3 Calculation of cross sections

The details of the CCC approach to e-He scattering have been given by Fursa & Bray (1995). The method is based on the close-coupling (CC) antisymmetric expansion of the total wave function using the He target states. These states are constructed using the frozen-core approximation where all two-electron configurations have one of the electrons described by the He⁺ 1s wave function. The other electron function is obtained as a linear combination of an orthogonal Laguerre basis upon diagonalization of the target Hamiltonian. The strength of this approach is that convergence, in say the cross section of interest, may be studied systematically by increasing the Laguerre basis sizes. One disadvantage is that the energy levels for the low-lying states are not as accurate as they could be if the frozen-core approximation was relaxed a little. We have not done so here because such a relaxation results in many more states to be used in the CC formalism requiring some judicious truncating.

In performing the presented calculations a total of 89 states were used that span the discrete spectrum reasonably accurately for states with principal quantum number $n\le6$. In addition, the target one-electron continuum was very accurately described. Integrated cross sections were obtained for the $n_i\to n_{\rm f}$ transitions with $n_i\le5$. The strength of the CCC formalism is to be able to solve large-scale CC equations at any incident energy. However, presently this has to be done from scratch for each energy. Hence, the R-matrix approach is much more efficient in generating cross sections on a fine energy mesh. We have chosen a sufficiently fine mesh so that the relevant integrals, over the energy, could be evaluated to a satisfactory accuracy. The present calculations have been carried out for electron energies that extend as far as 500 eV relative to the ground state. There is no difficulty in extending the calculations as far as say 1 keV.