Generalized Levinson theorem: Applications to electron-atom scattering.

Abstract

A recent formulation provides an absolute definition of the zero-energy phase shift \ensuremath{\delta} for multiparticle single-channel scattering of a particle by a neutral compound target in a given partial wave l. This formulation, along with the minimum principle for the scattering length, leads to a determination of \ensuremath{\delta} that represents a generalization of Levinson's theorem. In its original form that theorem is applicable only to potential scattering of a particle and relates \ensuremath{\delta}/\ensuremath{\pi} to the number of bound states of that l. The generalized Levinson theorem relates \ensuremath{\delta}/\ensuremath{\pi} for scattering in a state of given angular momentum to the number of composite bound states of that angular momentum plus a calculable number that, for a system described in the Hartree-Fock approximation, is the number of states of that angular momentum excluded by the Pauli principle. Thus, for example, for electron scattering by Na, with its (1s${)}^{2}$(2s${)}^{2}$(2p${)}^{6}$3s configuration and with one L=0 singlet composite bound state, \ensuremath{\delta} would be \ensuremath{\pi}+2\ensuremath{\pi} for s-wave singlet scattering, 0+3\ensuremath{\pi} for s-wave triplet scattering, and 0+\ensuremath{\pi} for both triplet and singlet p-wave scattering; the Pauli contribution has been listed first. The method is applicable to a number of ${\mathit{e}}^{\ifmmode\pm\else\textpm\fi{}}$-atom and nucleon-nucleus scattering processes, but only applications of the former type are described here. We obtain the absolute zero-energy phase shifts for ${\mathit{e}}^{\mathrm{\ensuremath{-}}}$-H and ${\mathit{e}}^{\mathrm{\ensuremath{-}}}$-He scattering and, in the Hartree-Fock approximation for the target, for atoms that include the noble gases, the alkali-metal atoms, and, as examples, B, C, N, O, and F, which have one, two, three, four, and five p electrons, respectively, outside of closed shells. In all cases, the applications provide results in agreement with expectations. \textcopyright{} 1996 The American Physical Society.