a charged particle by a neutral polarizable system. The long-range components of the resultant effective one-body (generalized optical-model) potential, which cause the phase shift to have a rapidly varying energy dependence, are taken account of exactly by solving a one-body problem numerically. The short-range potential components generate only a slowly varying energy-dependent effect on the phase shift, and this effect can be accounted for by a few terms in a power series in k². The procedure is here extended to the case for which the polarizable system is itself charged. An MERT expansion is derived for the difference δ(k) between the total phase shift η(k) and the phase shift ρ(k) due to the long-range tail alone; both η(k) and ρ(k) are defined relative to pure Coulomb scattering. With the strongly energy-dependent Coulombic and other long-range effects accounted for exactly by the numerical solution of a one-body scattering problem, low-energy scattering data can be matched by the proper choice of the coefficients of just the first few terms in the MERT expansion; these terms will then determine the scattering in the (experimentally inaccessible) energy domain extending down to zero energy. For a repulsive Coulomb field, the leading term in η(k) is determined exactly for all L by the Born approximation; for V(r)=(2μ²)(-β²r⁴), where α=β²²|μe²Z₁Z₂|=β²a₀ is the electric-dipole polarizability of the target, tanη(k)=β²a₀³k⁵15 for ka₀L1.