7 - Calculation of scattering amplitudes

Summary

The background for the details of multichannel scattering calculations has now been established. We consider methods based on the integral-equation formulation of chapter 6. These momentum-space methods have proved accurate at all energies in a sufficient variety of situations to justify the belief that they can be generally applied. In some situations sufficient accuracy is achieved without resorting to the full power of the integral-equation solution. The methods used in these situations are distorted-wave methods. Their relationship to the full solutions will be examined in a simplified illustrative case. A brief outline will be given of alternative methods based on a coordinate-space formulation of the multichannel problem.

There are two characteristic difficulties of multichannel many-fermion problems. The first is that computational methods can of course directly address only a finite number of channels whereas the physical problem has an infinite number of discrete channels and the ionisation continuum. The second is that the electrons are identical so that the formulation in terms of one-electron states must be explicitly antisymmetric in the position (or momentum) and spin coordinates.

We first show how to set up the problem within the framework of formal scattering theory using antisymmetric products of one-electron states. The problem is then formulated in terms of the calculation of reduced T-matrix elements relating the absolute values of initial and final momenta in different angular-momentum states. This depends on a knowledge of the corresponding potential matrix elements, whose calculation we treat in detail. We then show how the target continuum is accounted for in the scattering formalism.