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2.2 The Elwert-Haug approximation

In the Elwert-Haug formulation, the interaction of the scattering electron with the static Coulomb field of the nucleus is included in the unperturbed Hamiltonian. However the exact wave functions in the Coulomb field are replaced by the approximate wavefunctions of Sommerfeld and Maue [9]. Bethe and Maximon [10] showed that these wavefunctions correctly represent all partial waves with $\kappa^2\gg(Z\alpha)^2$ in the exact (Darwin) solution. Using the Sommerfeld-Maue wavefunctions and making some further approximation, Elwert and Haug obtained an analytic but rather complicated expression for the triply differential cross section, involving hypergeometric functions, as a function of Z, the momentum vectors and energies. (In a recent paper [11] Haug has also derived an expression for the polarization correlation C200). We have found it more convenient in our calculations of the predictions of this theory to work from Elwert and Haug's results for the matrix element, computing the hypergeometric functions numerically. The Elwert-Haug result is expected to be valid in the point Coulomb potential for $\alpha Z \ll 1$ at all energies, or for all $\alpha Z$ at small scattering angles and sufficiently high incident and outgoing electron energy $E\gg mc^2$, and in suitable intermediate situations. In the case of the neutral atom, screening by the bound electrons is treated by taking the product of the form factor F(q), calculated as above, with the Elwert-Haug result for the matrix element, thus making an additional approximation reducing the region of validity.


next up previous
Next: 2.3 The exact partial Up: 2. Discussion of theories Previous: 2.1 Bethe-Heitler approximation
Eoin Carney
1999-06-14