Theory of Low-Energy Scattering of Electrons by Hydrogen Atoms

The theory of low-energy electron-hydrogen atom scattering is reformulated to include, in any truncated coupled-channel calculation, the contribution from continuum states due to exchange effects. The existence of such a contribution has been explicitly demonstrated by Castillejo, Percival, and Seaton, although the standard treatment does not include it. Our formulation of the problem employs the expansion of the exact wave function ${\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}$ in terms of the complete set of hydrogenic states ${{\ensuremath{\phi}}_{\ensuremath{\alpha}}}$ in the form ${\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}=\ensuremath{\Sigma}{\ensuremath{\alpha}}^{}{\ensuremath{\phi}}_{\ensuremath{\alpha}}(1){{\ensuremath{\psi}}_{\ensuremath{\alpha}}}^{\ifmmode\pm\else\textpm\fi{}}(2)$. Although this expansion is not termwise symmetric or antisymmetric, the entire sum is made so by use of the integral equation for ${\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}$. The boundary conditions satisfied by the scattering coefficients ${{\ensuremath{\psi}}_{\ensuremath{\alpha}}}^{\ifmmode\pm\else\textpm\fi{}}$ are automatically specified in this approach. Furthermore, it is now clear that the entire, exact elastic or inelastic amplitude is contained in the scattering wave function ${{\ensuremath{\psi}}_{\ensuremath{\alpha}}}^{\ifmmode\pm\else\textpm\fi{}}$. The elastic wave function ${{\ensuremath{\psi}}_{0}}^{\ifmmode\pm\else\textpm\fi{}}$ (and also the set of open-channel wave functions) is shown to obey both homogeneous (optical potential) and inhomogeneous equations. The equivalence of these equations with each other and with the equation found using Feshbach's projection-operator method is demonstrated. A set of homogeneous equations are found for the case where only certain chosen channels are kept and the rest are truncated. This latter set of equations is identical to those obtained by Hahn, O'Malley, and Spruch as an adjunct to calculations establishing bounds on scattering parameters. Both the exact and the approximate sets of equations for the ${{\ensuremath{\psi}}_{\ensuremath{\alpha}}}^{\ifmmode\pm\else\textpm\fi{}}$ differ from the corresponding equations used in all previous coupled-channel calculations. Using our results, the relation between the Born-Oppenheimer amplitude and first-order exchange amplitude of Bell and Moiseiwitsch is discussed. Each is shown to be a different first-order approximation to the exact amplitude, although the Born-Oppenheimer amplitude is seen to be an approximation to the amplitude derived from the inhomogeneous equation. Some simple single-channel calculations of phase shifts and scattering lengths have been carried out. The triplet phase shifts and scattering length derived from the approximate homogeneous equation are identical to those of the standard static exchange approximation, a result expected on the basis of the antisymmetry of the ${\ensuremath{\Psi}}^{\ensuremath{-}}$. However, the singlet phase shifts and scattering length differ from those of the standard static exchange approximation. We find that for ${k}^{2}\ensuremath{\le}0.09$, the singlet phase shifts are larger than those calculated in the $1s\ensuremath{-}2s$ strong-coupling approximation, and that the singlet scattering length is 7.85, where quantities are given in atomic units. For ${k}^{2}g0.09$, the singlet phase shifts are less than those of the $1s\ensuremath{-}2s$ case and approach those of the static exchange calculation. Reasons for this behavior are discussed.