On the use of the Breit—Pauli approximation in the study of relativistic effects in electron-atom scattering

Abstract

The electron-atom scattering problem is formulated by using the Breit-Pauli hamiltonian, and the Kohn variational principle is derived for this hamiltonian. Two distinct types of relativistic corrections are considered separately: (1) relativistic corrections due to the motion of the colliding electron and its interaction with the target; (2) relativistic corrections due to breakdown of LS -coupling in the target. In both of these cases it is shown that within the Breit-Pauli approximation a collision strength may be written Q ^vel ( i,j ) = Q ^nr (i,j) + a ² C ⁽²⁾ _rel ( i,J ),where Q ^rel is the collision strength including relativistic corrections and Q ^nr is the non-relativistic collision strength. The quantities C ⁽²⁾ _rel are contributions of orders a ² and a ⁴ respectively, relative to Q ^nr . In the case of corrections of type (1), consistency problems render it difficult to calculate the term a ⁴ C ⁽⁴⁾ _rel reliably. O n the other hand, strong semi-empirical evidence suggests that in the case of corrections of type (2), the a ⁴ correction can be reliably estimated within the framework of existing theory. By means of Racah algebra it is demonstrated that fine structure interactions between colliding electron and target give no contributions of order a ² provided that that Q ^rel ( i,j ) is summed over the fine structure levels of the initial and final target terms.Breakdown of L-S -coupling in the target (due to fine structure interactions among the the target electrons) gives contribution of order a ² to the total collision strength. However, these contributions do not vanish when the collision strengths are summed over the fine structure levels of the initial and final terms. Asymptotic expansions for the dependence of Q ^rel upon the nuclear charge Z of the target are derived for corrections of types (1) and (2). The present work is discussed in relation to recent work by Carse & Walker (1973) and Walker (1974), who have studied the studied the electron-hydrogen scattering problem in a formulation based upon the Dirac equation. Practical procedures for carrying out calculations in the framework of the present theory are discussed, and one such procedure is formulated in some detail.