Double Ionization of Helium by Protons and Electrons at High Velocities

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Double ionization of helium at high projectile velocities, $v$, is considered in terms of two mechanisms. In the shakeoff mechanism the ratio of double- to single-ionization cross sections, $\frac{{\ensuremath{\sigma}}^{\mathrm{ii}}}{{\ensuremath{\sigma}}^{i}}$, is independent of $v$. In the two-step mechanism it is shown that $\frac{{\ensuremath{\sigma}}^{\mathrm{ii}}}{{\ensuremath{\sigma}}^{i}}\ensuremath{\sim}{({v}^{2}\mathrm{ln}v)}^{\ensuremath{-}1}$. Combining amplitudes gives a reasonable fit to the observed velocity dependence as well as an explanation of the observed factor-of-2 differences between double ionization of helium by protons and electrons near $v\ensuremath{\simeq}10{v}_{\mathrm{Bohr}}$.