The Scattering of Slow Electrons by Neutral Atoms

A treatment of the $N+1\mathrm{}$ electron Schroedinger equation describing the elastic scattering of an electron beam by an atom with $N$ electrons in the outer shell which, in first approximation, leads to the equation $({\nabla }^2+k^2+U(r\left)\right)f(x, y, z)=0,$ widely used for the computation of low velocity elastic scattering cross-sections, in which $\left(\frac{1}{2}\right)k^2$ is the kinetic energy of the incident electron and $(-\frac{1}{2})U\left(r\right)\mathrm{}$ is the interaction energy of the atom and the incident electron including terms arising from the distortion of the atom by the field of the electron. The treatment is based on a wave function antisymmetric in the space-spin coordinates of all the electrons. A discussion of exchange interference and its application by Oppenheimer to supply a qualitative explanation of the Ramsauer effect. It is found that the exchange scattering amplitude as given by Oppenheimer requires modification. The modification greatly reduces the value of the exchange term in the elastic scattering amplitude. The tentative conclusion is reached that exchange interference is of minor importance in the complete explanation of the Ramsauer effect. A derivation of the relation $\int F^{*}\mathrm{Fd}\Omega =\frac{2\pi i}{k{\left(F\right(\frac{z}{r})-F^{*}(\frac{z}{r}\left)\right)}_{\frac{z}{r}=1\mathrm{}}}$ in which $F\left(\frac{z}{r}\right)$ is the elastic scattering amplitude and $\int F^{*}\mathrm{Fd}\Omega$ is the total scattering cross-section. A simple generalization of this relation for electron energies great enough to produce excitation. Development of a new method of solving the scattering equation $({\nabla }^2+k^2+U(r\left)\right)f(x, y, z)=0\mathrm{}$ and application to computation of scattering amplitudes and cross-sections.