Positron-Atom Scattering by the Kohn and Harris Methods

The $S$-wave scattering parameters have been calculated for the ${e}^{+}\ensuremath{-}\mathrm{H}$ and ${e}^{+}\ensuremath{-}\mathrm{He}$ systems below the positronium-formation threshold. The phase shifts were computed using the Harris variational method, while at zero energy the scattering lengths were obtained by the Kohn variational method. For hydrogen, a generalized Hylleraas wave function with three nonlinear parameters was used, giving an upper-bound scattering length $a\ensuremath{\le}\ensuremath{-}2.10278$, agreeing with Schwartz's result $a\ensuremath{\le}\ensuremath{-}2.10$. An extrapolation procedure gives an estimate of $a=\ensuremath{-}2.1036\ifmmode\pm\else\textpm\fi{}0.0004$. At nonzero energies, agreement with the lower-bound results of Bhatia, Temkin, Drachman and Eiserike, to better than 0.004 rad is achieved. Since the exact atomic-helium wave function is not known, the situation for helium is less clear; various model-target wave functions are examined using a generalized Hylleraas function with two nonlinear parameters. For the exponential model used by Drachman, the scattering is in close agreement with previous results, but the annihilation-rate parameter ${Z}_{\mathrm{eff}}$ is higher by 15% in poorer agreement with experiment at zero energy.