Higher Born Approximations for Elastic Electron Scattering from Atoms. A theorem analogous to the unitarity relation by Glauber and Schomaker is derived for Born’s approximations. For a given potential Φ (r) =μ (ħ2/2 m) V (r) of the scattering atom the n-th approximation fn for the scattering amplitude is defined as the function resulting from the n-th iterative step of Born’s approximation.| If | fn |2 is expanded into a series of powers of the parameter μ, this series up to terms in μn+1 is defined as the n-th approximation for the differential cross section dσn/dΩ, and not |fn|2 itself. In this case an extension of the optical theorem Im fn+1 (0) = (k/4π) σn can be derived.For a potential Φ(r)=-(Z e2/4π ε0 r)·exp(-r/R) ∑ airi exact analytic expressions for the scattering amplitude fn and the differential cross section dσn/dΩ can be given in first and second Born approximations.An important example is the hydrogen-like atomic potential in which, in contradistinction to Wentzel’s model, the electron density σ(r) is limited even for r → 0. For this model, the differential scattering cross section in second Born approximation exceeds the values of the first approximation for all scattering angles, while experimental data indicate a correction of the opposite sign. An estimate of the cross sections in third approximation shows that they are again below’ the first approximation. These results indicate that the Born’s first approximation is not only less complicated but even more accurate than the second one. The absolute values of the scattering amplitudes in first and second approximations diverge considerably even at high energies, mainly due to the imaginary part Im f2(ϑ) of /2(ϑ). Therefore, conclusions on the phase η(ϑ) from Im f2(ϑ) do not seem to be reliable.
DIFFERENTIAL CROSS SECTIONS FOR ELASTIC ELECTRON SCATTERING BY A CALCIUM ATOM AT LOW ENERGY RANGE.
