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 $k^2$. The procedure is here extended to the case for which the polarizable system is itself charged. An MERT expansion is derived for the difference $\delta \mathrm{}\left(k\right)\mathrm{}$ between the total phase shift $\eta \mathrm{}\left(k\right)\mathrm{}$ and the phase shift $\rho \mathrm{}\left(k\right)\mathrm{}$ due to the long-range tail alone; both $\eta \mathrm{}\left(k\right)\mathrm{}$ and $\rho \mathrm{}\left(k\right)\mathrm{}$ 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 $\eta \mathrm{}\left(k\right)\mathrm{}$ is determined exactly for all $L$ by the Born approximation; for $V\left(r\right)=\left(\frac{2\mu }{{\hslash }^2}\right)(-\frac{{\beta }^2}{r^4})$, where $\alpha =\frac{{\beta }^2{\hslash }^2}{|\mu e^2{Z}_1{Z}_2}|={\beta }^2{a}_0$ is the electric-dipole polarizability of the target, $tan\eta \mathrm{}\left(k\right)=\frac{{\beta }^2{a_0}^3{k}^5}{15}$ for $k{a}_{0}L\ll 1$.