Impact-Parameter Expansion of High-Energy Elastic-Scattering Amplitudes

An impact-parameter expansion of high-energy scattering amplitudes is formulated, and the unitarity condition at large but finite energies is discussed. Applications are then made to high-energy $\pi p$ and $\mathrm{pp}$ elastic scattering with particular reference to the Van Hove model. This model is shown to give a good description of ${\pi }^{-}p$ scattering at 17.0 BeV/c scattering momentum; the addition of a short-range real part gives a qualitative description of $\mathrm{pp}$ scattering at momenta between 12 and 20 BeV/c.