Inversion Problem for Ion-Atom Differential Elastic Scattering

An inversion procedure to recover the intermolecular potentials from low-energy (4-14 eV) differential scattering data for the systems He${\mathrm{H}}^{+}$, Ne${\mathrm{H}}^{+}$, Ar${\mathrm{H}}^{+}$, and Kr${\mathrm{H}}^{+}$ is presented. The first step in this procedure is to construct the phase shifts {$\eta \mathrm{}\left(l\right)\mathrm{}$} from the experimental differential cross section $\sigma \left(\theta \right)$ and, for reasons of intuition, the corresponding classical deflection function. This is accomplished by employing a new technique developed recently by Remler which involves singularities in the $S$ matrix. The Remler method is as accurate as, but less cumbersome than, the frequently employed standard partial-wave sum and in addition leads to an intuitive connection between $\eta \mathrm{}\left(l\right)\mathrm{}$ and $\sigma \left(\theta \right)$. Having found {$\eta \mathrm{}\left(l\right)\mathrm{}$}, the intermolecular potential may be determined by means of the transformation method of Vollmer. To test the entire inversion procedure, the intermolecular potential obtained from the ${\mathrm{H}}^{+}$ + He scattering data is compared to the ab initio calculation by Wolniewicz of the ground-state potential for He${\mathrm{H}}^{+}$. This method should be applicable to any spherically symmetric scattering system.