The motion of electrons in the static fields of hydrogen and helium

1. The scattering of electrons by atoms was first investigated by Born,* who adopted a method similar to that used by Huyghens and Kirchoff for the scattering of light by obstacles. Physically, Born’s approximation represents the integral of the amplitude of the secondary wavelets scattered from the incident plane wave only, and neglects the distortion of the plane wave by the atom. The perturbation of the incident wave will be smaller the larger the energy of the wave, so one would expect Born’s theory to be valid for high velocity impacts. Criteria for its validity have been given by Mott and Möller. The Born theory certainly breaks down for sufficiently low velocities as the cross-section curves obtained fall uniformly with increase of velocity of the incident electron beam, whereas the rare gas and alkali metal cross-sections (obtained from experiments) exhibit maxima and minima. At an impact there are present the incident waves and those scattered, both elastically and inelastically, and all these will interact. Born’s first approxi­mation neglects the effect of the scattered waves and the next approximation is obtained by neglecting only the inelastically scattered waves, and calculating exactly the elastic scattering of the electrons by the static field. This is done by the method of partial cross-sections which was developed by Faxen and Holtsmark and applied to the rare gases. The method consists in resolving the incident electrons into beams with different angular momenta, √{l (l +1)}h /2π, and a numerical solution of the wave equation is found which gives asymptotically the sum of an incident plane wave and a scattered spherical wave. The analysis is similar to that used by Rayleigh for the scattering of sound waves by obstacles. The incident electrons withl = 0 make head-on collisions, and these are most important for slow electrons and weak fields. For strong fields and fast particles, the main contribution to the scattering may come from large values ofl , when the calculation of scattered intensities by this method is not so convenient. In these cases the Born approximation is usually satisfactory. The agreement between Holtsmark’s calculations and experiment is reasonably good, showing that the distortion of the wave is of fundamental importance in the scattering of slow electrons.