Higher Born Approximations for the Coulomb Scattering of a Spinless Particle

Abstract

The Coulomb scattering amplitude of a Klein-Gordon particle is computed exactly to the third order in $\ensuremath{\beta}=\frac{Z{e}^{2}}{\ensuremath{\hbar}v}$ by perturbation theory using the potential $\frac{\mathrm{Ze}{e}^{\ensuremath{-}\ensuremath{\lambda}r}}{r}$ in the limit $\ensuremath{\lambda}\ensuremath{\rightarrow}0$. It contains only finite terms except for terms containing powers of $\mathrm{ln}(\frac{|\mathrm{Q}|}{\ensuremath{\lambda}})$ which can be shown to arise from the expansion of a common phase factor. Thus, first- and second-order corrections to the relativistic Rutherford cross section are obtained.Our results are compared to the exact scattering amplitude in the form of the partial-wave summation, expanded in powers of $\ensuremath{\beta}$, which we have succeeded in summing to third order.