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

Abstract

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, $(n_0,l_0,m_0,)\to \mathrm{}(n,l,m,)\mathrm{}$ 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 $1s\to 1s$ and $1s\to 2p,m \mathrm{}(m=+1,0,-1\mathrm{})\mathrm{}$ are evaluated from the general expression, and found to be in agreement with previous work except for the exchange amplitude for $1s\to 2p,\pm 1$. The threshold behavior of the cross sections for arbitrary inelastic processes is found to be proportional to the ($2\left|\Delta m\right|+1\mathrm{}$) 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.