Asymptotic Behavior of the Born Scattering Amplitudes in Collisions of Electrons with Hydrogen Atoms

The inelastic Born amplitudes for scattering of an electron by a hydrogen atom are obtained, using spherical coordinates, for processes from any initial state to any final state of a hydrogen atom, (n₀,l₀,m₀,)→(n,l,m,) in terms of differential operators which depend on the quantum numbers of the states involved, and which act on certain simple functions of electron momenta. The amplitudes for 1s→1s and 1s→2p,m (m=+1,0,-1) are evaluated from the general expression, and found to be in agreement with previous work except for the exchange amplitude for 1s→2p,±1. The threshold behavior of the cross sections for arbitrary inelastic processes is found to be proportional to the (2|Δm|+1) th power of the momentum of the outgoing electron. From the investigation of the high-energy behavior of the amplitudes, it is also found that while the Ochkur relation is valid, so that the direct scattering mode is dominant over the exchange mode for large-angle scattering, the exchange scattering is dominant for very-small-angle scattering processes, involving large principal and angular-momentum quantum numbers.