Potentials for High-Energy Scattering from Hydrogenlike Atoms

Abstract

The time-dependent equation $\frac{i\hslash v\partial \varphi (\mathcal{r}, R)}{\partial Z}=\exp \left(\frac{i{H}_{e}Z}{\hslash v}\right)V(\mathcal{r}, R)\exp (-\frac{i{H}_{e}Z}{\hslash v})\varphi (v, R), t=\frac{Z}{v}$ describing a fast particle perturbing a bound system was solved by Glauber in the diabatic approximation of setting $H_e$ equal to zero. The nonlocal potential $\exp \left(\frac{i{H}_{e}Z}{\hslash v}\right)V(\mathcal{r}, R)\times \exp (-\frac{i{H}_{e}Z}{\hslash v})$ is thus approximated by a local or diagonal form $V(\mathcal{r}, R)$. In this paper the local approximation is retained. The bound system is assumed to consist of a single electron attached to a fixed point with wave function $\exp (-\frac{\mathcal{r}}{a_0})$. It is then shown how the diabatic approximation can be relaxed by modifying $V$ to include the effect of $H_e$ to order $v^{-1\mathrm{}}$. In particular in the impulse approximation, scattering is described by a static local potential $V_L(\mathcal{r}, R)=\frac{1}{R}-\frac{\left[\mathrm{erf}\right(i^{\frac{1}{2}}\alpha +\epsilon \left)\right]}{|\overrightarrow{\mathrm{R}}-\overrightarrow{\mathrm{r}}|}$, where ${\alpha }^2=\frac{{|\overrightarrow{\mathrm{R}}-\overrightarrow{\mathrm{r}}|}^2}{\mathrm{}\left(2cZ\right)\mathrm{}}$, and $c=\frac{\hslash }{\left(mv{a}_0\right)}$. An analytic form is given for $\varphi (\mathcal{r} ;\overrightarrow{\mathrm{B}}, \infty )$. The binding of the electron neglected in the impulse approximation can be taken into account by changing $V_L$ to $V_L(\mathcal{r}+x, R)$, where $x=e^{\mathrm{icZ}}-1\mathrm{}$. The scattering problem is thus reduced to quadratures.