Integral Equation Methods and the Born Approximation
We have seen in Chapter 2 that at low incident electron energies it is appropriate to use a partial wave expansion of the wave function describing the collision. In this energy region only a few partial wave phase shifts are significantly different from zero, and these can be obtained easily by numerically integrating the radial equation (9) subject to the asymptotic boundary conditions defined by equations (13) or (55). At high energies, on the other hand, this procedure breaks down because of the large number of partial waves which are needed in order to determine the cross section accurately. It is then necessary to obtain an expression for the complete scattering amplitude. In this chapter we derive an integral form for the scattering amplitude and we then evaluate it by an iterative method that gives rise to the Born series. We then consider conditions for the convergence of this series. Finally, we evaluate the first and second Born approximations in the case of some simple potentials of interest in electron-atom scattering and we also consider the Born exchange amplitude for electron-hydrogen-atom elastic scattering.
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